Lesson 3 of 5

Pythagoras' theorem.

About 25 minutes · 5 practice questions

Read the full article: Pythagoras and geometry

In a right triangle the squares on the two legs exactly fill the square on the hypotenuse: how to find a missing side, and how to tell whether an angle is right.

Step 1 of 5 · Squares on the sides

Step 1 of 5

Squares on the sides

In a right triangle the two sides that make the right angle are called the legs, $a$ ⁠ and $b$ ⁠, and the side opposite it is the hypotenuse, $c$ ⁠. The hypotenuse is always the longest side. A square stands on each side, and every cell in it is one unit of area. Predict what happens when the pieces of the two smaller squares slide into the large one.

To the question

Step 2 of 5

Finding the hypotenuse

Pythagoras’ theorem: in a right triangle $a^2 + b^2 = c^2$ ⁠. The areas add, not the sides. To get back from the area $c^2$ ⁠ to the length $c$ ⁠, take the square root: the number whose square is the area.

To the question

Step 3 of 5

When the root is not whole

In most triangles the hypotenuse is not a whole number. $3$ ⁠, $4$ ⁠, $5$ ⁠ and $6$ ⁠, $8$ ⁠, $10$ ⁠ are special cases worth remembering. When the root is not whole, you can at least say which two whole numbers it lies between, and get an approximate value on a calculator.

To the question

Step 4 of 5

Finding a leg

The theorem also helps when the hypotenuse is known and a leg is missing: subtract instead of adding, $b^2 = c^2 - a^2$ ⁠. Here Side a is locked and you set Side b. The sum at the top of the board shows $c^2$ ⁠ at every moment.

To the question

Step 5 of 5

Is the angle right?

The theorem works backwards too: if $a^2 + b^2 = c^2$ ⁠, the angle between $a$ ⁠ and $b$ ⁠ is a right angle. Move Corner angle and watch the sum: when the angle is wider, $c^2$ ⁠ is more than $a^2 + b^2$ ⁠, and when it is sharper, $c^2$ ⁠ is less.

To the question

Interactive scene

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Each step sets up the Scene for you, and a control marked “fixed” stays put for that step.

To the experiment

Practice · Predict, then look

A square stands on each side of the right triangle. The squares on $a$ ⁠ and $b$ ⁠ are cut into pieces and slid into the square on $c$ ⁠. What will happen?