Additional topics in math

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Congruence and similarity

Congruence and similarity on the SAT test

SAT Subscore: Additional topics in math

Congruence and similarity questions include congruent angles and similar triangles topics.

Similar triangles have the same angle measures and their corresponding side lengths are related by a constant ratio.

Sum of angles in a triangle: The sum of the measures in degrees of the angles of a triangle is 180.

Angle measures in triangles: In an isosceles triangle the angles opposite the two equal sides are equal; the angles of an equilateral triangle are equal to 60°.

Congruent angles are angles that have the same angle measure.

Vertical Angles are the angles opposite each other when two lines cross, vertical angles are equal.

Supplementary angles are those angles that measure up to 180 degrees. Angles that lie on the same side of a straight line, are always supplementary.

Alternate angles are angles located in opposite positions relative to a transversal intersecting two parallel lines. Alternate angles are equal.

Corresponding angles are angles located in the same position relative to parallel lines intersected by a transversal. Corresponding angles are equal.

Two intersecting lines create vertical angles and supplementary angles.

Two parallel lines combined with two intersecting lines form 2 similar triangles and equal alternate angles.

A parallel line inside a triangle forms 2 similar triangles and equal corresponding angles.

Continue reading this page for detailed explanations and examples.

Angle relationships inside a triangle

Sum of angles in a triangle:
The sum of the measures in degrees of the angles of a triangle is 180. (This statement is provided at the beginning of each SAT math section).
If we mark the angles as x°, y° and z° then x°+y°+z°=180°.

Angle measures in an isosceles triangle:
An isosceles triangle is a triangle that has two equal sides, the angles opposite the two equal sides are equal.

Angle measures in an equilateral triangle:
An equilateral triangle is a triangle with all three sides of equal length, the angles of an equilateral triangle are equal to 60°.

The figure below presents a triangle and its angles.

Angle sum inside a triangle

Consider the following example:

In a triangle ABC given that the angle A is equal to 50° and AB=BC.

What is the value of the angles B and C?

The sum of the measures of the angles of a triangle is 180°, therefore the angles A+B+C=180°.
We are given that AB=BC, therefore the angle B is equal to the angle C.
We are given that A=50°.

Therefore, we can mark by x the value of angle B and C and solve an equation with 1 variable:
x+x+50°=180°
2x=180°-50°
2x=130°
x=65°

Checking the answer:
65°+65°+50°=180°
180°=180°

Consider the following example:

In a triangle ABC given that the angle A is equal to 60° and the angle B is twice bigger than the angle C.

What is the value of the angles B and C?

The sum of the measures of the angles of a triangle is 180°, therefore the angles A+B+C=180°.
We are given that A=60°.
We are given that the angle B is twice bigger than the angle C.

Therefore, we can mark by x the value of angle C and solve an equation with 1 variable:
2x+x+60°=180°
3x=120°
x=40° therefore C=40° and D=40°*2=80°

Checking the answer:
40°+80°+60°=180°
180°=180°

Angle relationships between intersecting lines- vertical and supplementary angles

Two intersecting lines create vertical angles and supplementary angles.

Vertical Angles are the angles opposite each other when two lines cross (vertical means that they share the same vertex). Vertical angles are equal.

Supplementary angles are those angles that measure up to 180 degrees. Angles that lie on the same side of a straight line, are always supplementary.

The figure below shows the angles created between two intersecting lines.

Angle relationships- vertical and supplementary angles

The angles 1 and 3 (180-α) and the angles 2 and 4 (α) are vertical angles. All pairs of vertical angles are equal.

 

The angles 1 and 2, the angles 2 and 3, the angles 3 and 4 and the angles 4 and 1 are supplementary angles. The sum of each pair of supplementary angles equals to 180 degrees (180-α+α=180°).

Consider the following example:

In the figure below are presented 4 lines that cross each other: AB, FI, AH and BG. The triangle ABC is an isosceles triangle, so that AS=BC. The angle FEG is equal to 30° and the angle FDH is equal to 120°.

What is the value of the angle BAC?

We are given that the angle FEG is equal to 30°, therefore the angle CED is also equal to 30° (the angles are vertical, therefore they are equal).

We are given that the angle FDH is equal to 120°, therefore the angle CDI is equal to 180°-120°=60° (the angles are supplementary, therefore their sum is 180°).

