Understand the Proof of the Pythagorean Theorem

Students explore and explain at least one proof of the Pythagorean Theorem—such as the area-of-squares proof or the similar-triangles proof—understanding why a² + b² = c² must hold for any right triangle. They connect the algebraic identity to geometric area arguments.

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Grade 8 · Geometry: Pythagorean Theorem · question 1 of 3

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20 rungs. A star at the top.

Choose the square whose area the others make.

How it is taught

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Squares on the two legs add to the square on the longest side.

The side of that big square is the longest side of the triangle.

Read it: Squares on a right triangle's sides

Draw a square on each side of a right triangle. With legs 18 and 24, the squares on the legs have areas 324 and 576, and together they make 900, the area of the square on the longest side.

Find the area of the square on each leg: the side times itself.
Add the two leg areas: that is the area of the square on the longest side.
For a side, find the number that times itself makes its square's area.

A square's area is its side times itself: 18 × 18 = 324 and 24 × 24 = 576.

In a right triangle, the squares on the two legs exactly fill the square on the longest side: 324 + 576 = 900.

30 × 30 = 900, so the longest side is 30. This is the Pythagorean theorem.

In a right triangle, the squares on the legs add up to the square on the longest side.

Adding the sides: 18 + 24 = 42.

The areas add, not the sides: 324 + 576 = 900, and 30 × 30 = 900, so the longest side is 30.

The squares on a right triangle's legs have areas 81 and 1600. The square on its longest side has area ___. — 1681. The leg squares add up to the square on the longest side: 81 + 1600 = 1681.

3-4-5 right triangle

Here's a three-four-five right triangle. Let's discover its secret together.

Grade 8 · Why a² + b² = c²

Here's our right triangle. The little square at the corner shows where the two short sides meet at a right angle.

We call the two short sides the legs. Let's say one leg is three units long and the other is four units long.

The longest side, opposite the right angle, is called the hypotenuse — it's five units long.

Now build a square right on the leg of length three. Its area is three times three — that's nine.

Build another square on the leg of length four. Its area is four times four — that's sixteen.

And one more square on the hypotenuse. Its area is five times five — twenty-five.

Watch what happens when we add the two leg-squares: nine plus sixteen equals twenty-five — exactly the hypotenuse square!

We can write this as a-squared plus b-squared equals c-squared. The two little squares always fit perfectly into the big one.

That's the Pythagorean Theorem — for any right triangle, the squares on the legs add up to the square on the hypotenuse.

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

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MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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