In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
If you draw squares on each side of a right triangle, the two smaller squares fill the big one exactly.
Watch a worked example first
Find the hypotenuse: legs 6 and 8
Watch the 6-8-10 right triangle solved — and learn to spot the 3-4-5 shortcut.
Read the transcript
Here's a right triangle. Its two legs measure six and eight, and we want the hypotenuse — the long side across from the right angle.
The Pythagorean theorem connects them: the square of one leg plus the square of the other leg equals the square of the hypotenuse.
So we square each leg. Six squared, and eight squared.
Six squared is thirty-six, eight squared is sixty-four, and together they make one hundred.
Now take the square root of one hundred. The hypotenuse is ten.
Here's the shortcut a strong solver spots: six, eight, ten is just three, four, five doubled. When you recognize a scaled Pythagorean triple, you can write the answer instantly, with no arithmetic.
One trap to avoid: don't stop at c-squared equals one hundred and write c equals one hundred. The theorem gives you c-squared, so you still have to take the square root.
And that's it. The hypotenuse is ten — a clean three, four, five triple in disguise. Spotting these by eye is a real competition skill.
Showing a random 20 of 50 problems.
Example 1
medium
Triangle ABC has A=(0,0), B=(5,0), C=(5,12). Find AC.
Example 2
medium
Find the distance between (−2,1) and (4,9).
Example 3
hard
A telephone pole is 24 ft tall. A support wire is anchored 7 ft from the base. How long is the wire?Find wire length w; 7-24-25 triple.
Example 4
medium
Find the length of a diagonal of a rectangular box with edges 3, 4, and 12.
Example 5
medium
A baseball diamond is a square with sides 90 ft. How far does the catcher (at home) throw to second base (diagonally opposite)?
Example 6
hard
A ladder 10 m long leans against a wall. The foot of the ladder is 6 m from the base of the wall. How high up the wall does the ladder reach?Ladder = hypotenuse; find height h.
Example 7
hard
A boat sails 9 km east then 12 km north. How far is it from its starting point?Find displacement d; 9-12-15 triple.
Example 8
challenge
Two ladders, 20 ft and 30 ft long, lean against opposite walls of an alley, crossing each other. The point where they cross is 8 ft above the ground. How wide is the alley? (Hard version — feel free to set up and observe.)
Example 9
easy
A square has side 6. Find the length of its diagonal.Square diagonal = hypotenuse; find d.
Example 10
medium
A right triangle has legs 9 and 40. Find the hypotenuse.Find hypotenuse c; 9-40-41 triple.
Example 11
medium
Find the perimeter of a right triangle with legs 9 and 12.Find perimeter
Example 12
easy
A right triangle has legs 3 and 4. Find the hypotenuse.Classic 3-4-5 triple: find c.
Example 13
hard
A right triangle has perimeter 30 and hypotenuse 13. Find its legs.
Example 14
medium
Find the distance from (−3,−4) to (0,0).
Example 15
medium
A rectangular field is 30 m by 40 m. Find the length of the diagonal path across it.Find diagonal d across the field.
Example 16
hard
A rectangular room is 12 ft long, 9 ft wide, and 8 ft tall. Find the space-diagonal length corner to opposite corner.
Example 17
easy
A TV screen is described as 16 inches wide and 12 inches tall. Find the diagonal.TV screen: find diagonal d.
Example 18
easy
A right triangle has legs 5 and 12. Find the hypotenuse.Find hypotenuse c; 5-12-13 triple.
Example 19
medium
Find the height of an equilateral triangle with side length 10.Find altitude h; answer is 5√3.
Example 20
easy
A right triangle has legs 1 and 1. Find the hypotenuse in simplest radical form.Find c in simplest radical form.