The sum of angles in a triangle is equal to 180°, therefore the sum of angles in the triangle CDE is 180° and the angle DCE=180°-60°-30°=90°.

We are given that the triangle ABC is an isosceles triangle, therefore the angles ABS and BAC are equal.
The sum of angles in a triangle is equal to 180°, therefore the sum of angles in the triangle ABC is 180°.
We calculated that the angle DCE is equal to 90° the angles ABS and BAC are equal.
From these 3 statements we can conclude that the angle BAC=ABC=(180°-90°)/2=90°/2=45°.

Angle relationships between intersecting and parallel lines- alternate angles

Two parallel lines combined with two intersecting lines form 2 similar triangles and equal alternate angles.

Similar triangles have the same shape, but not the same size (they have the same angle measures, but not the same side lengths).
Note that the corresponding side lengths of similar triangles are related by a constant ratio, which is called k. See further details below.

Alternate angles are angles located in opposite positions relative to a transversal intersecting two parallel lines. Alternate angles are equal.

In the figure below two parallel lines AB and DE (marked in blue) combined with two intersecting lines AE and BD form 2 similar triangles ABC and CDE.

The angles marked in x°, y° and z° are equal.

Angles between intersecting lines and parallel lines

The equal angles are marked in x°, y° and z°:

The equal angles z° are vertical angles that are formed by the lines AE and BD.

The equal angles x° are called alternate angles, they are formed by the parallel lines AB and DE and the line AE that crosses them.

The equal angles y° are called alternate angles, they are formed by the parallel lines AB and DE and the line BD that crosses them.

Since the angles x°, y° and z° are equal between the triangle ABC and CDE, the triangles are similar.  

 

Consider the following example:

In the figure below are presented 4 lines that cross each other: HI, DE, EF and BG. The lines HI and DE are parallel. The angle FAH is equal to 40° and the angle EDG is equal to α. In addition, AC=BC.

What is the value of α?

Angle relationships between intersecting lines and parallel lines question

We are given that the angle EDG is equal to α.

Angles that lie on the same side of a straight line, are always supplementary and their sum is equal to 180°. Therefore, the angles EDG and CDE are supplementary and EDG+ CDE=180°.

The angle CDE=180°-α.

Two parallel lines combined with two intersecting lines form 2 similar triangles. We are given that the lines HI and DE are parallel, therefore the triangles ABC and CDE are similar and the angles CDE and ABC are equal and their value is 180°-α (we found that CDE=180°-α).

Note that instead of using similar triangles we can use the statement that two parallel lines combined with two intersecting lines form 2 equal alternate angles. We are given that the lines HI and DE are parallel, therefore the angles CDE and ABC are equal (these angles are alternate) and their value is 180°-α (we found that CDE=180°-α).

We are given that the angle FAH is equal to 40°, therefore the angle BAC is also equal to 40° (the angles are vertical, therefore they are equal).

We are given that AC=BC, therefore the angle BAC=ABC.

We found that the angle BAC=ABC, the angle BAC=40° and the angle ABC=CDE=180°-α. Therefore, the angle ABC=CDE= BAC=FAH=180°-α=40°

We can solve the equation 180°-α=40°, getting α=180°-40°=140°

Angle relationships between parallel lines inside a triangle- corresponding angles

A parallel line inside a triangle (the line is parallel to one of the sides of the triangle) forms 2 similar triangles (a small triangle and a large triangle) and equal corresponding angles.

Corresponding angles are angles located in the same position relative to parallel lines intersected by a transversal. Corresponding angles are equal.

In the figure below a parallel line EB inside a triangle ACD (EB is parallel to CD) forms 2 similar triangles (a small triangle ABE and a large triangle ACD).

 

Angle relationships between parallel lines inside a triangle- corresponding angles

The equal angles are marked in x° and y°:

The equal angles x° are called corresponding angles, they are formed by the parallel lines BE and CD and the line AD.

The equal angles y° are called corresponding angles, they are formed by the parallel lines BE and CD and the line AC.

Since the angles x°, y° and z° are equal between the triangle ABE and ACD, the triangles are similar.  

