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You know a triangle's angles always add to 180°. When one of those angles is exactly 90°, the sides get a rule of their own. A ladder leans against a wall, and the wall meets the ground at a right angle.
Which side is the hypotenuse?
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Pythagoras' Theorem
Apply Pythagoras' Theorem (a² + b² = c²) to calculate unknown side lengths in right-angled triangles, including in real-world and coordinate-geometry contexts
What a learner can do afterwards
- Calculate the hypotenuse of a right-angled triangle given the two shorter sides
- Find a shorter side given the hypotenuse and the other short side
- Use Pythagoras' Theorem to find the distance between two points on a coordinate grid
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A right-angled triangle has one corner that is exactly 90 degrees, like the corner of a book. The two sides that form that corner are called the legs. The third side sits opposite the right angle. It is called the hypotenuse, and it is always the longest side of the triangle.
Pythagoras' Theorem links all three sides. Call the legs a and b, and call the hypotenuse c. Then a² + b² = c². Squaring means multiplying a number by itself, so 5² is 25. In words: square each leg, add the squares, and the total is the square of the hypotenuse. This rule only works on right-angled triangles.
Try it together
A ladder leans against a wall and meets the ground at a right angle. Its foot is 9 m from the wall and its top reaches 12 m up. The ground and the wall are the legs. Square them: 9² = 81 and 12² = 144. Add the squares: 81 + 144 = 225. Take the square root: c = 15. The ladder is 15 m long.
To find a missing leg, turn the rule around: a² = c² − b². Say the hypotenuse is 25 cm and one leg is 7 cm. Square both: 25² = 625 and 7² = 49. Subtract the smaller square from the bigger one: 625 − 49 = 576. The square root of 576 is 24, so the missing leg is 24 cm. Always square the sides first. Using the plain lengths, like 25 − 7 = 18, gives a number that does not fit the triangle.
Try it together
The same rule measures distances on a grid. A treasure chest sits at (1, 2) and a player stands at (13, 18). Find the gaps by subtracting the coordinates: 13 − 1 = 12 across, and 18 − 2 = 16 up. Those gaps are the legs of a right triangle, and the straight-line distance is its hypotenuse. Square and add: 12² + 16² = 144 + 256 = 400. The square root of 400 is 20, so they are 20 units apart.
In a right-angled triangle, a² + b² = c², where c is the hypotenuse opposite the right angle. Add the squares of the legs to find the hypotenuse, subtract the squares to find a missing leg, and use the same rule for the distance between two points on a grid.
Adapted from OpenStax Algebra and Trigonometry (CC-BY 4.0), openstax.org · license
More to read
3 Ways to Use the Pythagorean Theorem
The Pythagorean Theorem only works for right triangles, so first check your triangle has a square corner. Call the two sides that form the corner a and b, and the slanted side opposite the corner c. The rule is: a squared + b squared = c squared. To find the slanted side, square a and b, add them, and take the square root. With sides 6 and 8 you get 36 + 64 = 100, so c = 10. To find a shorter side instead, subtract: with hypotenuse 13 and one side 5, work out 169 - 25 = 144, so the missing side is 12. Always finish by asking if the answer is sensible: the slanted side must come out longest.
3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
Triangles, rectangles, and the Pythagorean Theorem appear together in real problems. Start with triangle facts: the angles add to 180 degrees, and in a right triangle the two shorter sides are called legs while the longest side is the hypotenuse. The theorem says leg squared + leg squared = hypotenuse squared. Rectangles use the same idea: a diagonal cuts a rectangle into two right triangles, so a room 9 m by 12 m has a diagonal of 15 m, since 81 + 144 = 225. This section also trains a good habit: before you calculate, write down what each side represents, and after you calculate, check that the hypotenuse is the longest length.
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