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# Circle theorems on the SAT test

### SAT Subscore: Additional topics in math

Circle theorems topic includes two parts:
Calculating arc length and sector areas using central angles measured in degrees.
Calculating angle measures in degrees inside a circle.

The following formulas are provided at the beginning of each SAT math section:
Circumference of a circle formula is C=2πr.
Area of a circle formula is A=πr2.
Number of degrees of arc in a circle is 360.

Pi (π) is the ratio of the circumference of any circle to the diameter of that circle. Regardless of the circle’s size, this ratio will always equal π (approximately 3.14).

A triangle that one of its angles is a central angle has 2 radii as its sides and it is therefore an isosceles triangle (one of its angles is a central angle and the other 2 angles are equal).

The intersection point of the diameters is at the center of the circle and it divides each diameter to 2 radii. The intersection creates 4 central angles: 2 pairs of equal vertical angles and 4 pairs of supplementary angles (the sum of supplementary angles is 180°).

The relationship between central angle, arc length and sector area is given by the following ratios:

central angle    =             arc length              =     sector area
______________       _____________________          ____________
360°                 circle circumference            circle area

### Calculating circumference and area of a circle

The following formulas are provided at the beginning of each SAT math section:
Circumference of a circle formula is C=2πr.
Area of a circle formula is A=πr2.

Units of measurement:
Arc length and circumference measure distance in units such as inches or centimeters.
Area and sector measure area is square units such as square inches or square centimeters.
Angles are measured in degrees.

Consider the following example:

The radius of the circle is equal to 10 centimeters.

What are the circumference and the area of the circle?

Circumference of a circle:
The circumference formula is C=2πr.
C=2*10*π=20π

Area of a circle:
The area formula is A=πr2.
A=πr2=π*102=100π

### The relationship between central angle, arc length and sector area

A central angle has a vertex at the center of a circle O and sides located on the circle circumference in two points A and B. The central angle ∠AOB determines a portion of the circumference (an arc) AC and a potion from the area (sector).

A minor arc: Each central angle divides a circle into two arcs. The smaller of the two arcs is called the minor arc and the larger of the two arcs is called the major arc (a minor arc subtends an angle less than or equal to 180°). Note that when dealing with arcs, we always look at the minor arc.

The relationship between central angle, arc length and sector area is given by the following ratios:

central angle    =             arc length              =     sector area
______________       _____________________          ____________
360°                 circle circumference            circle area

This relationship presents 3 equal ratios:

Ratio 1: The angle as a portion of the total degrees of the circle (360°) =

Ratio 2: The arc length as a portion from the length of the circle circumference (C=2πr) =

Ratio 3: The sector area as a portion from the total area of the circle (A=πr2).

Note that we can calculate the denominators of the arc and the sector ratios given the value of the radius r.

In the figure below we have the following ratios:

∠AOC     =        arc AC          =    sector ACO area
_______      _______________        ________________
360°         circumference               circle area

________     ________________        _________________
360°            circumference             circle area

∠BOD     =         arc BD            =     sector BDO area
________      _______________          __________________
360°           circumference                circle area

∠BOC     =          arc BC           =     sector BCO area
________       ______________           _________________
360°          circumference                 circle area

Consider the following example:

A central angle of a circle is equal to 120 degrees and the radius of the circle is equal to 5 centimeters.

What is the arc length of the central angle?

Calculating the circumference of the circle:
C=2πr
C=2*5*π=10π

Calculating the arc length with the ratios:
Ratio 1: The angle as a portion of the total degrees of the circle (360°).
Ratio 2: The arc length as a portion from the length of the circle circumference (C=2πr).

central angle     =           arc length
______________        _____________________
360°                  circle circumference

Representing with the variable x the arc length and plugging the given values into the ratio:

120°   =     x
_____      _____
360°        10π

360x=1200π
x=1200/360π
x=120/36π
x=10/3π=3.33π
x=31/3π=31/3*3.14=10.47
The arc length is 10.47 centimeters.

120/360=1/3=0.33
3.33/10.47=0.32
The answers are different because of rounding differences.

Consider the following example:

The arc length of a circle is 2 centimeters and the radius of the circle is 1.6 centimeters.

What is the area of the sector of the arc?

Calculating the circumference of the circle:
C=2πr
C=2*1.6*π=3.2π=10.

Calculating the area of the circle:
The area formula is A=πr2.
A=π*1.62=2.56π=8.

Calculating the sector area with the ratios:
Ratio 1: The arc length as a portion from the length of the circle circumference (C=2πr).
Ratio 2: The sector area as a portion from the total area of the circle (A=πr2).

Representing with the variable x the sector area and plugging the given values into the ratio:

arc length              =     sector area
____________________        _____________
circle circumference           circle area

1.6   =   x
____      ___
10         8

1.6*8=10x
x=1.6*8/10
x=1.28
The sector area is 1.28 centimeters.

1.6/10=0.16
1.28/8=0.16

### Angle relationships in circle

This subject includes 2 angle types:

Angles of isosceles triangles in a circle.

Angles created by intersection of diameters in a circle.

