SAT Math — Geometry and Trigonometry

Chapter Overview & Weightage

Geometry and Trigonometry is one of the four content domains on the SAT Math section. It consistently accounts for roughly 15% of all Math questions — about 5–7 questions out of the 44 total.

The SAT provides a reference sheet with key formulas at the start of the Math section — including area formulas for circles, triangles, rectangles, and 3D shapes. You do not need to memorise these, but you must know when and how to apply them quickly. Trig ratios are not provided — those you must know.

Sub-topic	Approx. questions	Notes
Area and perimeter	1–2	Often embedded in word problems
Triangles (congruence, similarity)	1–2	Ratio and proportional reasoning
Circles	1–2	Arc length, sector area, equation of circle
Trigonometry	1–2	SOH-CAH-TOA, co-function identity
3D geometry	0–1	Volume of common solids

Key Concepts You Must Know

Triangles:

Sum of angles = 180°
Exterior angle = sum of two non-adjacent interior angles
Similar triangles: corresponding sides are proportional
Special right triangles: 30-60-90 (sides $1 : \sqrt{3} : 2$ ) and 45-45-90 (sides $1 : 1 : \sqrt{2}$ )
Pythagorean theorem: $a^2 + b^2 = c^2$

Circles:

Circumference $= 2\pi r$ , Area $= \pi r^2$
Arc length $= \frac{\theta}{360} \times 2\pi r$ (degrees) or $= r\theta$ (radians)
Sector area $= \frac{\theta}{360} \times \pi r^2$
Standard equation: $(x-h)^2 + (y-k)^2 = r^2$

Trigonometry:

SOH-CAH-TOA in right triangles
$\sin\theta = \cos(90° - \theta)$ — co-function identity
The SAT loves questions where you must find a trig ratio from given side lengths, not a calculator value

Important Formulas

\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}

When to use: Any right triangle problem with angles and sides.

\sin\theta = \cos(90° - \theta) \quad \text{and} \quad \cos\theta = \sin(90° - \theta)

When to use: SAT frequently gives sin of one angle and asks for cos of its complement — they’re equal.

(x-h)^2 + (y-k)^2 = r^2

Centre is $(h, k)$ , radius is $r$ . When to use: Coordinate geometry questions involving circles.

Solved Previous Year Questions

SAT PYQ 1 — Co-function identity

Q: In a right triangle, $\sin(x°) = \frac{4}{5}$ . What is $\cos(90° - x°)$ ?

Solution: By the co-function identity, $\cos(90° - x°) = \sin(x°) = \mathbf{\frac{4}{5}}$ .

This is a one-line answer. The SAT often makes it this clean — the trap is students over-thinking it and trying to find the angle.

SAT PYQ 2 — Circle equation

Q: A circle in the xy-plane has centre $(3, -2)$ and passes through $(7, -2)$ . What is the equation of the circle?

Solution: Radius $= |7 - 3| = 4$ (same y-coordinate, so horizontal distance).

(x - 3)^2 + (y + 2)^2 = 16

SAT PYQ 3 — Similar triangles

Q: In the figure, triangles ABC and DEF are similar. AB = 6, BC = 8, DE = 9. Find EF.

Solution: Corresponding sides are proportional:

\frac{DE}{AB} = \frac{EF}{BC} \implies \frac{9}{6} = \frac{EF}{8} \implies EF = \frac{9 \times 8}{6} = 12

Difficulty Distribution

Difficulty	Approx. share	What to expect
Easy	40%	Direct formula application
Medium	45%	Multi-step or multi-concept
Hard	15%	Unusual setup, algebraic reasoning required

Expert Strategy

SAT geometry rewards students who can draw the figure even when none is given. If the problem mentions a triangle or circle, sketch it immediately, label everything from the problem, and write down what you need to find. This alone prevents about half the errors.

Don’t reach for a calculator on trig questions first. SAT geometry trig questions almost always work out to “nice” values — fractions or small integers. If your calculation gives a messy decimal, re-read the question.

For circle equation questions, complete the square if the equation is given in general form ( $x^2 + y^2 + Dx + Ey + F = 0$ ). This is a required skill for harder SAT questions.

Use the reference sheet strategically — if you blank on a formula mid-exam, it’s there for you. But using it costs 15–20 seconds per lookup. Know the most-used formulas cold (triangle area, Pythagoras, trig ratios).

Common Traps

Trap 1: Using degrees in arc length/sector area when radians are needed. The SAT uses both — read carefully.

Trap 2: Forgetting that similar triangles require corresponding vertices in the right order. If triangle ABC ~ triangle DEF, then A corresponds to D, B to E, C to F. Mixing up the correspondence gives wrong ratios.

Trap 3: Confusing diameter and radius. If a question says a circle has diameter 10, $r = 5$ . Plugging in 10 where $r$ belongs is the #1 circle error.

Trap 4: Assuming a 45-45-90 or 30-60-90 triangle without checking. These special triangles must be explicitly identified or derivable from angle information given — never assume.