Pythagorean Theorem: 9 Solved Exercises, Step by Step

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 − b²).

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.

• Both legs known, hypotenuse wanted: add the squares, then take the square root. c = √(a² + b²).
• Hypotenuse and one leg known, other leg wanted: subtract, then take the square root. a = √(c² − b²).

Getting these two backwards is the single most common source of wrong answers. Before you compute anything, decide which side is the hypotenuse: if the unknown is the longest side you add, otherwise you subtract.

Pythagorean triples: the shortcuts worth memorising

Some right triangles have three whole-number sides. These sets are called Pythagorean triples, and textbook exercises use them constantly because the answers come out clean.

Triple	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
20, 21, 29	400 + 441 = 841

Every multiple of a triple is also a triple: from 3-4-5 you get 6-8-10, 9-12-15, 30-40-50. So if an exercise gives you legs of 30 and 40, you can write 50 immediately without squaring anything.

The converse: is this triangle right-angled?

The theorem also runs backwards. Given three sides, call the longest one c and test whether c² = a² + b². If the two sides match, the triangle has a right angle opposite c. This is the converse of the Pythagorean theorem, and the comparison also tells you more:

• a² + b² = c² → right triangle.
• a² + b² < c² → obtuse triangle (the angle opposite c is wider than 90°).
• a² + b² > c² → acute triangle.

Two applications that come up constantly

Distance between two points. On a coordinate plane the gap between A(x₁, y₁) and B(x₂, y₂) is just the hypotenuse of a right triangle whose legs are the horizontal and vertical differences. That is where the distance formula comes from: d = √((x₂ − x₁)² + (y₂ − y₁)²). It is not a new rule to learn, only the theorem in different clothing.

Diagonals of rectangles and boxes. A diagonal splits a rectangle into two right triangles, so the diagonal of a rectangle with sides a and b is √(a² + b²). In three dimensions you apply the theorem twice and get the space diagonal of a box: d = √(a² + b² + c²). The same reasoning gives you the height of an equilateral triangle and the apothem of a regular polygon, which is why the theorem turns up all over area and perimeter problems.

Three mistakes to avoid

1. Subtracting when you should add, or the reverse. Identify the hypotenuse first.
2. Forgetting the final square root. c² = 100 does not mean c = 100.
3. Rounding too early. Keep radicals like 5√2 exact until the last step, and only then approximate. If you are rusty on squares and roots, review the laws of exponents first.

The nine exercises below

They follow the same pattern at increasing difficulty: whole-number legs, a missing leg, the diagonal of a rectangle, a real ladder problem, a radical to simplify, a two-step perimeter question, the converse test, the distance between two points, and finally the diagonal of a box in three dimensions. Every answer has been verified by redoing the calculations.