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High school · Geometry

Grow squares on a right triangle

Compare the areas on two perpendicular legs with the area on the hypotenuse.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Grow squares on a right triangle. Hypotenuse length c: 5. Two colored leg regions: 0 square units. Colored hypotenuse region: 0 square units3² + 4² = 25a = 3b = 4c ≈ 50% of full depthTwo leg areas together = hypotenuse area
The triangle is right-angled in the Euclidean plane. Regions at partial progress are strips, becoming squares only at full progress. The drawing illustrates the area identity; it is not a dissection proof or a claim for arbitrary triangles. The whole diagram rescales to fit the screen.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Hypotenuse length c
5
Two colored leg regions
0 square units
Colored hypotenuse region
0 square units

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

For a right triangle, the square built on the hypotenuse has the same area as the squares on the two legs combined. Here every attached square grows through the same fraction of its final depth. At every stage the two leg areas together match the hypotenuse area; at full depth, the familiar squared-length formula appears.

A relationship to keep

a² + b² = c², with c = √(a² + b²)

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Find the right angle

    The small corner mark identifies the perpendicular legs a and b. The sloping side c lies opposite the right angle and is the hypotenuse.

  2. STEP 2

    Grow equal fractions

    Move the timeline to extend colored regions outward from all three sides. They use the same fraction of each square's final depth, so comparing their areas remains meaningful.

  3. STEP 3

    Compare complete squares

    At the end, all three squares are complete. The areas a² and b² add to c². A 3–4–5 triangle gives 9 + 16 = 25, even though 3 + 4 is not 5.

Your turn to explain

Make a prediction. Test your reasoning.

A right triangle has legs 5 and 6. What is its hypotenuse length?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

c = √(5² + 6²) = √61 ≈ 7.810 units. Add the squared lengths, then take the square root.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 8 · Geometry

Recover a right triangle's missing leg

When the hypotenuse is known, the Pythagorean theorem finds a missing leg by subtraction of squares.

Grade 8 · Geometry

Test whether three sides form a right triangle

The converse of the Pythagorean theorem recognizes a right triangle from a square-sum equality.