#### Angles of isosceles triangles in a circle

At the beginning of each SAT math section, it is provided that the number of degrees of arc in a circle is 360. In other words, the sum of central angle measures in a circle is equal to 360°.

A triangle that one of its angles is a central angle has 2 radii as its sides and it is therefore an isosceles triangle. Meaning that one of its angles is a central angle and the other 2 angles are equal.

The sum of angles in a triangle is 180°, if we are given the value of the central angle or the value of one of the equal angles, we can calculate the other angle values.

Consider the following example:

In the figure below, the point O is the center of a circle. The angle ∠AOB is equal to 100°.

What is the value of the angle ∠OAB?

Since the angle ∠AOB is a central angle, the sides OA and OB are radii of the circle and therefore the triangle ABO is an isosceles triangle.

In an isosceles triangle the angles opposite the two equal sides are equal, therefore ∠OAB=∠ABO.

The sum of angles in a triangle is equal to 180°, therefore ∠OAB+∠ABO+∠AOB=180°.

We are given that ∠AOB=100° and we know that ∠OAB=∠ABO, therefore 100°+∠OAB+∠ABO=180°.

Represent by x the equal angles ∠OAB and ∠ABO we get an equation that we can solve:
100+2x =180
2x=80
x=40
The value of the angle ∠OAB is 40 degrees.

40+40+100=180
180=180

Consider the following example:

In the figure below, the point O is the center of a circle.
The angle ∠AOB is twice bigger than the angle ∠COA.
The angle ∠COB is 5 times bigger than the angle ∠COA (the angle ∠COB is bigger than 180°).

What is the value of the angles ∠AOB and ∠COB?

Represent by x the angle ∠COA.

We are given that the angle ∠AOB is twice bigger than the angle ∠COA, therefore the angle ∠AOB=2x.
We are given that the angle ∠COB is 5 times bigger than the angle ∠COA, therefore the angle ∠COB=5x.

The sum of central angle measures in a circle is equal to 360° and we are given that the point O is the center of the circle, therefore we can write an equation and solve it:
x+2x+5x=360
8x=360
x=45

We know that the angle ∠AOB=2x, therefore the angle ∠AOB=90°.
We know that the angle ∠COB=5x, therefore the angle ∠COB=5*45=225 degrees.

#### Angles created by intersection of diameters in a circle

The intersection point of the diameters is at the center of the circle and it divides each diameter to 2 radii.

The intersection creates 4 central angles: 2 pairs of equal vertical angles and 4 pairs of supplementary angles (the sum of supplementary angles is 180°).

Given the value of one of the 4 central angles, we can calculate the other central angles.

In the figure below AB and CD are diameters of the circle.

What are the central and supplementary angles created by the diameters?

The intersection creates 4 equal radii AO, CO, DO and BO.

The intersection point O is at the center of the circle and it divides each diameter to 2 radii: the diameter AB is divided to radii AO and OB; the diameter CD is divided to radii CO and OD.

The intersection creates 4 central angles: ∠COA, ∠AOD, ∠DOB and ∠BOC.

The intersection creates 2 pairs of equal vertical angles: ∠COA=∠BOD; ∠COB=∠AOD.

The intersection creates 4 pairs of supplementary angles: ∠COA+∠AOD=180°; ∠AOD+∠DOB=180°;  ∠DOB+∠BOC=180°; ∠BOC+∠COA=180°.

Consider the following example:

In the figure below, O the is center of the circle and the chords AB and CD intersect at point O. The angle ∠BOD is equal to 40°.

What is the ratio between the angles ∠ACO and ∠ADO?

Given the value of one of the 4 central angles, we can calculate the other central angles:

We are given that ∠BOD=40°

∠AOC=∠BOD (vertical angles are equal), therefore, ∠AOC=40°.

∠BOD+∠AOD=180° (the sum of supplementary angles is 180°), therefore ∠AOD=180°-40°=140°.

Calculating the angles in triangle ACO:

We are given that O the is center of the circle and the chords AB and CD intersect at point O, therefore AB and CD are diameters and create 4 radii CO=BO=AO=DO.

Since CO=AO the triangle ACO is isosceles and the angles ∠ACO and ∠CAO are equal (represented by x).

The sum of angles in a triangle is equal to 180°, therefore ACO+CAO+AOC=180°.

Plugging the values we get an equation that we can solve:
x+x+40=180
2x=140
x=70
The angle ∠ACO is equal to 70 degrees.

Calculating the angles in triangle ADO:

We are given that O the is center of the circle and the chords AB and CD intersect at point O, therefore AB and CD are diameters and create 4 radii CO=BO=AO=DO.

Since AO=DO the triangle ADO is isosceles and the angles ∠DAO and ∠ADO are equal (represented by x).

The sum of angles in a triangle is equal to 180°, therefore DAO+ADO+AOD=180°.

We found that the angle ∠ACO is equal to 140 degrees.

Plugging the values, we get an equation that we can solve:
x+x+140=180
2x=40
x=20
The angle ∠ADO is equal to 20 degrees.

The ratio between the angles ∠ACO and ∠ADO is 70/20=7/2=31/2

Note that we need to write the value of an angle ∠ACO in the nominator and not the opposite.