Find a diagonal with the Pythagorean theorem
Use a right-triangle relationship and check which side is the hypotenuse.
Supplied materials
- Paper or a note app; a calculator is optional.
- Fictional rectangular panel: 3 units wide and 4 units tall. Its diagonal makes a right triangle with the sides. For a right triangle, a² + b² = c², where c is opposite the right angle.
Learn the idea
The hypotenuse is the side opposite the right angle and is the longest side. In a rectangle, the diagonal is the hypotenuse of the triangle made by a width and a height.
Square the two perpendicular side lengths, add, then take the positive square root. Adding the side lengths gives a route along two edges, not the diagonal. This rule applies to right triangles, not every triangle.
- Label the width and height a and b, and the diagonal c.
- Calculate 3² + 4².
- Find the positive length whose square equals that sum.
- Repeat with a 6-by-8 panel.
See the method in action
3² + 4² = 9 + 16 = 25, so c = √25 = 5 units. The diagonal is longer than either side but shorter than the 3 + 4 = 7 unit path along the edges.
Check your reasoning
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See the answer and explanation
10 units
6² + 8² = 36 + 64 = 100, so the diagonal is √100 = 10 units. The squared quantity 100 is not itself the length.
Take it further
Check a 5-by-12 panel: its diagonal is 13. Then explain why 5, 12 and 14 cannot be the sides of a right triangle with 14 as hypotenuse.
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Longer courses for learners ages 14+
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