The hypotenuse
x² = 12² + 5² = 144 + 25 = 169, x = √169 = 13 cm. Level 1 uses Pythagorean triples, level 2 lengths with decimals and level 3 roots that are not exact, rounded.
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Teaching guide: how to practise Pythagoras' theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This free Pythagorean theorem worksheet generator builds A4 worksheets with triangles and shapes drawn to scale, to find the hypotenuse and the shorter sides, test whether a triangle is right-angled, use the theorem in rectangles, rhombuses, triangles and trapeziums, solve word problems and understand the picture of squares on the sides. For Grades 8 and 9, with an answer key that shows every step.
Student worksheet 
Answer key A shorter side
When the hypotenuse is known, the missing side is the root of the difference: 13² = x² + 5², x² = 169 − 25 = 144, x = 12 cm.
Is it right-angled?
The converse of the theorem: compare the square of the longest side with the sum of the squares of the other two. Near misses (6, 8, 11) make pupils calculate; at level 3, acute, right or obtuse.
Plane shapes
The diagonal of a rectangle, the height of an isosceles or equilateral triangle, the side of a rhombus from half its diagonals and the slanted side of a trapezium: the answer key marks the right-angled triangle in each shape.
Word problems
Ladders, screens, ramps, kites and short cuts: pupils sketch the triangle, decide whether the hypotenuse or a shorter side is unknown and interpret the answer.
Squares on the sides
The classic picture of the theorem: the areas of the squares on the two shorter sides add up to the area of the square on the hypotenuse. A good way in before the formula.
Frequently asked questions about Pythagoras' theorem worksheets
Which grade learns Pythagoras' theorem?
Grade 8 (Year 9): the theorem, Pythagorean triples, the converse and its use in plane shapes and word problems. In Grade 9 it is used with roots that are not exact and to classify triangles by their angles; the levels of the worksheets follow that order.
What is a Pythagorean triple?
Three whole numbers that satisfy the theorem, such as 3, 4 and 5 (9 + 16 = 25) or 5, 12 and 13. Multiples of a triple are triples too (6, 8 and 10). Level 1 uses them so the root is exact and pupils can focus on the method.
How do you tell whether a triangle is right-angled?
With the converse of the theorem: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled. If it is less, the triangle is acute-angled, and if it is greater, obtuse-angled.
How are answers rounded?
Roots that are not exact are rounded to 1 or 2 decimal places, as you choose, and the instruction says so. Only the final root is rounded, and answers never sit close to a half, so a calculator and the answer key agree.
Are the drawings to scale?
Yes. Every triangle and shape is drawn in the proportions of its lengths with the right-angle mark, and the labels are placed clear of the lines.
What is in the answer key?
Every step: half the base or diagonal where needed, the theorem with the values substituted, the squares, the sum or difference, the root and the answer with its unit; in the shapes, the right-angled triangle is marked.
What other worksheets go well with these?
The powers and square roots worksheets to practise squares and roots, the area and perimeter worksheets, where many shapes need a height found with Pythagoras, and the PISA-style maths problems. Find every resource under maths worksheets.
Is any pupil data sent anywhere?
No. Worksheets are generated in your browser and templates are saved in your browser's memory.
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