Consider the following example:

In the figure below are presented 4 lines that cross each other: AD,AC,BF and CE. The lines BF and CE are parallel. The angle BAF is equal to 20° and the angle ADE is equal to α. In addition, AF=AB.

What is the value of α?

Calculating side lengths with triangles similarity

Similar triangles have the same shape, but not the same size (they have the same angle measures, but not the same side lengths).

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k.

Note that:
Similar triangles have the same angle measures.
If 2 lines are parallel, the triangles that are formed by them are similar.

The figure below shows two similar triangles: ABC and DEF. The triangle ABC is similar to the triangle DEF.

Finding side lengths with triangles similarity

Similar triangles have the same angle measures, therefore the angle A is equal to the angle D; the angle B is equal to the angle E and the angle C is equal to the angle F.

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k, therefore the following ratios between pairs of the sides exist:
DE=k*AB (marked in orange in the figure above)
DF=k*AC (marked in green in the figure above)
EF=k*BC (marked in black in the figure above)

We can also present all the ratios as being equal to k:

k =  DE   =   DF   =   EF
       ____     ____      ____
       AB        AC        BC

Note that we could write the ratio in an opposite way by switching between the numerator and the denominator (so that the numerator will include the sides of the small triangle and the denominator will include the sides of the big triangle). The parameter k will be smaller than 1 and equal to 1 divided by the k parameter above.
For example: if the ratio k between the sides of the big triangle divided by the sides of the small triangle is 2 (like 4 divided by 2), then the ratio k between the sided of the small triangle divided by the sides of the big triangle will be equal to 1/2 (like 2 divided by 4).

Consider the following example:

In the figure below is given that AB=5 centimeters; CD=27 centimeters; BE is parallel to CD; AC is 3 times bigger than AB.

What is the value of BE?

An example- Calculating side lengths with triangles similarity

We are given that AC is 3 times bigger than AB, therefore we can write an equation AC=AB*3.

We are given that AB=5 centimeters, therefore AC=AB*3=5*3=15. Meaning that AC/AB=3.

A parallel line inside a triangle forms 2 similar triangles. We are given that BE is parallel to CD, therefore the triangles ABE and ACD are similar.

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k.

We found AC/AB=3, therefore k=3.

K is also equal to BE/CD, therefore we can write an equation CD/BE=3.

We are given that CD=27 centimeters, therefore we can write an equation 27/BE=3.

27/BE=3

BE=27/3

BE=9 centimeters

Calculating side lengths with triangles similarity when one side is common for both triangles

In these questions one side is common for both triangles, therefore we need the length of two sides instead of three sides to calculate the length of the fourth side using the similar triangles ratio.  

Consider the following example:

The figure below presents two similar triangles, ABC and BDC.

The length of AB is 20 centimeters and the length of BC is 15 centimeters.

What is the length of DC?

Example -calculating side lengths with triangles similarity when one side is common for both triangles

In this example the side BC is common for both the triangles. Therefore we can you it twice in the similar triangles ratio.

The similar triangles ratio is

k   =   AB     =    BC
          ____        ____
           BC           DC

k   =   20   =  15
         ____     ____
          15        DC

4  =  15
__      __
3      DC

4DC=45

DC=45/4=11 1/4

The length of the side DC is 11 1/4 centimeters.

Note that we could write the ratio in an opposite way by switching between the numerator and the denominator and get the same answer:

k    =   BC     =    DC
           ____         ____
           AB            BC

K   =   15   =    DC
         _____      _____
           20          15

 3   =   DC
___      ___
 4         15

4DC=45

DC=45/4=11 1/4

The ratio k in this case will be 15/20=3/4, which is 1 divided by the previous ratio k=4/3.

Right triangle trigonometry and word problems

Right triangle trigonometry and word problems on the SAT test

SAT Subscore: Additional topics in math

Right triangle is a triangle with a right angle (equal to 90°). The side opposite the right angle (the longest side of the right triangle) is called a hypotenuse.

Right triangle trigonometry and right triangle word problems require calculating side lengths and angle measures in right triangles.

Pythagorean theorem:

Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides: a2 + b2 = c2

Pythagorean triples are combinations of side lengths a, b and c that satisfy the Pythagorean theorem. If you remember the triples values, you know the size of the third side without the need to calculate it. The most common Pythagorean triples are: 3, 4 and 5;    5, 12 and 13;    7, 24 and 25.  

