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Pythagorean theorem reference

Pythagorean theorem reference and worked examples

A six-page printable guide to right-triangle parts, square areas, missing sides, irrational lengths and the coordinate distance formula.

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Read the diagrams and worked examples. Identify the hypotenuse each time, and explain why each calculation uses addition or subtraction.

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Reference handout
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Six pages, from square areas to coordinate distance

The handout introduces the right triangle and its three side squares, then works through finding a hypotenuse, finding a leg, handling an irrational length and calculating the distance between two points. Each page has its own diagram and explanation.

Use it beside the matching practice and print the pages you need. The explanations are already included, so there is no separate answer key.

Learning objective

What this activity builds

Connect the area relationship a² + b² = c² to practical missing-side and coordinate-distance calculations.

Worked example

Why do the numbers in the Pythagorean theorem appear as squares?

They describe the areas of squares built on the three sides.

For a right triangle with legs 3 and 4, those square areas are 9 and 16. Their sum is 25, which is the area of the square on the hypotenuse. Its side length is 5, not 25.

Teaching notes

Make the practice count

  • Use one page beside the corresponding practice rather than introducing all six at once.
  • Distinguish side length from square area in every diagram.
  • Compare a missing-hypotenuse example with a missing-leg example.
  • Use the final coordinate example to connect algebra with a visible right triangle.

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6 reference pages
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Math SheetNameScoreTeacherDatePythagorean Theorem ReferenceUse these six pages to connect right triangles, square areas, unknown sides, irrational roots, and coordinate distance.The theorem and its partsIn a right triangle, the hypotenuse is opposite the right angle. If the legs are a and b and the hypotenuse is c, then asquared plus b squared equals c squared.34590°a2 + b2 = c2For legs 3 and 4, the hypotenuse is 5.math-sheet.comPage 1 / 6
Math SheetNameScoreTeacherDatePythagorean Theorem ReferenceUse these six pages to connect right triangles, square areas, unknown sides, irrational roots, and coordinate distance.Squares on the three sidesA square built on each side has area equal to that side length squared. The two leg-square areas add to thehypotenuse-square area.91625A(a) + A(b) = A(c)9 + 16 = 25Areas 9 and 16 combine to make area 25.math-sheet.comPage 2 / 6
Math SheetNameScoreTeacherDatePythagorean Theorem ReferenceUse these six pages to connect right triangles, square areas, unknown sides, irrational roots, and coordinate distance.Finding the hypotenuseSquare both legs, add, and take the positive square root. A length cannot be negative.68?c2 = 62 + 82c2 = 36 + 64 = 100c = 100 = 10With legs 6 and 8: c squared is 36 plus 64, so c is 10.math-sheet.comPage 3 / 6
Math SheetNameScoreTeacherDatePythagorean Theorem ReferenceUse these six pages to connect right triangles, square areas, unknown sides, irrational roots, and coordinate distance.Finding a legStart with the hypotenuse square and subtract the known leg square. Then take the positive square root.?513a2 = 132 - 52a2 = 169 - 25 = 144a = 144 = 12With c equal to 13 and one leg 5: the other leg is 12.math-sheet.comPage 4 / 6
Math SheetNameScoreTeacherDatePythagorean Theorem ReferenceUse these six pages to connect right triangles, square areas, unknown sides, irrational roots, and coordinate distance.Irrational lengths and roundingSome radicands are not perfect squares. Keep the exact root during the work, then round only the final length to fourdecimal places.23?c2 = 22 + 32 = 13c = 13 ~ 3.6056With legs 2 and 3, the exact length is the positive root of 13, approximately 3.6056.math-sheet.comPage 5 / 6
Math SheetNameScoreTeacherDatePythagorean Theorem ReferenceUse these six pages to connect right triangles, square areas, unknown sides, irrational roots, and coordinate distance.Distance on a coordinate planeHorizontal and vertical distances are absolute differences. They form the legs of a right triangle between the two points.-5-5-4-4-2-200224455P1P2dx = 4 - (-2) = 6dy = 5 - 1 = 4d2 = 62 + 42 = 52d = 52 ~ 7.2111From (-2, 1) to (4, 5), the leg lengths are 6 and 4; the distance is the positive root of 52, approximately 7.2111.math-sheet.comPage 6 / 6

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