Maths · Grades 7–9
Free Pythagorean theorem worksheets
Five printable Pythagorean theorem worksheets for grades 7 to 9: find a missing hypotenuse or leg in labelled right triangles, whole-number and decimal sides, mixed units, and two composite-shape problems. Full answer keys on this page.
Nine labelled triangles, mixed units
Find BC, PQ, UW and six more sides in cm, in, mm, yd, ft and m. Four hypotenuses, five legs; round to the nearest tenth.
Nine labelled triangles, metric version
The same nine triangles with cm, m and mm on the answer lines. Use as a retake or a second test version.
Whole set
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Homework: triples, a rectangle and a trapezoid
Eight problems with whole-number triples (3-4-5, 5-12-13, 10-24-26), a √137 hypotenuse, a rectangle diagonal and a trapezoid.
Find the missing side (10 triangles, irrational answers)
Ten unlabelled right triangles with whole-number sides and non-integer answers. Practice square roots and rounding.
Decimal legs, find the hypotenuse (6 problems)
Six triangles with decimal legs in cm, in, mi and ft; the hypotenuse is always the unknown. Gentlest sheet in the set.
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The Pythagorean theorem is the first formula most students use to work out a length they cannot measure. These five worksheets give 42 right triangles plus a rectangle and a trapezoid, ordered so that students meet "find the hypotenuse" first, then "find a leg", then problems where they have to decide which is which. Every answer is listed on this page, rounded to the nearest tenth, so marking takes minutes.
What's in this set
- Labelled triangles, mixed units (9 problems). Triangles ABC, PQR, UVW and so on with sides in cm, in, mm, yd, ft and m. Four ask for the hypotenuse, five for a leg. Students must write the answer for a named segment (BC, PQ, UW), which rehearses geometric notation.
- Labelled triangles, metric version (9 problems). The same nine triangles with all lengths in cm, m and mm and the unit printed on the answer line. Use it as the retake, or give sheet 1 to one half of the class and sheet 2 to the other for a no-copying test.
- Homework sheet (8 problems). Whole-number triangles including the 3-4-5, 5-12-13, 6-8-10 and 10-24-26 triples, one hypotenuse given as a square root (√137), a rectangle with a 15-unit diagonal, and a trapezoid whose slanted side must be found by drawing in a right triangle. "Show all work for credit" is printed on the sheet.
- Find the missing side (10 problems). Ten unlabelled triangles with whole-number sides that do not give whole-number answers (for example 19 and 17, or 5 and 13). This is the sheet for practicing square roots on a calculator and rounding.
- Decimal sides, find c (6 problems). Six triangles with decimal legs (3.4 cm and 4.5 cm; 12.9 mi and 12.6 mi) where the hypotenuse is always the unknown. The gentlest sheet; use it first with a class that is anxious about the topic.
What students will practice
- Applying a² + b² = c² to find an unknown hypotenuse or leg (Common Core 8.G.B.7)
- Identifying the hypotenuse as the side opposite the right angle, whatever way the triangle is turned
- Rearranging to a² = c² − b² when a leg is missing, the step where most errors happen
- Evaluating square roots and rounding to the nearest tenth (8.EE.A.2)
- Recognizing Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 10-24-26) to save time
- Drawing a right triangle inside a rectangle or trapezoid to find a diagonal or slanted edge
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How to use these worksheets in class
- Day 1: hypotenuse only. Start with sheet 5 (decimals, always find c). Every problem uses the formula the same way, so students build confidence with the calculator routine: square, square, add, square root, round.
- Day 2: which side is missing? Sheet 4 mixes legs and hypotenuses with no letters to help. Before anyone calculates, have students mark the hypotenuse on all ten triangles and write "add" or "subtract" next to each. Then compute.
- Day 3: notation and units. Sheet 1 (or 2) names the vertices. Students must translate "BC = ?" into "the side between B and C" and carry the unit through. Rotate the page for triangles like number 8, where the right angle sits at the bottom.