Trigonometric ratios (sine, cosine and tangent):

Trigonometric ratios (functions) represent connections between angle degrees and side lengths in a right triangle:

  • The sine of an angle (sin) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, to the length of the hypotenuse. 
  • The cosine of an angle (cos) in a right triangle is defined as the ratio of the length of the side that is adjacent to the angle, to the length of the hypotenuse. 
  • The tangent of an angle (tan) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, side that is adjacent to the angle. 

Complementary angles are two angles with the sum of 90 degrees. Sine of an angle (α) in a right triangle is equal to cosine of its complementary angle (90-α). 

Similar triangles have the same angle measures and their corresponding side lengths are related by a constant ratio therefore they also have similar sine, cosine and tangent.

Special right triangles:

Special right triangles are right triangles whose sides are in a particular ratio. 

  • In a 30°, 60°, 90° right triangle the side opposite the 30° angle is half the length of the hypotenuse and the side opposite to 60° angle is equal to the length of a side opposite to 30° angle multiplied by √3.
    In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.
  • In a 45°, 45°, 90° right triangle the sides opposite the 45° angles are equal and the hypotenuse is equal to the side opposite to 45° angle multiplied by √2.
    In 45°, 45°, 90° triangle the sides are s, s and s√2.

Continue reading this page for detailed explanations and examples.

Pythagorean theorem in right triangles

Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. (The hypotenuse is the side opposite the right angle).

The formula of the Pythagorean theorem is a2 + b2 = c2. This formula is provided at the beginning of each math section followed by a diagram.  

The following figure presents a right triangle and the Pythagorean theorem a2 + b2 = c2.

Pythagorean theorem in a triangle

Calculating side lengths with Pythagorean theorem

If we are given the values of two sides of a right triangle, we can calculate the value of the hypotenuse with the Pythagorean theorem.

If we are given the values of one side and the hypotenuse of a right triangle, we can calculate the value of the second side with the Pythagorean theorem.

Consider the following example:

The length of the sides of a right triangle are 3 centimeters and 4 centimeters.

What is the length of the hypotenuse?

The formula of the Pythagorean theorem is a2+b2=c2.
a=3, b=4, c=?
32 + 42 = c2
c2=9+16
c2=25
c=5
The length of the hypotenuse is 5 centimeters.

Checking the answer:
32+42=52
9+16=25
25=25

Consider the following example:

The length of a side of a right triangle is 4 centimeters and the length of the hypotenuse is 8 centimeters.

What is the length of the other side?

The formula of the Pythagorean theorem is a2+b2=c2.
a=4, b=?, c=8
42+b2=82
16+b2=64
b2=64-16
b2=48
b=√48
b=√(4*12)
b=√4*√12
b=2√12=6.93

Checking the answer:
a2+b2=c2
42+(2√12)2=82
16+4*12=64
16+48=64
64=64

Pythagorean triples in a right triangle

Pythagorean triples are combinations of side lengths a, b and c that satisfy the Pythagorean theorem. If you remember the triples values, you know the size of the third side without the need to calculate it.

The most common Pythagorean triple is 3, 4 and 5.
Additional Pythagorean triples are:
5, 12 and 13
7, 24 and 25  

Note that each multiplication of the triple also satisfies the Pythagorean theorem.
For example: The triple 9, 12 and 15 is the triple 3, 4 and 5 multiplies by 3. Therefore, the triple 9, 12 and 15 is a Pythagorean triple.
We can check this statement by plugging into the formula of the Pythagorean theorem:
a2+b2=c2
92+122=152
81+144=225
225=225

Consider the following example:

The length of a side of a right triangle is 7 centimeters and the length of the hypotenuse is 25 centimeters.

What is the length of the other side?

The triple 7, 24 and 25 is a Pythagorean triple, therefore the length of the second side is 24 centimeters.

Checking the answer:
a2+b2=c2
72+b2=252
b2=252-72
b2=625-49
b2=576
b=24
The length of the other side is 24 centimeters.

Consider the following example:

The length of the sides of a right triangle are 10 centimeters and 24 centimeters.

What is the length of the hypotenuse?

