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Math · Checkpoint 22 of 30 · Geometry and Trigonometry · ~13 min
Circles, area, volume & trig
Circle equations, area and volume, SOH-CAH-TOA, and special right triangles.
The 30-second version
a circle is (x − h)² + (y − k)² = r² — center (h, k), radius r, and the right side is r squared. Trig on the SAT is right-triangle trig: SOH-CAH-TOA, plus the fact that sine of an angle equals cosine of its complement. Volume formulas are given to you in the test's reference sheet — the skill is picking radius vs. diameter correctly, not memorizing.
Learn it
Circle equations
(x − 3)² + (y + 2)² = 25: center (3, −2) — flip the signs — radius √25 = 5. When the equation arrives expanded (x² + y² − 6x + 4y − 12 = 0), complete the square: group x's and y's, add (half the coefficient)² to both sides: (x − 3)² + (y + 2)² = 25. Same circle. This is the hardest routine skill in the domain and worth a few practice reps on its own.
Area and volume
Circle area πr², circumference 2πr. Prisms and cylinders: base area × height. Cones and pyramids: ⅓ of that. Spheres: (4/3)πr³. The reference sheet in Bluebook lists these — questions test setup: "a cylindrical tank 10 ft across" means r = 5, and units may need converting before you cube them.
Right-triangle trig
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Label the sides relative to the angle in question first. The complementary identity — sin(x°) = cos(90° − x°) — gets its own questions: "if sin A = 0.6, what is cos(90° − A)?" Answer: 0.6, no calculation.
Special right triangles
45-45-90 sides ratio 1 : 1 : √2. 30-60-90 sides ratio 1 : √3 : 2 (shortest opposite 30°). These appear both openly and hidden — a square's diagonal makes 45-45-90; an equilateral triangle's height makes 30-60-90.
Common traps
- Radius vs. diameter — "12 inches across" is r = 6. The most common geometry error on the test.
- Reading r² as r. If the equation ends "= 36," the radius is 6.
- Center signs. (x + 4)² means h = −4.
Try it
What are the center and radius of the circle (x + 1)² + (y − 4)² = 49?
Template: (x − h)² + (y − k)² = r². (x + 1) = (x − (−1)) → h = −1; (y − 4) → k = 4; r = √49 = 7. A flips both signs; C forgets the square root. Both mistakes have dedicated wrong answers on the real test.
Free resources for this checkpoint
Khan Academy Official Digital SAT Prep (opens in a new tab)
Free structured lessons, video instruction, and practice exercises for both SAT sections, built with College Board as the official practice partner. After a Bluebook practice test, you can link your results to get personalized practice recommendations.
Official Practice Partner
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Practice next
Question Bank filter: Math → Geometry and Trigonometry → Circles, Area and volume, and Right triangles and trigonometry. Then head into the Desmos checkpoints — graphing circle equations makes them visual.
Practice these skillsOriginal Digital SAT-style questions on what this checkpoint covers.