The Pythagorean Theorem in 3D
The longest straight line inside a box runs from one bottom corner to the opposite top corner. Find it by using the Pythagorean theorem twice.
Know these first
This unlocks
- More lessons that build on this are coming.
What you'll be able to do
- Find the space diagonal of a box with d² = l² + w² + h².
- Decide whether a long object fits inside a box.
- Work backward to a missing dimension from the diagonal.
- Find shortest paths across a box's surface by unfolding it flat.
The big idea
Start with the floor of the box. Its diagonal is the hypotenuse of a right triangle with legs l and w, so (floor diagonal)² = l² + w².
Now stand that floor diagonal up against the box's height. They make a second right triangle, and its hypotenuse is the space diagonal: d² = (l² + w²) + h². So d² = l² + w² + h².
If you must stay on the surface - like an ant walking - unfold two faces flat into one rectangle. The shortest path is that rectangle's diagonal. There are three ways to unfold; check them all.
Words you need
- Space diagonal
- A segment from one corner of a box through the inside to the opposite corner.
- Face diagonal
- A diagonal across one flat face of the box.
- Net
- A box unfolded flat into connected rectangles.
Worked examples
Example 1
A box is 3 by 4 by 12. Find its space diagonal.
- Floor diagonal: 3² + 4² = 25, so it's 5.
- Space diagonal: 5² + 12² = 169, so d = 13.
- Shortcut: d² = 3² + 4² + 12² = 169.
Answer: 13
Example 2
A box is 2 by 3 and its space diagonal is 7. Find its height.
- d² = l² + w² + h², so 49 = 4 + 9 + h².
- h² = 49 − 13 = 36.
- h = 6.
Answer: 6
Example 3
Will a 14-inch rod fit inside a box that is 6 by 8 by 10 inches?
- The longest straight line inside is the space diagonal.
- d² = 36 + 64 + 100 = 200, so d = √200 ≈ 14.14 inches.
- 14.14 is more than 14, so the rod fits.
Answer: Yes, it fits
Common mistakes
Using only two dimensions. That gives a face diagonal. The space diagonal uses all three: l² + w² + h².
Forgetting the square root. l² + w² + h² is d², not d.
Checking only one way to unfold. For a path on the surface, try all three unfoldings - the shortest isn't always the obvious one.
Practice: 20 questions
14 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- Find the space diagonal of the box.
- Find the space diagonal of the box.
- A pencil box is 6 by 8 by 24 inches. What is the longest pencil that fits inside? Word problem
- A room is 12 feet long, 16 feet wide and 15 feet tall. How far is it from one floor corner to the opposite ceiling corner? Word problem
- A cube-shaped crate has 4-foot sides. How long is its space diagonal, to the nearest tenth? Word problem
Tier 2 Intermediate One more step to think through
- A box is 4 by 4 and its space diagonal is 9. Find its height.
- Find the space diagonal to the nearest tenth.
- A box's floor is 9 by 12. Find the diagonal of the floor.
- Will a 30-inch umbrella fit inside a suitcase that is 24 by 16 by 8 inches? Word problem
- A box is 12 cm long and 4 cm wide, and its space diagonal is 13 cm. How tall is it? Word problem
- A fish tank is 30 by 12 by 16 inches. How long is its space diagonal, to the nearest tenth of an inch? Word problem
- Will a 3-foot pole fit inside a box that is 2 by 2 by 1 feet? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- An ant walks on the outside of the box from one corner to the opposite corner. Find the shortest path, to the nearest tenth.
- Can a 50 cm baton fit inside a box that is 40 by 30 by 10 cm? Word problem
- A shipping box has a 9 by 12 inch floor and is 8 inches tall. What is the longest straight rod that fits inside? Word problem
- A 26-inch rod fits exactly from corner to corner in a box that is 6 inches by 8 inches. How tall is the box? Word problem
- A party hat is a cone with radius 5 cm and height 12 cm. How long is its slanted side? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- An ant crawls on the outside of a box that is 6 by 4 by 3 inches, from one corner to the farthest corner. What is its shortest path, to the nearest tenth? Word problem
- A square pyramid has a 10 m base and is 12 m tall. How long is each slanted edge from the top to a base corner, to the nearest tenth? Word problem
- A spider walks on the outside of a cube with 3-inch sides, from one corner to the opposite corner. What is its shortest path, to the nearest tenth? Word problem
Answers
Tier 1 Foundations
- 7. d² = 2² + 3² + 6² = 49, so d = 7.
- 3. d² = 1² + 2² + 2² = 9, so d = 3.
- 26. d² = 6² + 8² + 24² = 676, so d = 26.
- 25. d² = 12² + 16² + 15² = 625, so d = 25.
- 4√3 ≈ 6.9. d² = 4² + 4² + 4² = 48, so d = 4√3 ≈ 6.9.
Tier 2 Intermediate
- 7. h² = 9² − 4² − 4² = 49, so h = 7.
- √110 ≈ 10.5. d² = 5² + 6² + 7² = 110, so d = √110 ≈ 10.5.
- 15. c² = 9² + 12² = 225, so c = 15.
- No, it doesn't fit. The longest straight line inside is the space diagonal: √(24² + 16² + 8²) ≈ 29.93, which is shorter than 30.
- 3. h² = 13² − 12² − 4² = 9, so h = 3.
- 10√13 ≈ 36.1. d² = 30² + 12² + 16² = 1300, so d = 10√13 ≈ 36.1.
- Yes, it fits. The longest straight line inside is the space diagonal: √(2² + 2² + 1²) = 3, which is at least 3.
Tier 3 Advanced
- 2√13 ≈ 7.2. Unfold two faces into one flat rectangle; the path is its diagonal. Of the three ways to unfold, the 4 by (2 + 4) rectangle gives the shortest: 2√13 ≈ 7.2.
- Yes, it fits. The longest straight line inside is the space diagonal: √(40² + 30² + 10²) ≈ 50.99, which is at least 50.
- 17. d² = 9² + 12² + 8² = 289, so d = 17.
- 24. h² = 26² − 6² − 8² = 576, so h = 24.
- 13. c² = 5² + 12² = 169, so c = 13.
Tier 4 Challenge
- √85 ≈ 9.2. Unfold two faces into one flat rectangle; the path is its diagonal. Of the three ways to unfold, the 6 by (4 + 3) rectangle gives the shortest: √85 ≈ 9.2.
- about 13.9 m. The base's diagonal is 10√2, so the center is 5√2 from each corner. The edge is √(12² + (5√2)²) = √194 ≈ 13.9 m.
- 3√5 ≈ 6.7. Unfold two faces into one flat rectangle; the path is its diagonal. Of the three ways to unfold, the 3 by (3 + 3) rectangle gives the shortest: 3√5 ≈ 6.7.
Stuck on this topic?
Trouble here often traces back to an earlier skill. Our free check-in finds the exact gap. A tutor then works on it with your child.
Find gaps free More practice