The triple 5, 12 and 13 is a Pythagorean triple, the given sides 10 and 24 are the triple sides 5 and 12 multiplied by 2. Therefore, the length of the hypotenuse is 13 multiplied by 2=26 centimeters.

Checking the answer:
a2+b2=c2
102+242=262
100+576=676
676=676 

Calculating triangles side lengths with Pythagorean theorem and triangles similarity

Similar triangles have the same shape, but not the same size (they have the same angle measures, but not the same side lengths).

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k.

 Note that:
Similar triangles have the same angle measures.
If 2 lines are parallel, the triangles that are formed by them are similar.

Given parallel lines and a right triangle, we can calculate side lengths using triangles similarity combined with the Pythagorean theorem.

Calculating side ratio with triangles similarity

Consider the following example:

In the figure below the lines BE and CD are parallel. The angle measures and the side lengths (in centimeters) are given in the figure.

What are the lengths of the sides of triangle ACD?

Calculating right triangles side lengths

Calculating the length of AB:

Since BE is parallel to CD, the triangles ABE and ACD are similar. Therefore, the angle ACD is equal to the angle ABE and equal to 90 degrees.

Since the angle ABE is equal to 90 degrees, the triangle ABE is a right triangle. Therefore, we can apply the Pythagorean theorem in the triangle ABE:

The sides 3, 4 and 5 are Pythagorean triple, or we can solve the Pythagorean theorem:
a2+b2=c2
32+42=c2
c2=9+16
c2=25
c=5
AB=5 centimeters

Calculating the length of CD and DE:
AC=BC+AB
AC=5+5=10 centimeters

Since the lines BE and CD are parallel, the triangles ACD and ABE are similar. Therefore, the ratio of the related sides is k=AC/AB=10/5=2 (the side of the big triangle divided by the side of the small triangle is equal to 2).

Since k=2, AD/AE=2 and CD/BE=2.

Since AE=4 and k=2, AD/4=2, AD=8 centimeters and DE=AD-AE=8-4=4 centimeters.
Since BE=3 and k=2, CD/3=2, CD=6 centimeters.

Checking by calculating Pythagorean theorem in the big triangle ACD:
a2+b2=c2
62+82=102
This is a Pythagorean triple 3, 4 and 5 multiplied by 2.

Trigonometric ratios (sine, cosine and tangent)

Trigonometric ratios (functions) represent connections between angle degrees and side lengths in a right triangle.

The sine of an angle (sin) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, to the length of the hypotenuse. In the figure below in a right triangle ABC, sine (A)= BC/AC.

The cosine of an angle (cos) in a right triangle is defined as the ratio of the length of the side that is adjacent to the angle, to the length of the hypotenuse. In the figure below in a right triangle ABC, cosine (A)= AB/AC.

The tangent of an angle (tan) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, side that is adjacent to the angle. In the figure below in a right triangle ABC, tangent (A)= BC/AB.

In the figure below we see a right triangle ABC.
sin (A)=opposite/hypotenuse=BC/AC
cos (A)=adjacent/hypotenuse=AB/AC
tan (A)=opposite/adjacent=BC/AB

Sine and cosine of complementary angles

Complementary angles are two angles with the sum of 90 degrees.

Complementary angles in a right triangle: Since one angle in a right triangle is equal to 90 degrees and the sum of angles of a triangle is 180 degrees, the two other acute angles are complementary.

Sine of an angle in a right triangle is equal to cosine of its complementary angle. Meaning that given two complementary angles α and 90-α in a right triangle, sinα=cos(90-α).

Showing the connection between sine and cosine of complementary angles:
If we represent by α the measure of one angle in a right triangle, then the other angle measure is 90-α (complementary angles).

The figure below presents a right triangle ABC with complementary angles α and 90-α.

sinα=BC/AC

cos(90-α)=BC/AC

We see that sinα=cos(90-α)

Sine and cosine equality of complementary angles

Consider the following example:

What is sin (60) is equal to:
A. cos(30)
B. tan(60)
C. sin(30)
D. cos(60)

We know that sinα=cos(90-α), therefore, sin60=cos(90-60)=cos(30).
The answer A is correct.

Calculation of side lengths with sine, cosine and tangent

To calculate a side length, we need to know the value of the trigonometric function (sine, cosine or tangent) and the value of the other side.

