Pythagoras’ Theorem definition
Pythagoras’ theorem is a rule used with right-angled triangles. It connects the lengths of the three sides of the triangle. The theorem says that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
The formula is usually written as a squared plus b squared equals c squared, where c is the hypotenuse and a and b are the two shorter sides. For example, if a right-angled triangle has shorter sides of 3 cm and 4 cm, then c squared equals 3 squared plus 4 squared. This gives c squared equals 9 plus 16, so c squared equals 25. The hypotenuse is 5 cm.
Pythagoras’ theorem is used in geometry, trigonometry, coordinate geometry, vectors, bearings, construction, 3D shapes and real-life measurement. At GCSE level, students use it to find missing sides, check whether a triangle is right-angled, calculate distances on grids and solve worded problems. At A-Level, the same idea appears in vectors, mechanics, coordinate geometry and more advanced spatial problems.
When finding the hypotenuse, add the squares of the two shorter sides and then take the square root. For example, if the shorter sides are 6 cm and 8 cm, then c squared equals 36 plus 64, which equals 100. The square root of 100 is 10, so the hypotenuse is 10 cm. This is a common exam calculation.
Pythagoras’ theorem can also be rearranged to find a shorter side. If the hypotenuse is 13 cm and one shorter side is 5 cm, then the missing side squared equals 13 squared minus 5 squared. This gives 169 minus 25, which equals 144. The missing side is the square root of 144, so it is 12 cm. Subtraction is used here because the hypotenuse is already known.
The theorem only works directly for right-angled triangles. This is very important. If a triangle does not have a right angle, students need other methods such as the cosine rule or sine rule. In exam questions, the right angle may be marked with a small square, or the question may describe a vertical wall, horizontal floor or perpendicular line. Students should always identify the right angle first.
Pythagoras’ theorem is useful in real life because many measurement problems involve right angles. It can be used to calculate the length of a ladder leaning against a wall, the diagonal of a rectangle, the distance across a field, or the diagonal inside a box. In coordinate geometry, it can find the distance between two points by forming a right-angled triangle from horizontal and vertical changes.
Related ideas include right-angled triangles, hypotenuse, square numbers, square roots, trigonometry, cosine rule, distance between points, bearings and 3D Pythagoras. A clear definition of Pythagoras’ theorem helps students understand when the formula applies and why the longest side must be treated carefully. In exams, label the hypotenuse first, square the lengths accurately and round the final answer only when needed.