The Pythagorean Theorem (explained in 2 minutes)
Right-angled triangles, the hypotenuse and a² + b² = c², with a worked example: sides of 3 cm and 4 cm give a hypotenuse of 5 cm.
- Subject: Maths
- Age group: 11-14
What this video covers
- A right-angled triangle has one angle measuring exactly ninety degrees.
- The two shorter sides are the legs, and the longest side opposite the ninety-degree angle is the hypotenuse.
- The square of one leg plus the square of the other leg equals the square of the hypotenuse.
- Squaring means multiplying a number by itself, and all three lengths must use the same unit.
- For a triangle with legs of three centimetres and four centimetres, the rule sets their squares equal to the square of the hypotenuse.
- Three squared is nine, four squared is sixteen, and their sum makes the hypotenuse squared equal twenty-five.
- The needed number is the number that multiplied by itself equals twenty-five.
- Five is the hypotenuse because five times five equals twenty-five, matching nine plus sixteen.
- When a leg is missing, subtract the known leg's square from the hypotenuse's square, then find the number whose square gives the result.
- The rule applies only to triangles with a ninety-degree angle and can find difficult-to-measure distances such as a ladder or a garden diagonal.
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Transcript
Let’s begin with the shape at the heart of this rule: a right-angled triangle. It has one angle that measures exactly ninety degrees, like the corner of a book.
The two shorter sides meet at that corner. The longest side is opposite the ninety-degree angle, and it has a special name: the hypotenuse. The other two sides are called the legs.
The Pythagorean theorem says that, in every right-angled triangle, the square of the length of one leg, plus the square of the length of the other leg, equals the square of the hypotenuse.
To square a number means to multiply it by itself. So four squared is four times four, which equals sixteen. This relationship works with any units, such as centimetres or metres, as long as all three lengths use the same unit.
Here is a worked example. Suppose the legs measure three centimetres and four centimetres. We want to find the hypotenuse. The rule gives three squared plus four squared equals the square of the hypotenuse.
Three squared is nine, and four squared is sixteen. So the square of the hypotenuse equals twenty-five.
Which number times itself equals twenty-five?
The answer is five, because five times five is twenty-five. Therefore, the hypotenuse is five centimetres. We can check by calculating five squared: it is twenty-five, matching nine plus sixteen.
Sometimes the missing length is a leg instead of the hypotenuse. In that case, find the square of the hypotenuse, subtract the square of the known leg, and then find the number whose square gives your answer.
Remember, this rule only applies when the triangle has a ninety-degree angle. It is useful for finding distances that are difficult to measure directly, such as the length of a ladder or the diagonal across a rectangular garden.