Matematica

Pythagorean Theorem: Solved Exercises From Easy to Hard

What the Pythagorean theorem is and how to use it to find the hypotenuse or a leg, step by step, with five solved exercises of increasing difficulty and verified answers.

Recommended for: Grade 8 · Grade 9

The Pythagorean theorem states that in a right triangle the square on the hypotenuse equals the sum of the squares on the two legs: c² = a² + b². From this you get the hypotenuse with c = √(a² + b²) and a leg with a = √(c² − other leg²).

Hypotenuse and legs: which is which

In a right triangle the hypotenuse is the side opposite the right angle and is always the longest side. The other two sides, the ones that form the 90° angle, are called the legs. The theorem ties these three sides together with a single relationship between their squares.

The two typical questions

Almost every exercise asks one of two things. If you know both legs and want the hypotenuse, add their squares and take the square root. If instead you know the hypotenuse and one leg and want the other leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root.

How to work without mistakes

The method is always the same. First identify the right angle and the hypotenuse. Then decide whether you are looking for the hypotenuse (add) or a leg (subtract). Finally square the sides, add or subtract, and take the square root.

In the exercises below you will find the same pattern at increasing difficulty: it starts with whole-number legs, moves to a real ladder problem and a radical to simplify, and ends with a two-step exercise involving the hypotenuse and the perimeter. Every answer is verified by redoing the calculations.

Solved exercises

1. In a right triangle the two legs measure 6 cm and 8 cm. How long is the hypotenuse? base

Show solution
  1. The hypotenuse is the side opposite the right angle: find it with c² = a² + b².
  2. Square the legs: 6² = 36 and 8² = 64.
  3. Add them: 36 + 64 = 100.
  4. The hypotenuse is the square root: c = √100 = 10.

Answer: c = 10 cm

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

Show solution
  1. To find a leg, subtract: b² = c² − a².
  2. Square the known sides: 13² = 169 and 5² = 25.
  3. Subtract: 169 − 25 = 144.
  4. The leg is the square root: b = √144 = 12.

Answer: b = 12 cm

3. A ladder leans against a wall. Its foot is 1.5 m from the wall and it reaches 2 m up the wall. How long is the ladder? intermedio

Show solution
  1. Wall and floor form a right angle, so the ladder is the hypotenuse.
  2. Square the two distances: 1.5² = 2.25 and 2² = 4.
  3. Add them: 2.25 + 4 = 6.25.
  4. Take the square root: √6.25 = 2.5.

Answer: The ladder is 2.5 m long

4. An isosceles right triangle has two equal legs, each 5 cm long. How long is the hypotenuse? intermedio

Show solution
  1. The two legs are equal: c² = 5² + 5².
  2. Compute: 25 + 25 = 50.
  3. c = √50 ; simplify the radical: √50 = √(25 · 2) = 5√2.
  4. Approximate value: 5 · 1.414 ≈ 7.07.

Answer: c = 5√2 cm ≈ 7.07 cm

5. A right triangle has legs of 20 cm and 21 cm. Find the hypotenuse, then the perimeter. avanzato

Show solution
  1. Find the hypotenuse first: c² = 20² + 21² = 400 + 441 = 841.
  2. c = √841 = 29.
  3. The perimeter is the sum of the three sides: P = 20 + 21 + 29.
  4. P = 70.

Answer: Hypotenuse = 29 cm ; perimeter = 70 cm

FAQ

When can I use the Pythagorean theorem?

Only in right triangles, that is triangles with a 90° angle. It relates the two legs to the hypotenuse and does not apply to triangles without a right angle.

How do I know which side is the hypotenuse?

The hypotenuse is always the side opposite the right angle and it is the longest side of the triangle. The other two sides, which form the right angle, are the legs.

Can the theorem tell me whether a triangle is right-angled?

Yes. If the three sides satisfy c² = a² + b² (with c the longest side), the triangle is right-angled. This is the converse of the Pythagorean theorem.