What is the difference between arc length and sector area?

Arc length is the distance along the curve; sector area is the pie slice. With θ in radians, arc length is rθ and sector area is ½r²θ. With θ in degrees, multiply the full circumference or area by θ/360.

How do I convert between radians and degrees?

Multiply radians by 180/π to get degrees, and degrees by π/180 to get radians. So π/4 radians is 45°, and 120° is 2π/3 radians.

Which height goes in a pyramid's volume formula?

The perpendicular height, from the top straight down to the base. The slant height runs down the middle of a triangular face and is only for that face's area. In a right square pyramid, slant height² = height² + (half the base edge)².

SAT Math

SAT circles, area and volume: review and practice

These questions cover the area and volume of shapes and solids, circle equations in the xy-plane, arcs and sectors. Some also ask you to convert between degrees and radians. They're part of Geometry and Trigonometry, 5–7 scored questions per test (about 15% of Math). Bluebook's reference sheet gives the main area and volume formulas, so the work is picking the right measurement: radius or diameter, height or slant height.

Try the practice questions What's tested

By Marteen Tang, founder of Ashore · Updated Nov 9, 2026 · Sources at the bottom of the page

Where it shows up on the SAT

Circles, Area, and Volume on the SAT
	Digital SAT
Section	Math
Content domain	Geometry and Trigonometry
Share of the section	≈15% (the whole Geometry and Trigonometry domain)
Questions	5–7 scored questions per test

This is the share for all of Geometry and Trigonometry across both Math modules. The framework counts operational (scored) questions. Individual skills within the domain have no fixed question count. Source: College Board Assessment Framework, version 4 (September 2026), Table 16, printed page 75.

What the questions ask you to do

Find area, perimeter, surface area and volume, often in a real-world setting.
Find the center and radius of a circle from its equation, including by completing the square.
Find arc length and sector area from a central angle.
Convert angles between degrees and radians.
Combine shapes, like a pyramid's base and its four triangular faces, to find a missing measurement.

Read the center and radius from a circle equation

If x² and y² have a coefficient other than 1, divide the whole equation by it first, then complete the square.

Read the center and radius from a circle equation
Form	Center	Radius
(x − h)² + (y − k)² = r²	(h, k)	The positive square root of the right side.
x² + y² = r²	(0, 0)	r, with r > 0.
x² + y² + Dx + Ey + F = 0	(−D/2, −E/2)	√((D/2)² + (E/2)² − F).
ax² + ay² + dx + ey + f = 0, a ≠ 0	Divide by a, then use the general form.	Complete both squares after normalizing.

The right side has to come out positive for the graph to be a circle. If it's zero, the graph is a single point.

Common traps

The wrong answers on these questions are built around these mistakes.

Using the diameter as the radius in A = πr². A circle with diameter 10 has area 25π, not 100π.
Reading r² as r. If (x − 2)² + (y + 1)² = 49, the radius is 7, not 49.
Flipping the center's signs. (x + 4)² + (y − 3)² = 25 has center (−4, 3), not (4, −3).
Using slant height in a volume formula. A pyramid's volume uses the height straight down from the top to the base. Slant height belongs only in the area of a triangular face.
Using degrees in a radian formula. Arc length = rθ only works with θ in radians. For a 60° arc in a circle of radius 6, convert first: 60° = π/3, so the arc is 2π.
Stopping at an area when the question asks for a length. If a square base has area 81, the edge is 9.

6 practice questions with answers

From Ashore's SAT question bank, easiest first. Try each one before you open the answer.

Question 1 · Easy
A rectangular piece of felt has a width of 8 inches. Its length is 17 inches greater than its width. What is the area, in square inches, of the piece of felt?
- A136
- B200
- C400
- D625
Show the answer
B. The length is 8 + 17 = 25 inches, so the area is 8 × 25 = 200 square inches. The other values use 17 as the length (136), double the required product (400), or square the length instead of multiplying length by width (625).
Question 2 · Easy
The volume of a right square prism is 320 cubic centimeters, and its height is 5 centimeters. What is the length, in centimeters, of one edge of its base?
- A5
- B8
- C16
- D64
Show the answer
B. The base area is 320/5 = 64 square centimeters, so its side length is √64 = 8 centimeters. Choice A reports the height, choice C divides the base area by four, and choice D reports the base area instead of a length.
Question 3 · Easy
For angles F and G, the difference between their measures, in that order, is −7π/18 radians. Which expression represents this difference in degrees?
- A(−7π/18)(360/π)
- B(−7π/18)(180/π)
- C(−7/18)(180/π)
- D(−7π/18)(π/180)
Show the answer
B. Multiplying the complete difference by 180/π gives (−7π/18)(180/π) = −70 degrees, so choice B is correct. Choice A doubles the conversion factor, choice C removes the original π before converting, and choice D reverses the conversion factor.
Question 4 · Medium
The graph of the given equation x² + y² + 18x − 10y = 38 is a circle in the xy-plane. What are the coordinates of the center of the circle?
- A(−9, −5)
- B(9, −5)
- C(−18, 10)
- D(−9, 5)
Show the answer
D. Completing the squares gives (x + 9)² + (y − 5)² = 38 + 81 + 25 = 144, so the center is (−9, 5); choice C uses the negatives of the full linear coefficients, choice B reverses both center signs, and choice A reverses only the vertical sign.
Question 5 · Hard
For a right square pyramid, the areas of the base and the four triangular faces together total 19,200 square inches. The triangular faces account for 9,984 square inches of this total. What is the height, in inches, of the pyramid?
- A20
- B48
- C52
- D96
Show the answer
A. The base area is 19,200 − 9,984 = 9,216 square inches, so its side is √9,216 = 96 inches and the common face altitude is 9,984/(2 × 96) = 52 inches. The pyramid’s height is √(52² − 48²) = √400 = 20 inches; the other choices are half the base side (48), the face altitude (52), and the full base side (96).
Question 6 · Hard
Each of two right square pyramids has base area 144 square centimeters. Their perpendicular heights are 10 centimeters and 20 centimeters. Which of the following is closest to the difference between their total surface areas, in square centimeters?
- A221
- B240
- C365
- D645
Show the answer
A. The base side is 12, giving slant heights √136 and √436, so the area difference is 24(√436 − √136) ≈ 221.249, closest to 221 square centimeters. Choice B uses the perpendicular-height difference as the slant-height difference, choice C retains one extra base area, and choice D reports only the taller pyramid’s total area.

Using Desmos on these questions

Type a circle equation as given, like x² + y² + 8x − 6y = 0, and Desmos draws the circle so you can estimate the center and radius from the grid. Confirm by completing the square, since the choices use exact values. For arc length or sector area, type the formula with numbers, like 6 × π/3. For a solid, type the volume formula with the unknown, such as (1/3)x² × 12 = 400, and click the positive vertical line to read x.

Questions

Still have a question? support@tryashore.com

Complete the square for x and for y. For x² + y² + 8x − 6y = 0, add 16 and 9 to both sides to get (x + 4)² + (y − 3)² = 25, so the center is (−4, 3) and the radius is 5.

By Marteen Tang · Founder of Ashore

Marteen Tang founded Ashore and works on its SAT and AP lessons, practice and plans. Updated Nov 9, 2026.

Sources

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