Geometry & Trig · SAT Math

Right Triangles on the Digital SAT

Right triangles are the single most-tested geometry topic on the Digital SAT. Master the Pythagorean theorem, the two special right triangles (30-60-90 and 45-45-90), the common Pythagorean triples, and basic SOHCAHTOA — and you'll handle every right-triangle question on the test in under 90 seconds. 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~11 min read

The Pythagorean theorem

In any right triangle, the squares of the two legs sum to the square of the hypotenuse:

a² + b² = c²

a, b are the two legs · c is the hypotenuse (the side across from the right angle)

The hypotenuse is always the longest side and always opposite the 90° angle. Identify it first, then label the legs.

Example. A right triangle has legs of 5 and 12. What is the hypotenuse?

c² = 5² + 12² = 25 + 144 = 169. So c = √169 = 13.

Memorize the Pythagorean triples

The SAT recycles a small set of right-triangle side combinations with integer sides. Memorize these four — they appear constantly:

3 - 4 - 5 (and multiples: 6-8-10, 9-12-15)

5 - 12 - 13

8 - 15 - 17

7 - 24 - 25

When you spot two of these numbers in a problem, the third is instant — no Pythagorean calculation needed.

Example. A right triangle has legs 9 and 12. The hypotenuse is 15 (recognize this as 3-4-5 scaled by 3).

Special right triangles: 45-45-90 and 30-60-90

Two triangle types appear on the SAT so often that they get their own side-ratio formulas — memorize them:

45-45-90 triangle. Sides in ratio 1 : 1 : √2. If each leg is a, the hypotenuse is a√2.

30-60-90 triangle. Sides in ratio 1 : √3 : 2. The side opposite 30° is x, opposite 60° is x√3, and the hypotenuse (opposite 90°) is 2x.

Example. A 45-45-90 triangle has hypotenuse 10. What is the leg?

a · √2 = 10 → a = 10/√2 = 5√2 ≈ 7.07.

SOHCAHTOA: basic trigonometry

For any angle θ in a right triangle (other than the 90°), the three basic trig ratios are:

sin(θ) = Opposite / Hypotenuse (SOH)

cos(θ) = Adjacent / Hypotenuse (CAH)

tan(θ) = Opposite / Adjacent (TOA)

"Opposite" means the leg across from the angle. "Adjacent" means the leg next to it (not the hypotenuse).

Key SAT identity: in a right triangle, sin(θ) = cos(90° - θ). The sine of one acute angle equals the cosine of the other. The SAT loves this one.

Word problems and applications

Right-triangle word problems usually involve ladders, ramps, shadows, towers, or navigation. The setup is always the same:

  1. Draw the picture — a right triangle.
  2. Label what's given (sides, angles) and what's asked.
  3. Decide which tool: Pythagorean (no angles), special triangle (30/45/60), or SOHCAHTOA (other angles).
  4. Solve.

The distance formula between two points (x₁, y₁) and (x₂, y₂) is the Pythagorean theorem in disguise: d = √((x₂-x₁)² + (y₂-y₁)²). The horizontal and vertical differences are the legs.

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Solving with Desmos

Desmos isn't as central for right triangles as for algebra, but it still saves time on numeric work. Three techniques:

1. Compute sides with the Pythagorean theorem

For "hypotenuse if legs are 9 and 40": type sqrt(9² + 40²). Desmos returns 41 (recognize it as a Pythagorean triple).

2. Switch to degrees for trig (critical)

The SAT uses degrees by default but Desmos uses radians. Open graph settings (the wrench icon), set angle mode to Degrees. Otherwise sin(30) returns the wrong value.

3. Inverse trig for finding angles

If you know the side lengths and need the angle: type arctan(opposite / adjacent) (or arcsin / arccos depending on which sides you have).

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Calling the wrong side the hypotenuse

The hypotenuse is always opposite the right angle — usually the longest side. Mislabeling it makes a² + b² = c² fail.

Forgetting Pythagorean triples

If you see 5 and 13 as two sides, the third is 12 (5-12-13 triple). If you see 8 and 17, the third is 15. Memorizing these saves serious time.

Confusing 30-60-90 ratios with 45-45-90

45-45-90 is 1 : 1 : √2. 30-60-90 is 1 : √3 : 2. Swapping them produces wildly wrong answers. The 30-60-90 has all DIFFERENT sides; 45-45-90 has two EQUAL legs.