- Homework or assessment: sheet 3. The rectangle and trapezoid are the two problems worth discussing in class afterwards, because they require students to see a right triangle that is not drawn for them.
- Differentiation: students who struggle can do sheet 3's first five problems only (all Pythagorean triples with whole-number answers). Confident students can do sheet 1 without a calculator, estimating each square root between two integers first.
Answer keys (rounded to the nearest tenth)
- Sheet 1: 1) BC = 9 cm; 2) PQ = 5.4 in; 3) UW = 6.7 mm; 4) XZ = 14.3 yd; 5) EF = 6 ft; 6) JL = 21.4 m; 7) MN = 17.9 in; 8) ST = 3.9 cm; 9) CE = 13.0 yd.
- Sheet 2: the same values: 9 cm, 5.4 m, 6.7 mm, 14.3 mm, 6 cm, 21.4 m, 17.9 m, 3.9 mm, 13.0 cm.
- Sheet 3: 1) c = 10; 2) a = 10; 3) b = 12; 4) a = 4; 5) x = 5; 6) x = 8; 7) x = 9; 8) x ≈ 9.2 (right triangle with legs 7 and 6).
- Sheet 4: 1) 8.5; 2) 13.9; 3) 17.3; 4) 14.9; 5) 11.5; 6) 21.6; 7) 12.7; 8) 14.4; 9) 16.6; 10) 16.2.
- Sheet 5: 1) 5.6 cm; 2) 20.5 in; 3) 18.0 mi; 4) 13.4 mi; 5) 16.3 in; 6) 10 ft.
Common mistakes to watch for
- Adding when a leg is missing. Sheet 1 problem 1 (12 and 15, find the third side) is the classic: students write 12² + 15² and get 19.2 instead of 9. Have them label c before they write anything.
- Forgetting the square root. An answer of 81 for a side of 9 means the student stopped at c². Ask "Is 81 cm a sensible length for that triangle?"
- Rounding too early. On sheet 4, rounding 72 to 70 before taking the root gives 8.4 instead of 8.5. Keep full precision until the last step.
- Applying the theorem to a non-right triangle. Every triangle here is right-angled, but the trapezoid on sheet 3 needs a right triangle drawn in first. Students who write 3² + 6² for that problem have not found the correct horizontal leg (10 − 3 = 7).
Frequently asked questions
What grade are these Pythagorean theorem worksheets for?
Grade 8 is where the theorem is taught in the Common Core (8.G.B.6 to 8.G.B.8) and in Year 9 in England. Sheet 5 works for a grade 7 introduction; sheets 1 and 3 suit grade 8 and 9 revision, and any high-school geometry class starting right triangles.
Do students need a calculator?
For sheets 1, 2, 4 and 5, yes: most answers are irrational and must be rounded. Sheet 3 can be done without one because seven of its eight answers are whole numbers.
Are answer keys included?
The PDFs are single pages. Every answer is listed above under "Answer keys", rounded to the nearest tenth as the sheets ask.
Which Pythagorean triples appear?
3-4-5, 6-8-10, 5-12-13 and 10-24-26 on sheet 3, and 9-12-15 on sheets 1 and 2 (problem 1). Recognizing them lets students skip the calculator and check their rounded answers on other sheets.
What comes after these worksheets?
The converse (is a triangle with sides 7, 24, 25 right-angled?), distance between two points on a coordinate grid, and word problems such as ladders against walls and the diagonal of a room. Sheet 3's rectangle and trapezoid are the bridge to those.
Related algebra and geometry worksheets
Solving for a missing leg is really solving an equation, so pair these with the two-step equations worksheets. For the perfect squares students need to recognize on sight, the multiplication worksheets cover the times tables to 12. Younger classes learning shape names can start with the 3D shapes worksheets. More middle-school practice is collected on the grade 8 worksheets page.
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