Digital SAT Geometry and Trigonometry: a complete guide to area, angles, right triangles and circles

What Geometry and Trigonometry demands

Geometry and Trigonometry is about 15% of the Digital SAT Math section. It is formula-driven and high-yield, because the reference sheet supplies the measurement formulas and the questions reward recognising the right shape and relationship. This guide ties together the matching dot-point pages, each with its own practice: area and volume, lines, angles and triangles, right triangles and trigonometry, circles, and coordinate geometry and circle equations.

Area and volume

The reference sheet gives the area of a rectangle ($\ell w$), triangle ($\frac{1}{2}bh$), and circle ($\pi r^2$), and volumes of a prism ($\ell w h$), cylinder ($\pi r^2 h$), sphere ($\frac{4}{3}\pi r^3$), cone ($\frac{1}{3}\pi r^2 h$), and pyramid ($\frac{1}{3}\ell w h$). For composite figures, decompose into known shapes and add or subtract. When all dimensions scale by $k$, area scales by $k^2$ and volume by $k^3$.

Lines, angles and triangles

Vertical angles are equal, complementary sum to $90^\circ$, supplementary sum to $180^\circ$. A transversal across parallel lines makes corresponding and alternate angles equal and co-interior angles supplementary. A triangle's angles sum to $180^\circ$. Similar triangles have equal angles and proportional sides, which turns many problems into a proportion.

Right triangles and trigonometry

The Pythagorean theorem $a^2 + b^2 = c^2$ and the common triples ($3$-$4$-$5$, $5$-$12$-$13$) find missing sides. The special right triangles ($45$-$45$-$90$: $x : x : x\sqrt{2}$; $30$-$60$-$90$: $x : x\sqrt{3} : 2x$) give exact sides from one. The trig ratios are SOH-CAH-TOA (not on the reference sheet), and the two acute angles are complementary, so $\sin\theta = \cos(90^\circ - \theta)$.

Circles and radians

A circle has circumference $2\pi r$ and area $\pi r^2$. An arc is $\frac{\theta}{360}$ of the circumference and a sector is $\frac{\theta}{360}$ of the area. Convert angles with $360^\circ = 2\pi$ radians (degrees $\times \frac{\pi}{180}$). In the coordinate plane, a circle is $(x - h)^2 + (y - k)^2 = r^2$ with center $(h, k)$ and radius $r$; complete the square to recover them from an expanded equation.

How Geometry and Trigonometry is examined

• Area and volume. Standard formulas, composite figures, scaling by $k^2$ and $k^3$.
• Lines and angles. Vertical, complementary, supplementary; parallel lines and transversals; triangle angle sum; similar triangles.
• Right triangles and trig. Pythagorean theorem, special triangles, SOH-CAH-TOA, complementary sine and cosine.
• Circles. Circumference, area, arcs and sectors as fractions, radians.
• Coordinate geometry. Distance and midpoint; circle equations and completing the square.

Check your knowledge

Work these under timed conditions, then read the solutions.

1. A cylinder has radius 5 and height 8. Find its volume in terms of $\pi$. (2 marks)
2. Two angles are complementary; one is $35^\circ$. What is the other? (1 mark)
3. In a right triangle, the leg opposite $\theta$ is 8 and the hypotenuse is 17. Find $\sin\theta$. (2 marks)
4. A circle of radius 9 has a sector with central angle $40^\circ$. Find the sector area in terms of $\pi$. (2 marks)
5. Find the center and radius of $(x + 1)^2 + (y - 4)^2 = 49$. (2 marks)