Forgetting Desmos defaults to radians

If a problem says 'find sin(30°)' and you type sin(30) without switching Desmos to degrees, you get -0.988, not 0.5. Toggle degrees in Desmos settings before any trig question.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

  1. 13
  2. 15
  3. 18
  4. 21

Click to reveal solution →

Answer: (B) 15

Recognize 9-12-15 as 3-4-5 × 3. Hypotenuse = 15.

Or compute: c² = 9² + 12² = 81 + 144 = 225 → c = 15.

2

A 45-45-90 right triangle has legs of length 6. What is the length of the hypotenuse?

  1. 6
  2. 6√2
  3. 12
  4. 12√2

Click to reveal solution →

Answer: (B) 6√2

In a 45-45-90 triangle, hypotenuse = leg · √2. So 6√2 ≈ 8.49.

3

In a 30-60-90 triangle, the side opposite the 30° angle is 5. What is the side opposite the 60° angle?

  1. 5√2
  2. 5√3
  3. 10
  4. 10√3

Click to reveal solution →

Answer: (B) 5√3

Ratio is 1 : √3 : 2 (opposite 30° : opposite 60° : opposite 90°). With 1 → 5, the side opposite 60° is 5 · √3 = 5√3.

4

If a right triangle has one leg of length 7 and a hypotenuse of length 25, what is the length of the other leg?

  1. 18
  2. 20
  3. 24
  4. 26

Click to reveal solution →

Answer: (C) 24

Recognize 7-24-25 Pythagorean triple. Other leg = 24.

Or compute: 7² + b² = 25² → 49 + b² = 625 → b² = 576 → b = 24.

5

A right triangle has an angle of 30° and the side opposite that angle is 4. What is the length of the hypotenuse?

  1. 4
  2. 4√3
  3. 8
  4. 8√3

Click to reveal solution →

Answer: (C) 8

30-60-90 ratio: side opposite 30° : hypotenuse = 1 : 2. So hypotenuse = 2 · 4 = 8.

6

In right triangle ABC, angle C = 90°, sin(A) = 3/5. What is cos(A)?

  1. 3/5
  2. 4/5
  3. 5/3
  4. 5/4

Click to reveal solution →

Answer: (B) 4/5

If sin(A) = 3/5, then opposite = 3 and hypotenuse = 5. That's a 3-4-5 triangle, so the adjacent side is 4.

cos(A) = adjacent / hypotenuse = 4/5.

7

A 25-foot ladder leans against a vertical wall. If the bottom of the ladder is 7 feet from the wall, how high up the wall does the ladder reach?

  1. 18 ft
  2. 20 ft
  3. 22 ft
  4. 24 ft

Click to reveal solution →

Answer: (D) 24 ft

Ladder = hypotenuse (25), distance from wall = leg (7), height up wall = other leg.

Recognize 7-24-25 triple → height = 24 ft.

8

In a right triangle, sin(θ) = 5/13. What is tan(θ)?

  1. 5/12
  2. 12/5
  3. 5/13
  4. 13/5

Click to reveal solution →

Answer: (A) 5/12

sin(θ) = opposite/hypotenuse = 5/13. Recognize 5-12-13 triple — adjacent side is 12.

tan(θ) = opposite/adjacent = 5/12.

Frequently asked questions

What is the Pythagorean theorem?

+
In a right triangle, a² + b² = c², where a and b are the lengths of the two legs and c is the length of the hypotenuse (the side opposite the right angle). It only works in right triangles.

What are the most common Pythagorean triples on the SAT?

+
3-4-5, 5-12-13, 8-15-17, and 7-24-25 — and any multiples (like 6-8-10 or 9-12-15). Memorize these. If you spot two of the numbers in a problem, the third is instant.

What are the special right triangles?

+
Two special right triangles appear constantly. The 45-45-90 has sides in ratio 1 : 1 : √2. The 30-60-90 has sides in ratio 1 : √3 : 2 (opposite 30°, 60°, 90° respectively).

What is SOHCAHTOA?

+
A mnemonic for the three basic trig ratios: sin(θ) = Opposite/Hypotenuse, cos(θ) = Adjacent/Hypotenuse, tan(θ) = Opposite/Adjacent. The 'opposite' side is across from the angle; the 'adjacent' side is next to it (not the hypotenuse).

Why is sin(θ) = cos(90° − θ)?

+
Because the two acute angles of a right triangle are complementary (they add to 90°). What's the opposite side for one angle is the adjacent side for the other — so their sin and cos values swap. The SAT loves to test this identity.

Keep going

All SAT Math topics

Browse every Digital SAT math topic

Quadratics

Another high-frequency advanced math topic

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