How to solve SAT circle questions

Circle questions on the Digital SAT come in two types. Some work in the $xy$-plane, where you read or build the equation of a circle and find its center and radius. Others are classic geometry, where you find arc lengths, sector areas, and angle measures using central angles, inscribed angles, and tangent lines. Geometry and Trigonometry is about 15% of the Math section, or 5 to 7 of the 44 questions. Those questions are split across four skills, so expect 1 or 2 circle questions per test. Questions that require completing the square tend to be among the harder ones.

What the questions look like

Circle equations. You get an equation like $(x - 4)^2 + (y + 3)^2 = 49$, or an expanded version like $x^2 + y^2 - 8x + 6y - 24 = 0$, and find the center, the radius, or the diameter. Some ask whether a point lies inside, on, or outside the circle.

Arcs and sectors. You find an arc length or a sector area from a central angle given in degrees or radians, or work backward from an arc length to the angle.

Angle and tangent relationships. You use the facts that an inscribed angle is half the central angle that intercepts the same arc, and that a tangent line is perpendicular to the radius at the point where it touches the circle.

Step-by-step method

1. Read standard form directly. The equation $(x - h)^2 + (y - k)^2 = r^2$ has center $(h, k)$ and radius $r$. The signs inside the parentheses are the opposite of the center's coordinates.
2. Complete the square on expanded equations.
   • Group the $x$ terms and the $y$ terms, and move the constant to the right side.
   • For each group, take half the linear coefficient, square it, and add it to both sides.
   • Factor each group as a perfect square.
3. Take the square root of the right side to get the radius. If the right side is 49, the radius is 7.
4. Use the fraction of the circle for arcs and sectors. With a central angle of $\theta$ degrees:
   $$\text{arc length} = \frac{\theta}{360} \cdot 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \cdot \pi r^2$$
   With $\theta$ in radians, arc length is $r\theta$ and sector area is $\frac{1}{2}r^2\theta$.
5. Apply the angle facts. An inscribed angle equals half its intercepted arc's central angle. An angle inscribed in a semicircle is $90^\circ$. A radius drawn to a point of tangency meets the tangent line at $90^\circ$, which often creates a right triangle.

Worked example

Example 1. A circle in the $xy$-plane has equation $x^2 + y^2 - 8x + 6y - 24 = 0$. What are the center and radius of the circle?

Group the terms and move the constant:

$$(x^2 - 8x) + (y^2 + 6y) = 24$$

Half of $-8$ is $-4$, and $(-4)^2 = 16$. Half of $6$ is $3$, and $3^2 = 9$. Add both to each side:

$$(x^2 - 8x + 16) + (y^2 + 6y + 9) = 24 + 16 + 9$$
$$(x - 4)^2 + (y + 3)^2 = 49$$

The center is $(4, -3)$ and the radius is $\sqrt{49} = 7$.

Example 2. A circle has a radius of 9 centimeters. A central angle of the circle measures $\frac{2\pi}{3}$ radians. What are the length of the intercepted arc and the area of the sector?

With the angle in radians:

$$\text{arc length} = r\theta = 9 \cdot \frac{2\pi}{3} = 6\pi \text{ cm}$$

$$\text{sector area} = \frac{1}{2}r^2\theta = \frac{1}{2}(81)\left(\frac{2\pi}{3}\right) = 27\pi \text{ cm}^2$$

Check with fractions of the circle. An angle of $\frac{2\pi}{3}$ radians is $120^\circ$, which is $\frac{1}{3}$ of the circle. The circumference is $18\pi$ and the area is $81\pi$, and one third of each gives $6\pi$ and $27\pi$.

Common traps

• Keeping the signs from the equation. In $(x - 4)^2 + (y + 3)^2 = 49$, the center is $(4, -3)$, not $(-4, 3)$. Rewrite $y + 3$ as $y - (-3)$ if it helps.
• Reporting $r^2$ as the radius. The right side of the equation is the radius squared. A right side of 49 means a radius of 7 and a diameter of 14.
• Adding to only one side when completing the square. In Example 1, the 16 and 9 must be added to the right side too. Forgetting gives a radius of $\sqrt{24}$, a wrong choice.
• Mixing degree and radian formulas. Don't plug a radian measure into $\frac{\theta}{360}$, or a degree measure into $r\theta$. Convert first if needed: $\pi$ radians equals $180^\circ$.
• Setting inscribed and central angles equal. An inscribed angle that intercepts a $100^\circ$ arc measures $50^\circ$, not $100^\circ$.

Desmos tip

Type the circle's equation exactly as given, expanded or not, and Desmos draws it. You can read the center and radius from the grid as a check on your algebra. To confirm your standard form, graph it alongside the original. If you completed the square correctly, the two circles overlap exactly. For questions asking whether a point is inside the circle, type the point and see where it lands.