Consider the following example:

The figure above represents a right triangle ABC.

If sin(a)= 0.5 and AC=5 centimeters, what is the value of AC?

sin(A)=BC/AC

0.5=BC/5

BC=0.5*5=2.5 centimeters

Calculation of sine, cosine and tangent in similar triangles

Similar triangles have the same angle measures and their corresponding side lengths are related by a constant ratio therefore they also have similar sine, cosine and tangent.

We can calculate sine, cosine or tangent in one triangle and conclude that they are identical in all similar triangles.

Related corresponding side lengths lead to similar sine, cosine and tangent:
If the sides of triangle 1 are x, y and z and the sides of a similar triangle 2 are kx,ky and kz (multiplied by a factor of k) then:
The sine of an angle A in triangle 1 is sin A1=x/z
The sine of an angle A in triangle 2 is sin A2=kx/kx=x/z
The sine of triangle 1 is equal to the sine of triangle 2.

Consider the following example:

In the figure below the lines BE and CD are parallel. The angle measures and the side lengths (in centimeters) are given in the figure.

What is the cosine of angle E?

Sine, cosine and tangent in similar triangles

Special right triangles

Special right triangles are right triangles whose sides are in a particular ratio. Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.

The special right triangles with the side sizes length are given at the beginning of each SAT section.

30°, 60°, 90° triangle:
In a 30°, 60°, 90° right triangle the side opposite the 30° angle is half the length of the hypotenuse and the side opposite to 60° angle is equal to the length of a side opposite to 30° angle multiplied by √3.
In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

 45°, 45°, 90° triangle:
In a 45°, 45°, 90° right triangle the sides opposite the 45° angles are equal and the hypotenuse is equal to the side opposite to 45° angle multiplied by √2.
In 45°, 45°, 90° triangle the sides are s, s and s√2.

The following graphs present the special right triangles with the side sizes length.

Calculation of side lengths given angle measures in special triangles

Given the length of any side in a special right triangle, we can calculate the length of the two other sides.

If you identify that the angles of a right triangle have the measures of 30,60,90 or 45,45,90 you know the ratios between their side length according to the graph presented above:

In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

In 45°, 45°, 90° triangle the sides are s, s and s√2.

 

Consider the following example:

In a right triangle the measures of the angles are 30°, 60°, 90° and the hypotenuse is equal to 8 centimeters.

What are the lengths of the other sides?

In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

Given the hypotenuse is equal to 8 centimeters, we know that the side opposite to 30° angle is equal to 8/2=4 centimeters. The side opposite to 60° angle is equal to 4*√3=√16*√3=√48.

Checking with Pythagorean theorem:
a2+b2=c2
42+√482=82
16+48=64
64=64

Consider the following example:

In a right triangle the measures of the angles are 45°, 45°, 90° and the hypotenuse is equal to √40 centimeters.

What are the lengths of the sides?

In 45°, 45°, 90° triangle the sides are s, s and s√2.

We are given that the hypotenuse is equal to √40 centimeters, therefore s√2=√40. Continue solving:
s√2=√40
s=√40/√2
s=√20

Checking with Pythagorean theorem:
a2+b2=c2
√202+√202=√402
20+20=40
40=40

Calculation of angle measures given side lengths in special triangles

Given the side lengths of a special right triangle, we can calculate the angle measures.

 If the ratio between a side and a hypotenuse in a right triangle is 1/2 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the side used in the ratio is located opposite to the 30° angle).

If the ratio between a side and a hypotenuse in a right triangle is √3/2 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the side used in the ratio is located opposite to the 60° angle).

If the ratio between two sides in a right triangle is √3 or 1/√3 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the bigger side which is multiplied by √3 is located opposite to the 60° angle).

If a right triangle is an isosceles triangle, it is a special right triangle with angles 45°, 45° and 90°.

If the ratio between a side and a hypotenuse in a right triangle is 1/√2 we know that the triangle is a special right triangle with 45°, 45°, 90° angles.

Consider the following example:

The ratio between two sides in a right triangle is √27/9, what are the measures of the angles of the triangle?

Simplifying the ratio √27/9 gives us:
√27/9=√27/√81
√27/√81=√27/(√27*√3)
√27/(√27*√3)=1/√3

If the ratio between two sides in a right triangle is 1/√3 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the bigger side which is multiplied by √3 is located opposite to the 60° angle).

