Converse of the Pythagorean Theorem Worksheet

Sixteen printable items. Decide whether three lengths make a right triangle, then classify triangles as acute, right or obtuse from their sides alone. The answer key shows every computation.

By the PythagoreanTheorem.net teamUpdated 4 min read

This converse of the Pythagorean theorem worksheet asks one question 16 times: do these three side lengths make a right triangle? Students square the longest side and compare it with the sum of the squares of the other two. Equal means right; smaller means acute; larger means obtuse. The answer key with every computation prints on its own page.

Jump to the answer key

The rule students apply

The converse of the Pythagorean theorem says that if a triangle's sides satisfy a2 + b2 = c2, the angle across from c is a right angle, and c is the hypotenuse. The same comparison also sorts every other triangle. With c the longest side:

Classify by sidesa2 + b2 = c2: right. a2 + b2 > c2: acute. a2 + b2 < c2: obtuse.

The reason is intuitive: open the angle between a and b wider than 90° and the third side gets longer, so c2 grows past a2 + b2. The acute, right or obtuse lesson explains it with the law of cosines.

What is on the sheet

  • Problems 1–8: right triangle, yes or no? Includes familiar triples, near misses such as 10, 24, 25, lengths listed out of order (15, 8, 17), decimals (1.5, 2, 2.5), a square root (2, √5, 3), and one set of lengths that cannot form a triangle at all.
  • Problems 9–16: acute, right or obtuse? Each triangle is drawn to scale. Several look almost exactly like right triangles but are not, so the only reliable test is the arithmetic.

Most students finish in 20–30 minutes. A calculator helps with the larger squares in problems 3, 11 and 14 but is not essential.

The worksheet: is it a right triangle?

NameDateScore

Directions: Problems 1–8: decide whether the three lengths are the sides of a right triangle. Problems 9–16: classify each triangle as acute, right or obtuse. Show the squares you compare.

  • Sides 6,  8,  10
    Right triangle?  yes  /  no
  • Sides 5,  11,  12
    Right triangle?  yes  /  no
  • Sides 9,  40,  41
    Right triangle?  yes  /  no
  • Sides 15,  8,  17
    Right triangle?  yes  /  no
  • Sides 10,  24,  25
    Right triangle?  yes  /  no
  • Sides 1.5,  2,  2.5
    Right triangle?  yes  /  no
  • Sides 2,  √5,  3
    Right triangle?  yes  /  no
  • Sides 2,  3,  6
    Right triangle?  yes  /  no
  • Acute, right or obtuse?645
  • Acute, right or obtuse?1057
  • Acute, right or obtuse?371235
  • Acute, right or obtuse?1169
  • Acute, right or obtuse?15813
  • Acute, right or obtuse?292021
  • Acute, right or obtuse?26724
  • Acute, right or obtuse?6.42.56

Answer key

In every item the longest side is squared last and compared with the sum of the other two squares.

  1. 62 + 82 = 36 + 64 = 100
    102 = 100
    100 = 100 → yes, right
  2. 52 + 112 = 25 + 121 = 146
    122 = 144
    146 ≠ 144 → no
  3. 92 + 402 = 81 + 1,600 = 1,681
    412 = 1,681
    1,681 = 1,681 → yes, right
  4. 82 + 152 = 64 + 225 = 289
    172 = 289
    289 = 289 → yes, right
  5. 102 + 242 = 100 + 576 = 676
    252 = 625
    676 ≠ 625 → no
  6. 1.52 + 22 = 2.25 + 4 = 6.25
    2.52 = 6.25
    6.25 = 6.25 → yes, right
  7. 22 + (√5)2 = 4 + 5 = 9
    32 = 9
    9 = 9 → yes, right
  8. 2 + 3 = 5 < 6
    The two short sides cannot reach: no triangle at all, so not a right triangle.
  9. 42 + 52 = 16 + 25 = 41
    62 = 36
    41 > 36 → acute
  10. 52 + 72 = 25 + 49 = 74
    102 = 100
    74 < 100 → obtuse
  11. 122 + 352 = 144 + 1,225 = 1,369
    372 = 1,369
    1,369 = 1,369 → right
  12. 62 + 92 = 36 + 81 = 117
    112 = 121
    117 < 121 → obtuse
  13. 82 + 132 = 64 + 169 = 233
    152 = 225
    233 > 225 → acute
  14. 202 + 212 = 400 + 441 = 841
    292 = 841
    841 = 841 → right
  15. 72 + 242 = 49 + 576 = 625
    262 = 676
    625 < 676 → obtuse
  16. 2.52 + 62 = 6.25 + 36 = 42.25
    6.42 = 40.96
    42.25 > 40.96 → acute

Teacher notes

  • Find the longest side first. Lengths are not always listed in order (problem 4 starts with 15), and in problem 7 the longest side is 3, not √5, because √5 ≈ 2.24 (see square roots). Squaring whichever number comes last is the habit this sheet is designed to break.
  • Problem 8 is a trap on purpose. Since 2 + 3 < 6, the lengths cannot close into a triangle, so there is no triangle to test. It is a good moment to review the triangle inequality: the two shorter sides must add up to more than the longest.
  • Looks can deceive. The largest angles in problems 12, 13 and 16 are about 92.1°, 87.8° and 87.5°, close enough to 90° that no student can judge by eye. Problem 15 (7, 24, 26) is one unit away from the 7-24-25 triple and its largest angle is about 98.7°.
  • Problem 5 (10, 24, 25) is acute because the true hypotenuse for legs 10 and 24 would be 26, from the 5-12-13 triangle doubled; shortening it closes the angle.
  • Standards: Common Core 8.G.B.6 asks students to explain the theorem and its converse; classifying triangles by the inequality is a staple of high school geometry.

Students can confirm answers with the right triangle checker. The distance formula worksheet uses the converse again on a coordinate grid, and the worksheet generator makes new acute-right-obtuse sheets.

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