Additional topics in math on the SAT test
Included topics: volume word problems; Congruence and similarity; right triangle trigonometry and word problems
Additional topics in math include geometry, trigonometry, and problems with complex numbers.
10% of the questions on the SAT fall into the additional topics classification. The other questions are classified into the three subscores (heart of Algebra, passport to advanced mathematics and problem solving and data analysis).
Volume word problems SAT topic
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.
Congruence and similarity SAT topic
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.
Right triangle trigonometry and word problems SAT topic
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 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.