Volume word problems

Volume word problems on the SAT test

Additional topics in math

Volume word problems require making calculations of volumes of three-dimensional shapes using volume formulas. To calculate the volume, plug the given dimension into the relevant volume formula.

Volume formulas of five basic shapes:
The volume formulas of five basic shapes are given at the beginning of the math sections of the SAT exam: right rectangular prism, right circular cylinder, sphere, right circular cone and rectangular pyramid.
Right rectangular prism volume formula is V= lwh.
Right circular cylinder volume formula is V=πr2h.
A sphere volume formula is V= 4/3 πr3.
A right circular cone volume formula is V= 1/3 πr2h.
A rectangular pyramid volume formula is V=1/3 lwh.

Calculating the effect of changes in dimensions on volume: The power of the dimension determines the size of the change in the volume value.

  • If a dimension in the volume formula is raised to a first power, the volume changes by the same factor as the shape.
  • If a dimension in the volume formula is raised to a second power, when the shape changes by a factor the volume changes by a square of the factor.
  • If a dimension in the volume formula is raised to a third power, when the shape changes by a factor the volume changes by a third degree of the factor.

Comparing volumes of two shapes: In these questions we are given ratios between the dimensions of two shapes and we are required to compare their volumes. We need to calculate the total effect on the volume of all the ratios between the shapes. 

Continue reading this page for detailed explanations and examples.

Volume formulas worksheet

In the formulas below:
l=length, w=width, h=height, V=volume, A=area
Circle measures: π=3.14159, diameter=2*radius

The volume of a right rectangular prism

A right rectangular prism is a three-dimensional object with 6 faces, where all the 6 faces are rectangles. In a right rectangular prism, the angles between the base and sides are right angles.

The volume of a right rectangular prism is defined as the product of the area of one face (length*width) multiplied by its height.

Right rectangular prism volume formula is V= lwh.

Consider the following example:

Calculate the volume of a rectangular prism that has a height of 6 centimeters, a length of 3 centimeters and a width of 4 centimeters.

Right rectangular prism volume formula is V= lwh.

V=lwh=3*4*6=72 cubic centimeters.

The volume of a right circular cylinder

A right circular cylinder has 2 identical and parallel circular bases at the ends. The elements are perpendicular to the bases, therefore the cylinder is called right.

The volume of a right circular cylinder is defined as the product of the area of the circular base multiplied by the height of the cylinder.

Circular base area formula is A=πr2.

Right circular cylinder volume formula is V=πr2h.

Consider the following example:

Calculate the volume of a circular cylinder that has a radius of 5 centimeters and a height of 10 centimeters.

Right circular cylinder volume formula is V=πr2h.

V=πr2h=π*52*10=250π=785 cubic centimeters.

The volume of a sphere

A sphere is a three-dimensional object that has a surface of a ball, all the point on the surface of the sphere are lying at the same distance (the radius) from the center.

A sphere volume formula is V= 4/3 πr3.

Consider the following example:

Calculate the volume of a sphere that has a radius of 5 centimeters.

 A sphere volume formula is 4/3 πr3.

V=4/3 πr3=4/3*π*53= 4/3*53*π=167π =524 cubic centimeters.

The volume of a right circular cone

A right circular cone is a cone in which the center point of the circular base is joined with the vertex of the cone and forms a right angle (the height is perpendicular to the radius or the circle).

A right circular cone volume formula is V= 1/3 πr2h.

Consider the following example:

A cone has a height of 10 centimeters and a circular base with a radius of 3 centimeters. What is the volume of the cone?

A right circular cone volume formula is V= 1/3 πr2h.

V= 1/3 πr2h=1/3*π*32*10=30π=30*3.14159=94 cubic centimeters.

The volume of a rectangular pyramid

A rectangular pyramid is a pyramid that has four-sided base and a vertex.

A rectangular pyramid volume is defined as the product of the area of the base multiplied by the height of the pyramid (the height is the distance from the center point of the base to the vertex) divided by 3.

A rectangular pyramid volume formula is V=1/3 lwh.

Consider the following example:

A pyramid has a height of 10 centimeters a length of 3 centimeters and a width of 5 centimeters. What is the volume of the pyramid?

