3D Pythagorean Theorem Worksheet

Twelve printable problems on the diagonals of boxes and cubes and the heights and slant heights of cones and pyramids, with an answer key that shows every step.

By the PythagoreanTheorem.net teamUpdated 4 min read

This 3D Pythagorean theorem worksheet has 12 problems on boxes, cubes, cones and square pyramids. Each solid hides one or two right triangles: find the triangle, then use a2 + b2 = c2. Problems get harder in three levels, and the answer key at the end shows the full working and prints on its own page.

The answer key starts on a new page, so you can print the problems alone.
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Problems

Useful facts. Give exact answers in simplest radical form, and round decimals to the nearest hundredth.

  • Box l × w × h: space diagonal d = √l2 + w2 + h2.
  • Cone: the height h, the radius r and the slant height ℓ form a right triangle, so r2 + h2 = ℓ2.
  • Square pyramid: the height meets the center of the base at a right angle. The slant height runs down the middle of a face; a lateral edge runs from the apex to a corner.
  • Level 1 A box is 6 cm long, 2 cm wide and 3 cm tall. Find its space diagonal d. 6 cm3 cm2 cmd
  • Level 1 A cube has edges of 5 cm. Find its space diagonal d, exact and rounded. 5 cmd
  • Level 1 A cone has radius 5 and height 12. Find its slant height ℓ. 125ℓ
  • Level 1 A square pyramid has a base edge of 6 m and a height of 4 m. Find its slant height ℓ. 6 m4 mℓ
  • Level 2 A box is 16 in long, 12 in wide and 15 in tall. Find (a) the base diagonal f and (b) the space diagonal d. 16 in15 in12 indf
  • Level 2 A cone has a slant height of 10 cm and a radius of 6 cm. Find its height h. h6 cm10 cm
  • Level 2 A square pyramid has a base edge of 10 ft and a slant height of 13 ft. Find its height h. 10 fth13 ft
  • Level 2 A cone has a base diameter of 14 in and a slant height of 25 in. Find its height h. h14 in25 in
  • Level 2 A storage box measures 60 cm × 40 cm × 50 cm. Will a 90 cm rod fit inside it? Explain. 60 cm50 cm40 cm?
  • Level 3 A box is 12 m long and 9 m wide, and its space diagonal is 17 m. How tall is it? 12 mh9 m17 m
  • Level 3 The space diagonal of a cube is 10 cm. Find the edge length x. x10 cm
  • Level 3 A square pyramid has a base edge of 8 and a height of 7. Find the length e of a lateral edge. 87e

Answer key

  1. d = √62 + 22 + 32 = √36 + 4 + 9 = √49 = 7 cm.
  2. Face diagonal: √52 + 52 = √50. Then d = √50 + 52 = √75 = 5√3 ≈ 8.66 cm.
  3. Radius, height and slant height form a right triangle: ℓ = √52 + 122 = √169 = 13.
  4. The slant height lands at the middle of a base edge, 3 m from the center: ℓ = √32 + 42 = √25 = 5 m.
  5. (a) f = √162 + 122 = √400 = 20 in. (b) d = √202 + 152 = √625 = 25 in.
  6. h = √102 − 62 = √100 − 36 = √64 = 8 cm.
  7. Half the base edge is 5 ft: h = √132 − 52 = √169 − 25 = √144 = 12 ft.
  8. The radius is half the diameter, 7 in: h = √252 − 72 = √625 − 49 = √576 = 24 in.
  9. The longest rod lies along the space diagonal: √3,600 + 1,600 + 2,500 = √7,700 ≈ 87.75 cm. That is less than 90 cm, so the rod does not fit.
  10. h2 = 172 − 122 − 92 = 289 − 144 − 81 = 64, so h = 8 m. (Or: base diagonal √144 + 81 = 15, then h = √172 − 152 = 8.)
  11. For a cube, d2 = x2 + x2 + x2 = 3x2. So 3x2 = 100, x2 = 100 over 3 and x = 10 over √3 = 10√3 over 3 ≈ 5.77 cm.
  12. The lateral edge lands at a corner, half a base diagonal from the center. The base diagonal is 8√2, so the half is 4√2 and its square is 32. e = √72 + 32 = √81 = 9.

Teacher notes

Standard and timing. The worksheet targets 8.G.B.7 (unknown side lengths in three dimensions) and takes most students 35 to 45 minutes. Problems 1 to 4 need one right triangle, 5 to 9 need two triangles or careful reading, and 10 to 12 work backward from a diagonal. Review the idea first with the Pythagorean theorem in 3D lesson if the class has not seen it. The sheet also follows on naturally from the 3D extension problem at the end of day 2 of the lesson plan.

Errors to watch for.

  • Using the diameter as the radius in problem 8 (which gives √625 − 196 = √429 ≈ 20.71).
  • Mixing up the slant height and the lateral edge of a pyramid (problems 4, 7 and 12). Have students trace both on a paper model.
  • Stopping at a face diagonal. In problem 9 the largest face diagonal is only √6,100 ≈ 78.10 cm; the space diagonal is longer than any face diagonal.
  • Rounding the first diagonal before the second step. Keep √50 exact in problem 2.

Support and extension. For support, let students use the facts box and check each answer with the 3D diagonal calculator. For extension, point out that problems 1, 5 and 10 are boxes whose edges and diagonal are all whole numbers (2-3-6-7, 12-15-16-25 and 8-9-12-17); ask students to find another one, then read about Pythagorean quadruples and how the same formula keeps working in four or more dimensions. A hands-on version of problem 9 is activity 10 on the activities page. For mixed 2D practice, use the main Pythagorean theorem worksheets; the teaching guide has more on misconceptions and sequencing.

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