The Pythagorean Theorem
In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse — the side opposite the right angle.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
The theorem applies only to right triangles. The hypotenuse is always the longest side, opposite the $90°$ angle. Solve for whichever side is missing.
How to solve one, step by step
Example: legs $3$ and $4$. Find the hypotenuse.
- $3^2 + 4^2 = 9 + 16 = 25$.
- Hypotenuse: $\sqrt{25} = 5$.
The mistakes that cost points
- Using it on non-right triangles. The relationship only holds when there's a $90°$ angle.
- Leaving the hypotenuse squared. After $c^2 = 25$, take the square root to get $c = 5$.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A right-angled triangle has a hypotenuse of $20$ feet and one shorter side of $12$ feet. What is the area, in square feet, of this triangle?
- A$96$
- B$48$
- C$240$
- D$192$
Show solution
Answer: A, $96$. STEP 1: Identify the sides. The hypotenuse is $20$ (the longest side, opposite the right angle). One leg is $12$. The other leg is unknown.
STEP 2: Use Pythagoras to find the missing leg. $\text{missing leg}^2 = 20^2 - 12^2 = 400 - 144 = 256$, so missing leg $= \sqrt{256} = 16$.
STEP 3: Compute the area. Area $= \frac{1}{2} \cdot 12 \cdot 16 = 96$ square feet.
Medium
A right-angled triangle has a hypotenuse of $2\sqrt{15}$ inches and one shorter side of $2\sqrt{3}$ inches. What is the area, in square inches, of this triangle?
- A$24$
- B$12$
- C$2\sqrt{15} + 6\sqrt{3}$
- D$12\sqrt{5}$
Show solution
Answer: B, $12$. STEP 1: Identify the sides. The hypotenuse is $2\sqrt{15}$. One leg is $2\sqrt{3}$. The other leg is unknown.
STEP 2: Use Pythagoras. $\text{missing leg}^2 = (2\sqrt{15})^2 - (2\sqrt{3})^2 = 60 - 12 = 48$, so missing leg $= \sqrt{48} = 4\sqrt{3}$.
STEP 3: Compute the area. Area $= \frac{1}{2} \cdot 2\sqrt{3} \cdot 4\sqrt{3} = \frac{1}{2} \cdot 8 \cdot 3 = 12$ square inches.
Hard
A right-angled triangle has a hypotenuse of $A$ inches and one shorter side of $15$ inches. Which expression represents the area, in square inches, of this triangle?
- A$15\sqrt{A^2 - 225}$
- B$A + 15 + \sqrt{A^2 - 225}$
- C$\frac{15}{2}\sqrt{A^2 - 225}$
- D$15A$
Show solution
Answer: C, $\frac{15}{2}\sqrt{A^2 - 225}$. STEP 1: Identify the sides. The hypotenuse is $A$. One leg is $15$. The other leg is unknown.
STEP 2: Use Pythagoras. $\text{missing leg}^2 = A^2 - 15^2 = A^2 - 225$, so missing leg $= \sqrt{A^2 - 225}$.
STEP 3: Compute the area. Area $= \frac{1}{2} \cdot 15 \cdot \sqrt{A^2 - 225} = \frac{15}{2}\sqrt{A^2 - 225}$ square inches.
Find your exact gaps
Take a free, full-length practice test and see precisely which question types trip you up.
Practice this for freeNo credit card. Register once, take as many tests as you like.