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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, so the skill is picking radius vs. diameter correctly, not memorizing.
Learn it
Circle equations
(x − 3)² + (y + 2)² = 25: center (3, −2) after flipping 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, so 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, because graphing circle equations makes them visual.
Practice these skillsOriginal Digital SAT-style questions on what this checkpoint covers.