Circle geometry | Plane geometry | ACT Math

The geometry mentioned in this chapter must be memorized. Many more advanced topics of geometry and coordinate geometry involving circles will build off of the principles taught in this chapter.

Core principles of circle geometry

Radius

The radius ( $r$ ) of a circle is the distance from the center of the circle to the edge of the circle. A circle is perfectly round, so the radius can touch any point on the edge and still be the same distance from the center.

Diameter

The diameter ( $d$ ) of a circle is the distance from one point on the edge of a circle, through the center, and to the opposite edge of the circle. The diameter must pass through the center. This ensures that the diameter will be the same regardless of where the points lie on the edge.

The diameter is equal to twice the radius:

$d = 2 * r$

Area

Finding the area of a circle requires knowledge of the radius. In fact, the radius is all we need to find the area of a circle. The area of a circle is given by:

$A = π r^{2}$

Where $A$ is the area, $r$ is the radius of the circle, and $π$ (pi) is a constant equal to about $3.14$ . Typically, there is a button on a calculator for pi.

On the test, questions may not use a number for pi but instead will include the symbol in the answer. So, if our radius were equal to $2$ and a question asked us to find the area, answers could look like either of the following:

$A A = 4 π \approx 12.57$

Circumference

The circumference of a circle is the distance around the outside of the circle. You can think of it as the perimeter of the circle. This may also be calculated just with the radius of a circle. It is given by the following equation:

$C = 2 π r$

In this equation, $C$ stands for the circumference of the circle and $r$ represents the radius of the circle.

Arc length

The arc length of a circle measures the length of a portion of the circumference. Arc length is represented in equations with the variable $s$ . We find the arc length relative to an angle that we refer to as $θ$ (theta). To determine the arc length relative to an angle $θ$ , we use the following formula:

With $θ$ in degrees (not radians):

$s = 2 π r (θ /360)$

You can easily extend this formula to be used for radians, given that a full circle in radians is $2 π$ .

With $θ$ in radians (not degrees):

$s = 2 π r (θ / (2 π))$

This simplifies to:

$s = r θ$

Angle measurements

The total sum of angle measurements within a circle is 360 degrees (or $2 π$ radians). This means that all the angles must add up to equal 360 degrees inside the circle. If you are missing one angle, you can solve for it by remembering this rule as you would for a parallelogram.

Diagram

Below is an image showing the core principles of circle geometry:

$Circle geometry showing diameter, radius, and arc length$

Examples:

Find the circumference of a circle that has a radius of $3$ .

(spoiler)

$C C C = 2 π r = 2 π * 3 = 6 π$

The circumference of the circle is equal to $6 π$

Here’s another:

If a circle has a diameter of $8$ , what is its area?

(spoiler)

First, let’s remember our formula for the area of a circle.

$A = π r^{2}$

We must solve for the radius since we are given the diameter:

$d r r r = 2 * r = d /2 = 8/2 = 4$

Now we can solve for the area of the circle:

$A A A = π r^{2} = π * 4^{2} = 16 π$

The area of the circle is $16 π$ .

Let’s do one more focused on angles:

What is the measure of angle $X$ ?

$Circle angles sum to 360$

(spoiler)

We know the sum of all the angles must equal 360 degrees and that the red angle symbolizes a right angle (90 degrees).

$70 + 70 + 90 + X X X = 360 = 360 - 70 - 70 - 90 = 130$

The measure of angle $X$ is 130 degrees.

Key points

Radius. The distance from the center to the edge of the circle.

Diameter. The distance from one end of the circle to the other, passing through the center. Diameter is twice the size of the radius.

Area. $A = π r^{2}$ where $A$ is area, $π$ (pi) is a constant ( $3.14$ ), and $r$ is radius.

Circumference. The distance around the circle; it can be thought of as the perimeter of the circle. $C = 2 * p i * r$ where $C$ is circumference and $r$ is radius.

Arc length. The arc length of an angle $θ$ is given by the formula $s = 2 π r (θ /360)$ for degrees, or $s = r * θ$ for radians.

Angle measurements. The sum of all angles inside a circle is 360 degrees or $2 π$ radians.

The geometry mentioned in this chapter must be memorized. Many more advanced topics of geometry and coordinate geometry involving circles will build off of the principles taught in this chapter.

Core principles of circle geometry

Radius

Diameter

The diameter is equal to twice the radius:

$d = 2 * r$

Area

Finding the area of a circle requires knowledge of the radius. In fact, the radius is all we need to find the area of a circle. The area of a circle is given by:

$A = π r^{2}$

Where $A$ is the area, $r$ is the radius of the circle, and $π$ (pi) is a constant equal to about $3.14$ . Typically, there is a button on a calculator for pi.

$A A = 4 π \approx 12.57$

Circumference

$C = 2 π r$

In this equation, $C$ stands for the circumference of the circle and $r$ represents the radius of the circle.

Arc length

With $θ$ in degrees (not radians):

$s = 2 π r (θ /360)$

You can easily extend this formula to be used for radians, given that a full circle in radians is $2 π$ .

With $θ$ in radians (not degrees):

$s = 2 π r (θ / (2 π))$

This simplifies to:

$s = r θ$

Angle measurements

Diagram

Below is an image showing the core principles of circle geometry:

$Circle geometry showing diameter, radius, and arc length$

Examples:

Find the circumference of a circle that has a radius of $3$ .

(spoiler)

$C C C = 2 π r = 2 π * 3 = 6 π$

The circumference of the circle is equal to $6 π$

Here’s another:

If a circle has a diameter of $8$ , what is its area?

(spoiler)

First, let’s remember our formula for the area of a circle.

$A = π r^{2}$

We must solve for the radius since we are given the diameter:

$d r r r = 2 * r = d /2 = 8/2 = 4$

Now we can solve for the area of the circle:

$A A A = π r^{2} = π * 4^{2} = 16 π$

The area of the circle is $16 π$ .

Let’s do one more focused on angles:

What is the measure of angle $X$ ?

$Circle angles sum to 360$

(spoiler)

We know the sum of all the angles must equal 360 degrees and that the red angle symbolizes a right angle (90 degrees).

$70 + 70 + 90 + X X X = 360 = 360 - 70 - 70 - 90 = 130$

The measure of angle $X$ is 130 degrees.

Key points

Radius. The distance from the center to the edge of the circle.

Diameter. The distance from one end of the circle to the other, passing through the center. Diameter is twice the size of the radius.

Area. $A = π r^{2}$ where $A$ is area, $π$ (pi) is a constant ( $3.14$ ), and $r$ is radius.

Circumference. The distance around the circle; it can be thought of as the perimeter of the circle. $C = 2 * p i * r$ where $C$ is circumference and $r$ is radius.

Arc length. The arc length of an angle $θ$ is given by the formula $s = 2 π r (θ /360)$ for degrees, or $s = r * θ$ for radians.

Angle measurements. The sum of all angles inside a circle is 360 degrees or $2 π$ radians.