Legs 6 cm and 8 cm
- a² + b² = c² gives 36 + 64 = c².
- c² = 100 cm², so c = 10 cm.
- The lengths are 6, 8 and 10; the square areas are 36, 64 and 100.
The Pythagorean area relationship
See why the squares belong in the theorem.
Identify the right angle before naming the sides
Explain why the squares on the legs have the same total area as the square on the hypotenuse.
In a right triangle, the two sides meeting at the right angle are the legs. The side opposite the right angle is the hypotenuse; it is the longest side. Rotating the triangle does not change these roles. The hypotenuse is not automatically the horizontal, vertical or bottom side.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
The theorem compares areas of squares
For legs a and b and hypotenuse c, a² + b² = c². A square constructed on each leg has area a² or b²; together those areas equal the area of the square on the hypotenuse. The statement adds square areas, not side lengths. It applies to right triangles, not every triangle.
Move from a model to a justified equation
For legs 3 and 4, the smaller square areas are 9 and 16. Their sum is 25, giving hypotenuse √25 = 5. Changing a leg changes both its square area and the resulting hypotenuse. The attached-square diagram uses actual geometry; the written labels state which side each area belongs to.
Math AnticsWatch · pause · explain
Math Antics — The Pythagorean Theorem
Pause before the calculation. Identify the legs, hypotenuse and three square areas.
Open on YouTube →Worked examples
A triangle is rotated on the page
- Find the right-angle marker.
- Name its adjacent sides a and b, and the opposite side c.
- Use a² + b² = c² regardless of orientation.
Guided Practice
A right triangle has legs 5 and 12. What are the three square areas?
A first hint
The hypotenuse-square area is the sum of the two leg-square areas.
Compare your reasoning
25 and 144 are the smaller areas. Their sum is 169, so the hypotenuse is 13.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
0 of 8 questions completed
This round saves on this device. It is practice, not a school mark.
Connections
If every side of a right triangle doubles, explain why every attached square has four times the area while the theorem remains true.
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