Using the Pythagorean Theorem
In a right triangle, the two short sides decide the long side. The hard part is rarely the formula - it's spotting the right triangle hiding inside the problem.
Know these first
This unlocks
What you'll be able to do
- Find a missing side of a right triangle, with exact and rounded answers.
- Decide whether a triangle is a right triangle from its side lengths or coordinates.
- Find the hidden right triangle in a real situation - poles, tents, ladders, ramps.
- Break a harder figure into right triangles and use the theorem more than once.
The big idea
A right triangle has two legs that meet at the 90° corner and a hypotenuse across from it - always the longest side. Build a square on each side: the two small squares together have exactly the area of the big one. a² + b² = c².
Easy problems hand you the right triangle. Harder ones hide it. A cable between two poles of different heights, the height of a tent, a ladder sliding down a wall: in each one you have to draw the right triangle yourself before the formula can help.
The theorem works backward too. If a² + b² = c² for a triangle's sides, the triangle must have a right angle. That's how builders check that a corner is square.
Words you need
- Legs
- The two sides that form the right angle.
- Hypotenuse
- The side across from the right angle - the longest side.
- Converse
- The theorem run backward: if a² + b² = c², the triangle is a right triangle.
- Simplified radical
- A square root with no perfect-square factor left inside, like 6√2 instead of √72.
- Altitude
- A segment from a corner that meets the opposite side at 90°.
Worked examples
Example 1
The legs of a right triangle are 6 and 8. Find the hypotenuse.
- Square each leg: 6² = 36 and 8² = 64.
- Add: 36 + 64 = 100.
- Take the square root: c = √100 = 10.
Answer: 10
Example 2
The legs are 5 and 10. Find the hypotenuse exactly, then to two decimal places.
- c² = 5² + 10² = 25 + 100 = 125.
- 125 = 25 × 5, and √25 = 5, so √125 = 5√5.
- 5√5 ≈ 11.18.
Answer: 5√5 ≈ 11.18
Example 3
Two poles stand 15 m apart. One is 4 m tall and the other is 12 m tall. How long is a cable stretched between their tops?
- Draw a level line from the top of the short pole to the tall pole. That makes a right triangle with the cable as its hypotenuse.
- One leg is the gap: 15 m. The other is the difference in heights: 12 − 4 = 8 m.
- Cable² = 15² + 8² = 225 + 64 = 289, so the cable is √289 = 17 m.
Answer: 17
Common mistakes
Adding the hypotenuse in. c is always the side across from the right angle. Only the two legs get added.
Assuming every triangle is a right triangle. a² + b² = c² only holds with a 90° angle. If you're not told, test it with the converse first.
Using the whole base for an isosceles triangle's height. The altitude splits the base in half. Use half the base as the leg.
Using the pole heights as legs. In a two-pole problem, the vertical leg is the difference in heights, not either height.
Practice: 20 questions
13 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Find the hypotenuse.
- Find the missing leg.
- A 17-foot ladder leans against a wall with its foot 8 feet from the wall. How high up the wall does it reach? Word problem
- A rectangular field is 40 m long and 30 m wide. How long is the diagonal path across it? Word problem
- A drone flies 120 m east, then 50 m north. How far is it from where it started? Word problem
Tier 2 Intermediate One more step to think through
- The legs are both 6. Find the hypotenuse in simplified radical form, then to two decimal places.
- Triangle ABC has vertices A(0, 0), B(3, 4) and C(7, 1). Is it a right triangle? If so, where is the right angle?
- A triangle has sides 9, 40 and 41. Is it a right triangle?
- A TV is advertised as 50 inches - that's its diagonal. The screen is 40 inches wide. How tall is it? Word problem
- A square courtyard has 20 m sides. A path runs straight from one corner to the opposite corner. Give its length in simplified radical form and to two decimal places. Word problem
- A ramp rises 2 feet over a level distance of 15 feet. How long is the ramp, to the nearest tenth of a foot? Word problem
- A carpenter builds a frame with sides 20, 48 and 52 inches. Is the corner between the 20-inch and 48-inch sides square? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- An isosceles triangle has equal sides of 13 and a base of 10. Find its height.
- Two poles stand 15 feet apart. One is 6 feet tall and the other is 14 feet tall. How long is a rope tied between their tops? Word problem
- A tent's front is an isosceles triangle with slanted sides of 10 feet and a base of 12 feet. What is the area of the tent's front? Word problem
- A 10-foot ladder leans on a wall with its foot 6 feet out. The top slides 2 feet down the wall. How far does the foot of the ladder move out? Word problem
- A storage box is 3 feet long, 4 feet wide and 12 feet tall. What is the longest straight pole that fits inside, corner to opposite corner? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Triangle ABC has AB = 13, AC = 20 and BC = 21. The altitude AD meets BC at D. Find BD and the altitude AD.
- A rectangular garden's diagonal is 29 m. Its length is 1 m more than its width. Find the length and width. Word problem Above grade level
- A round garden of radius 5 m sits in the middle of a circular plaza of radius 13 m with the same center. A straight path crosses the plaza and just touches the garden's edge. How long is the path? Word problem Above grade level
Answers
Tier 1 Foundations
- 15. c² = 9² + 12² = 225, so c = 15.
- 24. b² = 25² − 7² = 576, so b = 24.
- 15. b² = 17² − 8² = 225, so b = 15.
- 50. c² = 40² + 30² = 2500, so c = 50.
- 130. c² = 120² + 50² = 16900, so c = 130.
Tier 2 Intermediate
- 6√2 ≈ 8.49. c² = 6² + 6² = 72, so c = 6√2 ≈ 8.49.
- Yes - right angle at B. AB² = 25, BC² = 25, AC² = 50. The two shorter add to the longest (25 + 25 = 50), so it's a right triangle.
- Yes, it is a right triangle. Check the longest side: 9² + 40² = 1681 and 41² = 1681, which match.
- 30. b² = 50² − 40² = 900, so b = 30.
- 20√2 ≈ 28.28. c² = 20² + 20² = 800, so c = 20√2 ≈ 28.28.
- √229 ≈ 15.1. c² = 2² + 15² = 229, so c = √229 ≈ 15.1.
- Yes, it is a right triangle. Check the longest side: 20² + 48² = 2704 and 52² = 2704, which match.
Tier 3 Advanced
- 12. The altitude cuts the base in half (5). h² = 13² − 5² = 144, so h = 12.
- 17. The right triangle has legs 15 (the gap) and 8 (the difference in heights): 15² + 8² = 289, so the cable is 17.
- 48. Height: h² = 10² − 6² gives h = 8. Area = ½ × 12 × h = 48.
- 2. Start: height = √(10² − 6²) = 8. After sliding, height = 6 and the base = √(10² − height²) = 8. The foot moves out 2.
- 13. d² = 3² + 4² + 12² = 169, so d = 13.
Tier 4 Challenge
- BD = 5, AD = 12. Both right triangles share AD: 13² − x² = 20² − (21 − x)². Solving gives x = 5, then AD² = 13² − 5², so AD = 12.
- The legs are 20 and 21. x² + (x + 1)² = 29² simplifies to 2x² + 2x − 840 = 0. The positive solution is x = 20, so the legs are 20 and 21.
- 24. The small radius meets the chord at 90° at its midpoint. Half-chord² = 13² − 5² = 144, so half the chord is 12 and the chord is 24.
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