The SAT Circle Guide: Radius, Diameter, Arc Length, and Equations

A circle has 360 degrees and zero sides, but that is not the most special thing about it. Other shapes, like an oval, can also have 360 degrees and no sides.

The property that really separates a circle from an oval is this: every point on the edge of a circle is the same distance from the center. That distance is called the radius.

If you draw from the center to any point on the circle, that segment has the same length. There are infinitely many points on the edge, and all of them are equidistant from the center.

The diameter is the radius's older brother. It goes through the center of the circle and touches two points on the circle. The diameter is double the radius.

The two foundational formulas are circumference and area. Circumference is the length around the circle. Area is the space inside the circle.

The circumference formula is C = pi d. Since the diameter is 2r, you can also write circumference as C = 2 pi r.

The area formula is A = pi r^2. I recommend memorizing these. The old memory trick I learned was, cherry pies are delicious and apple pies are too. It is silly, but I still remember it from middle school, so it did its job.

These formulas matter because a lot of the later circle questions build on them. Arc length, sectors, radius questions, and graph questions all come back to knowing what the radius, diameter, circumference, and area represent.

The equation of a circle on a graph is (x - h)^2 + (y - k)^2 = r^2. This equation lets you plot a circle on the coordinate plane.

To draw a circle, you really need two pieces of information: where the center is and how big the radius is. That is exactly what the equation gives you.

The center is (h, k). The radius is r. If the equation says (x - 2)^2 + (y - 2)^2 = 25, the center is (2, 2) and the radius is 5.

One thing students miss is that the signs inside the parentheses look opposite. If you see x - 2, the h value is positive 2. If you see y + 3, that means y - negative 3, so the k value is negative 3.

An arc is a piece of the circle's edge. If you imagine a slice of pizza, the crust of that slice is the arc. The distance along that crust is the arc length.

The formula is: arc length = circumference times arc degree over 360.

This formula is not random. A full circle has 360 degrees. If your arc uses 180 degrees, it is half the circle, so the arc length is half the circumference. If it uses 90 degrees, it is one fourth of the circle, so the arc length is one fourth of the circumference.

For example, if a 180-degree arc comes from a circle with circumference 100 pi, the arc length is 50 pi because 180 over 360 is one half.

The SAT can also ask you to convert between radians and degrees. Radians and degrees measure the same thing, just like inches and feet both measure length. They are different units for angles.

Radian measurements often have pi in them, like pi over 2, pi over 4, or pi over 8. Degree measurements look like 90 degrees, 45 degrees, or any other degree value.

To convert from radians to degrees, multiply by 180 over pi. To convert from degrees to radians, multiply by pi over 180.

If you forget which fraction to use, think about what needs to cancel. If you start with pi over 2 radians and want degrees, the pi has to disappear. That means you need a pi in the denominator, so you multiply by 180 over pi. The result is 90 degrees.

If you start with 45 degrees and want radians, you need to introduce pi, so you multiply by pi over 180. That gives 45pi over 180, which reduces to pi over 4.

A tangent line is a line that touches a circle at exactly one point. The useful SAT fact is that the tangent line is perpendicular to the radius drawn to the point of tangency.

That means if a line touches the circle at one point, and you draw a segment from that point back to the center, the angle between the tangent line and that radius is 90 degrees.

This becomes powerful on coordinate geometry questions. Perpendicular lines have negative reciprocal slopes. If one line has slope 2, the perpendicular slope is negative one half. If one line has slope negative 2, the perpendicular slope is one half.

So if the SAT gives you the center of a circle and the point where a tangent line touches the circle, you can find the slope of the radius, then flip and negate it to get the tangent line's slope.

One example asks about a circle in the xy-plane with an equation like (x + 3)^2 + (y - 1)^2 = 25, then asks which point does not lie in the interior of the circle.

You can solve this algebraically by plugging in each point and checking whether the left side is less than 25. But you can also use Desmos. Type the circle equation into Desmos, then plot the answer choices.

Any point clearly inside the circle is not the answer. The point outside the circle is the one that does not lie in the interior. In the example from the video, choice D was the point outside the circle.

This is one of those questions where the graphing calculator can save time, especially if the answer choices are just coordinate points.

Another example gives a circle with center O and points A and B on the circle. The measure of arc AB is 45 degrees and the length of the arc is 3 inches. The question asks for the circumference.

I still draw these out, even after doing SAT questions for years. Drawing the circle and marking the 45-degree arc makes the relationship obvious.

Use the formula: arc length = circumference times angle over 360. The arc length is 3, the angle is 45, and the circumference is unknown. So 3 = C times 45 over 360.

Since 45 over 360 is one eighth, we have 3 = C over 8. Multiply by 8 and the circumference is 24 inches.

If an angle has a measure of 9pi over 20 radians and the SAT asks for degrees, multiply by 180 over pi.

The pi cancels. Then you simplify 9 times 180 over 20. Since 180 over 20 is 9, the result is 81 degrees.

The important move is not memorizing that exact answer. The important move is knowing that radians with pi should lose the pi when you convert to degrees.

Another example gives an equation like 2x^2 - 6x + 2y^2 + 2y = 45 and tells you the graph is a circle. It asks for the radius.

You could complete the square, and that is a valid algebra method. But in the video I showed the Desmos approach. Type the equation into Desmos, then identify the top and bottom points of the circle.

I trust the top and bottom points more than random left and right-looking points because Desmos will often show exact extrema there. In the example, the top and bottom points were 10 units apart, so that distance is the diameter.

If the diameter is 10, the radius is 5. That is the answer.

A tangent line question might say a circle has center (-4, -6), and line k is tangent to the circle at (-7, -7). What is the slope of line k?

Before doing anything else, draw it. The tangent point is on the circle, and the center is inside the circle. Connect the center to the tangent point. That segment is a radius, and the tangent line is perpendicular to it.

Find the slope of the radius using change in y over change in x. From (-4, -6) to (-7, -7), the change in y is -1 and the change in x is -3, so the radius slope is 1 over 3.

The tangent line has the negative reciprocal slope. Flip 1 over 3 to get 3, then change the sign. The slope of line k is -3.

A final example gives a circle with a diameter whose endpoints are (2, 4) and (2, 14). The equation is written as (x - 2)^2 + (y - 9)^2 = r^2, and the question asks for r.

Because the x-coordinate is the same for both endpoints, the diameter is vertical. The y-values are 4 and 14, so the distance between them is 10.

That means the diameter is 10. The radius is half the diameter, so r equals 5.

You can also see why the center is (2, 9): it is halfway between y = 4 and y = 14. But the question only asks for r, so the radius is enough.