SAT Math Geometry and Trigonometry

Area, volume, Pythagorean theorem, special triangles, and trig basics

Geometry looks intimidating until you realize most SAT problems are recycled formulas: area, perimeter, volume, the Pythagorean theorem, two special triangles ( $30$ - $60$ - $90$ and $45$ - $45$ - $90$ ), and basic trigonometry. Expect 4–6 questions per section.

SAT trigonometry is minimal — sine, cosine, tangent in right triangles. You don't need identities, radians, or trig function graphs in any depth. Far less than a typical Polish high school covers, but you need the basics solid.

What the SAT actually tests

Areas: triangle ( $\frac{1}{2} bh$ ), rectangle ( $l w$ ), trapezoid ( $\frac{a + b}{2} \cdot h$ ), circle ( $π r^{2}$ )
Volumes: rectangular prism ( $l w h$ ), cylinder ( $π r^{2} h$ ), sphere ( $\frac{4}{3} π r^{3}$ )
Pythagorean theorem and triples ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ )
Special triangles: $30$ - $60$ - $90$ and $45$ - $45$ - $90$ side ratios
Sine, cosine, tangent in right triangles
Inscribed and central angles, tangent segments to circles

Key concepts

Pythagorean theorem

$a^{2} + b^{2} = c^{2}$ , where $c$ is the hypotenuse. Memorize the triples: $(3, 4, 5)$ , $(5, 12, 13)$ , $(8, 15, 17)$ .

45-45-90 triangle

Side ratio $1 : 1 : 2$ . If a leg is 1, the hypotenuse is $2$ . Isosceles right triangle.

30-60-90 triangle

Side ratio $1 : 3 : 2$ . Shortest side opposite 30°, longer leg opposite 60°, hypotenuse opposite 90°.

SOH-CAH-TOA

$sin = \frac{opposite}{hypotenuse}$ , $cos = \frac{adjacent}{hypotenuse}$ , $tan = \frac{opposite}{adjacent}$ .

Worked examples

Example 1

In a right triangle, one leg is 6 and the hypotenuse is 10. What is the other leg?

Solution

Pythagoras: $a^{2} + 36 = 100$ , so $a^{2} = 64$ , $a = 8$ . (Recognize $6$ - $8$ - $10$ = doubled $3$ - $4$ - $5$ .)

💡 Recognizing Pythagorean triples saves work. Learn: $(3, 4, 5)$ , $(5, 12, 13)$ , $(8, 15, 17)$ .

Example 2

In a right triangle, one acute angle is 30° and the hypotenuse is 12. What is the shorter leg?

Solution

$30$ - $60$ - $90$ triangle. Shorter leg (opposite 30°) is half the hypotenuse: $6$ .

💡 In a $30$ - $60$ - $90$ , the shorter leg = hypotenuse / 2.

Example 3

In a right triangle, $sin A = \frac{3}{5}$ . What is $cos A$ ?

Solution

$sin A = \frac{3}{5}$ means opposite = 3, hypotenuse = 5. Pythagoras: adjacent = $25 - 9 = 4$ . So $cos A = \frac{4}{5}$ .

💡 Given $sin = \frac{a}{c}$ , you have a $3$ - $4$ - $5$ triangle — find the remaining side with Pythagoras.

Common pitfalls

Mixing up area formulas. Triangle is $\frac{1}{2} bh$ , NOT $bh$ . Trapezoid is $\frac{a + b}{2} \cdot h$ .
Computing a leg instead of the hypotenuse (or vice versa) in Pythagoras. Hypotenuse is ALWAYS the longest side.
Dropping units or mixing them (cm vs m, degrees vs radians).
Trig: confusing sin with cos. SOH-CAH-TOA only helps once you identify which side is adjacent to the angle.

Exam strategy

Memorize the formulas — area, volume, Pythagoras, the two special triangles, SOH-CAH-TOA. These cover 90% of SAT geometry. When you see a triangle, ask first: is it right? If yes — Pythagoras or trig. If you see a 30°, 45°, or 60° angle, that's the SAT signaling a special triangle. For circles, memorize: inscribed angle = half the central angle subtending the same arc.

Frequently asked questions

What geometry formulas do I need for the SAT?

Areas: triangle (½bh), rectangle (lw), trapezoid ((a+b)/2 · h), circle (πr²). Volumes: rectangular prism (lwh), cylinder (πr²h), sphere (4/3 πr³). Pythagorean theorem. Side ratios for 30-60-90 (1:√3:2) and 45-45-90 (1:1:√2). SOH-CAH-TOA for trig.

Do I need to know radians for the SAT?

Yes, but only basic. Mostly converting between degrees and radians. The SAT does not test advanced trig identities.

Which Pythagorean triples should I memorize?

Three core triples: (3, 4, 5), (5, 12, 13), (8, 15, 17). Plus their multiples like (6, 8, 10). Recognition saves arithmetic.

Does the SAT test solid geometry?

Yes, but simply. Mostly volume and surface area of prisms, cylinders, and spheres. No complex cross-sections or pyramids.

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