Area of a Circle: Formula, Examples and More

If you are aiming for a perfect score on the math sections of the SAT or ACT, you are going to need to know a thing or two about circles.

While questions on circles will not make up the majority of your SAT or ACT Math questions, you should expect to see a small handful of them on the test. When you’re aiming for the 99th percentile, every question counts. As such, you should be familiar with how to calculate the area, circumference, radius, diameter, degrees, arcs, and points of a circle.

This guide will give you all the information you need to know about circles so that you can answer each and every circle-related question you face on test day.

What is a circle?

While it might seem self-explanatory, it is critical that you know exactly what a circle is before you take the SAT or ACT.

A circle refers to specific points on a plane that are also equidistant from a single point, known as the center of the circle.

This set of points forms the closed, round, two dimensional-shape that we call a circle.

Key terms

There are a few terms you will need to know about circles before you start measuring them or using any other circle formulas:

• Circumference (c): The distance around the circle
• Arc: A curved portion of a circle that lies between two set points
• Diameter (d): The length of a line segment that marks the largest distance between two points on a circle
• Radius (r): The length of a line segment that connects the center of the circle with a set point on a circle (half of the diameter)
• Pi (π): Pi, which is approximately equivalent to 3.1415926 is the ratio of the circumference of a circle to its diameter

Once you are familiar with these terms, you will be able to use various formulas to solve circle questions on the SAT and ACT Math sections.

Measuring a circle

Circles can be measured using two different units: radians and degrees.

A radian is a unit that determines the measurement of an angle. When an arc of a circle has the same length as the radius of that same circle, the angle of that arc is equivalent to one radian.

While you will need to learn about radians for geometry and other math classes, you will not typically use this measurement very often on the SAT or the ACT.

On standardized tests, you will largely rely on degrees instead.

A circle is 360 degrees. To measure a part of a circle, you will need to see how that part fits within the allotted 360 degrees.

For instance, if you were trying to measure the angle of the arc for one half of the circle, you would take 360 degrees and divide it by half to get 180 degrees. Likewise, if you were trying to measure the angle of the arc for a quarter of the circle, you would take 360 degrees and divide it by 4 to get 90 degrees.

Remember that a complete circle is always 360 degrees, and you will be able to measure the arcs with ease.

Key formulas

Now that you know key terms as well as the different ways to measure a circle, you will be able to understand the important circle formulas that you’ll need to know for the SAT and ACT.

While these key formulas will be provided for you, I recommend that you memorize them so that you don’t have to waste your limited time flipping to the reference section during the test.

Area

If you’re asked to find the area of a circle, use this formula:

• a=πr²

Arc

When measuring the circumference of the arc of a circle, you will need to take the arc degree and divide it by 360 (the total number of degrees in a circle). Then you need to multiply that by pi and the diameter of the circle. You can use the formula below:

• c arc= πd (arc degree/360)

You will also want to know that the number of radians of arc in a circle is 2π

Circumference

To find the circumference of a circle, you can use either of these two formulas:

• c=2πr
• c=πd

Diameter

When calculating the diameter of a circle, remember that it represents the longest possible line segment between two set points on the circle. You can calculate the diameter using the following formula:

• d=2r

Equation of a circle

The equation of a circle is calculated using this formula:

• (x-h)² + (y-k)² = r²

In this formula, (h,k) is the coordinates of the center of the circle.

Example questions

Here is an example of a question you may encounter on the SAT (taken from a College Board practice test):

Take a moment and try to solve this question on your own. 

Finished?

If you chose option “A,” you are correct!

Because the coordinates provided in this question for the center of the circle are (0,4), the equation for the circle is (x-0)² + (y-4)² = r²

Using this formula and the distance formula given the endpoint (4/3, 5), you will end up with r²= 25/9.

This would make option “A” the only possible answer option.

Here’s another sample question:

Take some time to answer this question, and look at the answer below when you are ready.

You should have chosen option “A” for this question.

Here’s why (according to the College Board): A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and angle AOC is a central angle of the circle. From the figure, the two diameters that meet to form angle AOC are perpendicular, so the measure of angle AOC is 90 degrees. This central angle intercepts minor arc AC, meaning minor arc AC has 90 degrees of arc. Since the circumference of the entire circle is 36, the length of minor arc AC is 90/360 x 36=9.

If you missed either (or both) of these questions, there’s no need to worry. With the right resources, you can master your knowledge of circles and other math concepts that will appear on the SAT and ACT.

You can learn more about circles and the different shapes you will encounter on the math sections of the SAT and ACT when you take prep courses through Prep Expert. With our prep courses, you can learn tips and tricks for memorizing formulas as well as strategies for tackling other math questions you will see on test day.

Learn more about Prep Expert or sign up for one of our SAT or ACT prep courses today when you visit our website.