SAT Math · Geometry and Trigonometry

The Pythagorean theorem

Q: How do you find the hypotenuse?

Square both legs, add them, and take the square root: c = √(a² + b²). For legs 6 and 8, c = √(36 + 64) = √100 = 10.

Q: How do you find a missing leg?

Subtract the square of the known leg from the square of the hypotenuse, then take the square root: a = √(c² − b²). With hypotenuse 13 and leg 5, the other leg is √144 = 12.

Q: What are Pythagorean triples?

Pythagorean triples are whole-number side lengths that fit a² + b² = c², such as 3, 4, 5 and 5, 12, 13. Any multiple of a triple, like 6, 8, 10, also works.

Q: Is the Pythagorean theorem on the SAT reference sheet?

Yes. The SAT reference sheet in Bluebook gives a² + b² = c², along with the side ratios for 45-45-90 and 30-60-90 special right triangles.

Q: How can Aristotle help with SAT geometry?

Aristotle is a voice AI SAT tutor. Your child talks through a geometry question out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including right triangles and trigonometry.

a² + b² = c² for any right triangle, plus the triples and special triangles that save time on SAT Math.

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Updated October 30, 2026

$a^2 + b^2 = c^2$

The two leg squares add up to the hypotenuse square: $9 + 16 = 25$ , so $c = 5$ .

What is the Pythagorean theorem?

a^2 + b^2 = c^2

In any right triangle the three sides are locked together: square the two legs, add them, and you get the square of the hypotenuse. The hypotenuse is the side opposite the right angle, and it is always the longest.

Legs a and b meet at the right angle; c is across from it.

It is literally about squares

9 + 16 = 25

Draw a real square on each side. The two leg squares, 9 and 16, together have exactly the area of the hypotenuse square, 25. That is why the formula squares each side before adding, and why a square root comes at the end.

Area 9 plus area 16 equals area 25.

Any distance on a grid is a hypotenuse

d = \sqrt{9^2 + 12^2} = 15

Between any two points on a grid hides a right triangle: the horizontal gap and the vertical gap are its legs, and the straight-line distance is its hypotenuse. The distance formula is the Pythagorean theorem wearing coordinates.

Across 9, up 12, so the straight line AB is 15.

Why the squares add up

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
Legs 3 and 4. What are the areas of the squares on them?
Answer
9 and 16.
Question 2
The square on the hypotenuse has area 25. What do you notice?
Answer
$9 + 16 = 25$ . The leg squares add up to the hypotenuse square, so $c = 5$ .
Question 3
Points $(1, 2)$ and $(5, 5)$ . Where is the right triangle?
Answer
The gaps 4 across and 3 up are the legs, so the distance is 5.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Question 2

The hypotenuse of a right triangle is 13 and one leg is 5. How long is the other leg?

Question 3

A rectangle is 6 units long and 3 units wide. What is the length of its diagonal?

Question 4

What is the distance between the points $(1, 2)$ and $(5, 5)$ ?

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The formulas

Pythagorean theorem

$a^2 + b^2 = c^2$

$c$ is always the hypotenuse, the side across from the right angle.

Finding a leg

$a = \sqrt{c^2 - b^2}$

Subtract, do not add, when the missing side is a leg.

45-45-90 triangle

$x,\ x,\ x\sqrt{2}$

An isosceles right triangle. The hypotenuse is $\sqrt{2}$ times a leg. On the reference sheet.

30-60-90 triangle

$x,\ x\sqrt{3},\ 2x$

The short leg is across from $30^\circ$ , and the hypotenuse is twice it. On the reference sheet.

Pythagorean triples worth knowing
Legs	Hypotenuse	Check
3, 4	5	$9 + 16 = 25$
5, 12	13	$25 + 144 = 169$
8, 15	17	$64 + 225 = 289$
7, 24	25	$49 + 576 = 625$
6, 8	10	3, 4, 5 doubled

Try it

Build a right triangle

Set the legs and get the hypotenuse, exact.

Leg a3

Leg b4

a² + b²3² + 4² = 9 + 16 = 25

Hypotenusec = √25 = 5

3, 4, 5 is a Pythagorean triple: all three sides are whole numbers.

Common mistakes

Adding when the missing side is a leg

A missing leg is $\sqrt{c^2 - b^2}$ . Adding gives a leg longer than the hypotenuse.

Calling the wrong side c

$c$ is always the side across from the right angle.

Using it on a triangle with no right angle

It only works for right triangles. Draw a height to make one.

Forgetting to take the square root

$a^2 + b^2$ gives $c^2$ , not $c$ . For legs 6 and 8 the hypotenuse is 10, not 100.

How it shows up on the SAT

Geometry and Trigonometry is about 15% of SAT Math. Expect a missing side, a triple like 5, 12, 13, or a distance on a graph.

What Aristotle tracks in right triangles and trigonometry

5 SAT Math skills, including:

Using the Pythagorean theorem in context
Solving problems with special right triangles
Solving problems with right triangle trigonometry
Using similarity to find sine, cosine, and tangent values

What is the Pythagorean theorem?

In any right triangle, the squares of the two legs add up to the square of the hypotenuse: a² + b² = c². The hypotenuse c is the longest side, across from the right angle.

How do you find the hypotenuse?

How do you find a missing leg?

What are Pythagorean triples?

Is the Pythagorean theorem on the SAT reference sheet?

How can Aristotle help with SAT geometry?

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