SAT Math geometry and trigonometry: circles, triangles and radians

What is on the reference sheet, and what you must know yourself

Reference sheet contents and skill list: College Board Assessment Framework for the Digital SAT Suite, checked 6 October 2026.

The key ideas

Right-triangle trigonometry on the SAT is mostly ratios. sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Because the two acute angles of a right triangle add to 90°, the sine of one equals the cosine of the other. Questions like 'if sin x° = cos 32°, what is x' rely on that single fact.

Similar triangles have equal angles and proportional sides. If one triangle is a scale copy of another with scale factor k, every length is multiplied by k, every area by k^2 and every volume by k^3. Angles do not change. Parallel lines cut by a transversal create equal alternate and corresponding angles, which is often how you prove two triangles are similar.

Circles are the most demanding topic. A circle with centre (h, k) and radius r has equation (x - h)^2 + (y - k)^2 = r^2. If the equation is given expanded, complete the square in x and in y to find the centre and radius. Arc length and sector area are fractions of the full circle: an angle of θ degrees gives θ/360 of the circumference or area. In radians, the full circle is 2π, so an angle of θ radians gives arc length rθ.

Area and volume questions often hide the formula inside a context, such as a cylindrical water tank or a cone-shaped cup. Identify the shape, find the formula on the reference sheet, and keep units consistent.

Worked example 1: the centre and radius of a circle

Question: The equation x^2 + y^2 - 6x + 8y - 11 = 0 represents a circle in the xy-plane. What is its radius?

1. Group x terms and y terms: (x^2 - 6x) + (y^2 + 8y) = 11.
2. Complete the square for x: x^2 - 6x = (x - 3)^2 - 9.
3. Complete the square for y: y^2 + 8y = (y + 4)^2 - 16.
4. Substitute: (x - 3)^2 - 9 + (y + 4)^2 - 16 = 11, so (x - 3)^2 + (y + 4)^2 = 36.
5. Answer: centre (3, -4) and radius sqrt(36) = 6. A common wrong answer is 36, the radius squared.

Worked example 2: right-triangle trigonometry

Question: In right triangle ABC, angle C is 90° and sin A = 5/13. What is cos A, and what is cos B?

1. sin A = opposite/hypotenuse, so take the side opposite A as 5 and the hypotenuse as 13.
2. Pythagoras gives the adjacent side: sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12.
3. cos A = adjacent/hypotenuse = 12/13.
4. Angles A and B are complementary (they add to 90°), so cos B = sin A = 5/13.
5. Answer: cos A = 12/13 and cos B = 5/13.

Worked example 3: arc length in degrees and radians

Question: A circle has radius 10. A central angle measures 72°. What is the length of the arc it cuts off, and what is the angle in radians?

1. The arc is 72/360 of the circumference, which is one fifth.
2. Circumference = 2π × 10 = 20π, so the arc length is 20π/5 = 4π (about 12.57).
3. Convert the angle: 72 × π/180 = 2π/5 radians.
4. Check with the radian formula: arc length = rθ = 10 × 2π/5 = 4π. The two methods agree.

Worked example 4: scale factors for area and volume

Question: Triangle PQR is similar to triangle XYZ, and each side of XYZ is 3 times the corresponding side of PQR. The area of PQR is 12. What is the area of XYZ?

1. The length scale factor is k = 3.
2. Areas scale by k^2 = 9.
3. Area of XYZ = 12 × 9 = 108.
4. If these were similar solids, volumes would scale by k^3 = 27. A common mistake is to multiply the area by 3.

Common mistakes that cost marks

• Giving r^2 instead of r for the radius of a circle.
• Sign errors when reading the centre: (x + 2)^2 means h = -2.
• Mixing degrees and radians, or leaving a calculator in the wrong mode when evaluating sine or cosine.
• Scaling areas or volumes by k instead of k^2 or k^3.
• Assuming a figure is drawn to scale when the question does not say so.
• Forgetting that a 30-60-90 triangle's shortest side is opposite the 30° angle.

Exam technique and how a tutor helps

Open the reference sheet once early in the test so you know where it is, but do not rely on it for the facts listed in the right-hand column of the table above. Sketch any figure that is described in words, and label it with the given values; most geometry errors come from picturing the shape wrongly. Desmos can graph a circle equation and show its centre visually, which is a quick check after completing the square.

Because this domain is only 5 to 7 questions, students aiming for the top of the range should make sure circles and radians are secure, since those are the topics least covered in some school courses. Students from British and IB schools usually know trigonometry well but may not have met circle equations or radians as recently as American-curriculum students.

In one-to-one lessons a tutor works on the shared whiteboard with the student, drawing and labelling figures together, which makes it obvious when the student is picturing a problem wrongly. The tutor then sets short timed sets on the student's weakest topics, typically circle equations, radian conversion and similarity, and returns to them until they are reliable.

Self-check: can you do these?

• Find the centre and radius of x^2 + y^2 + 4x - 10y + 20 = 0. (Answer: centre (-2, 5), radius 3)
• Convert 150° to radians. (Answer: 5π/6)
• A 45-45-90 triangle has legs of length 6. How long is the hypotenuse? (Answer: 6 sqrt(2))
• If cos x° = sin 40° and 0 < x < 90, what is x? (Answer: 50)
• Every edge of a cube is doubled. By what factor does the volume increase? (Answer: 8)