A rectangular pyramid volume formula is V=1/3 lwh.

V=1/3 lwh= 1/3*3*5*10=50 cubic centimeters.

Calculating dimensions given the volumes values of the shapes

To calculate an unknown dimension, plug the given dimensions and the volume values into the volume formula and solve.

Consider the following example:

A sphere has a volume of 33.5 cubic centimeters.

What is the radius of the sphere?

A sphere volume formula is V= 4/3 πr3.
4/3 πr3=33.5
πr3=33.5*3/4
πr3=25.125
3.14159r3=25.125
r3=25.125/3.14159
r3=8
r=3√8
r=2
The radius of the sphere is 2 centimeters.

Consider the following example:

A pyramid has a square base with a height that is 3 timed bigger than the length of the base.

If the volume of the pyramid is 27 cubic centimeters, what is its height?

x= the length and the width of the square pyramid
3x= the height of the pyramid

A rectangular pyramid volume formula is V=1/3 lwh.
V=1/3 lwh= 1/3 x*x*3x
1/3 x*x*3x=27
x3=27
x=3√27
x=3

Calculating the effect of changes in dimensions on volume

The power of the dimension determines the size of the change in the volume value:

If a dimension in the volume formula is raised to a first power, the volume changes by the same factor as the shape.
For example: A rectangular pyramid volume formula is V=1/3 lwh. The length, width and height in the formula are raised to a first power. If we double the length or the width or the height (one of them), then the volume will also be doubled.

If a dimension in the volume formula is raised to a second power, when the shape changes by a factor the volume changes by a square of the factor.
For example: A right circular cone volume formula is V= 1/3 πr2h. The radius of the circle is raised to a second power. If we double the radius, the volume will change by 22=4. The height in the formula is raised to a first power. If we double the height, then the volume will also be doubled.

If a dimension in the volume formula is raised to a third power, when the shape changes by a factor the volume changes by a third degree of the factor.
For example: A sphere volume formula is 4/3 πr3. The radius of the sphere is raised to a third power. If we double the radius, the volume will change by 23=8.

Consider the following example:

A circular cylinder has a radius of 5 centimeters and a height of 10 centimeters. What is the change in the volume if we double the radius and halve the height?

Right circular cylinder volume formula is V=πr2h.

We see in the formula that the radius is raised to a second power and the height is raised to a first power.
If we doble the radius, the volume will be multiplied by 22=4.
If we halve the height, the volume will be multiplied by 0.51.
The total change in the volume is 4*0.5=2.

We can check the answer by calculating the volumes in the two scenarios:
V=πr2h=π*52*10= 250π cubic centimeters.
V=πr2h=π*102*5= 500π cubic centimeters.
The volume value is doubled.

Comparing volumes of two shapes

In these questions we are given ratios between the dimensions of two shapes and we are required to compare their volumes.

We need to calculate the total effect on the volume of all the ratios between the shapes.

Consider the following example:  

A right circular cone has a volume of 200 cubic centimeters.
A right circular cone volume formula is V= 1/3 πr2h.

Which of the following cones has a higher volume than the given cone?
A. A cone with 3 times the radius and 1/10 times the height.
B. A cone with 2 times the radius and 1/4 times the height.
C. A cone with 1/2 times the radius and 3 times the height.
D. A cone with 1/3 times the radius and 10 times the height.

A right circular cone volume formula is V= 1/3 πr2h.

We see in the formula that the radius is raised to a second power and the height is raised to a first power.

Answer A: A cone with 3 times the radius and 1/10 times the height:
The total change is (32)*1/10=9/10 <1.
This cone has a smaller volume than the given cone.

Answer B: A cone with 2 times the radius and 1/4 times the height.
The total change is (22)*1/4=4/4=1
This cone has the same volume as the given cone.

Answer C: A cone with 1/2 times the radius and 3 times the height.
The total change is (1/2)2*3=(1/4)*3=2/4 <1.
This cone has a smaller volume than the given cone.

Answer D: A cone with 1/3 times the radius and 10 times the height.
The total change is (1/3)2*10=(1/9)*10=10/9 >1
This cone has a bigger volume than the given cone.

The correct answer is D– A cone with 1/3 times the radius and 10 times the height has a bigger volume than the given cone.