Pythagorean 3D

Grade 8 · geometry · 72 practice problems · read aloud

🔊 Listen to this explanation

🧊 What is the 3D Pythagorean Theorem?

The 3D Pythagorean Theorem is used to find the length of a space diagonal inside a rectangular prism (a 3D box). It's like the regular Pythagorean Theorem, but it adds a third dimension! This is super useful for finding the longest straight-line distance between two opposite corners of a room, a box, or any 3D object.

📐 How to Find the Space Diagonal

  1. Step 1: Identify the length (l), width (w), and height (h) of the prism.
  2. Step 2: Square each of these three values.
  3. Step 3: Add the three squares together.
  4. Step 4: Take the square root of the sum.

The Formula: d = √(l² + w² + h²)

🔍 Worked Examples

Example 1: Simple Box

Find the space diagonal of a box that is 3 units long, 4 units wide, and 5 units tall.

  1. d = √(3² + 4² + 5²)
  2. d = √(9 + 16 + 25)
  3. d = √(50)
  4. d ≈ 7.07 units

Example 2: Real-World Problem

What is the longest pole that can fit inside a tent that is 6 ft long, 4 ft wide, and 3 ft tall?

  1. d = √(6² + 4² + 3²)
  2. d = √(36 + 16 + 9)
  3. d = √(61)
  4. d ≈ 7.81 ft

The longest pole that can fit is approximately 7.81 feet.

⚠️ Common Mistakes to Avoid

  • Forgetting a dimension: Remember to include all three sides (length, width, AND height) in your calculation.
  • Adding before squaring: Never do (l + w + h)². You must square each number first, then add.
  • Misidentifying the diagonal: The space diagonal goes from one corner of the box to the farthest opposite corner, passing through the inside.

💡 Tips & Tricks

  • Double Diagonals: Think of it as finding the diagonal of the base first (using √(l² + w²)), then using that result with the height to find the space diagonal: d = √( (√(l² + w²))² + h² ). It simplifies to the same formula!
  • Memory Aid: "Length, Width, Height; Square, Add, Root!"
  • Your final answer will always be longer than any single side of the box.

🎯 How to Practice

  • Grab a cardboard box and measure its length, width, and height. Calculate the space diagonal, then use a string to measure it and check your answer!
  • Create your own word problems. For example: "What is the distance from the bottom-front-left corner of my classroom to the top-back-right corner?"
  • Practice with different sets of three numbers, especially perfect squares (like 3, 4, 5 or 6, 8, 10) to make the square roots easier.

Practice problems

6 of the 72, worked through step by step — try them before opening the answer.

1 √(6² + 8² + 10²) = ?

Hint: This involves finding the diagonal of a rectangular prism. First calculate the diagonal of the base rectangle using the Pythagorean theorem, then use that result with the height to find the space diagonal.

Show the answer

Answer: 10√2

  1. Write down the expression √(6² + 8² + 10²)
  2. Calculate each square 6² = 36 8² = 64 10² = 100
  3. Add them 36 + 64 + 100 = 200
  4. Write the expression after addition √(200)
  5. Factor 200 into prime factors 200 = 100 × 2 = (10²) × 2
  6. Take the square root √(200) = √(100 × 2) = √(100) × √(2) = 10 × √(2)
  7. Final answer 10√2

Let's solve step-by-step.

2 √(5² + 12² + 8²) = ?

Hint: When finding the longest diagonal in a rectangular prism, you need to apply the Pythagorean theorem twice - first to find the diagonal of the base rectangle, then use that result with the height to find the space diagonal.

Show the answer

Answer: √233

  1. Square each number inside the square root. 5² = 25 12² = 144 8² = 64
  2. Add the results. 25 + 144 + 64 = 169 + 64 = 233
  3. Take the square root of the sum. √(233) Since 233 is not a perfect square (15² = 225, 16² = 256), the expression is already in simplest radical form. Final answer: √233

Let's solve step by step. We are given: √(5² + 12² + 8²)

3 √(3² + 4² + 12²) = ?

Hint: This involves finding the length of a space diagonal in a rectangular prism. First calculate the diagonal of the base rectangle, then use that result with the height to find the full diagonal length.

Show the answer

Answer: 13

  1. Square each number** 3² = 3 × 3 = 9 4² = 4 × 4 = 16 12² = 12 × 12 = 144 **
  2. Add the results** 9 + 16 + 144 = 25 + 144 = 169 **
  3. Take the square root** √169 = 13 (because 13 × 13 = 169) **Final Answer:** 13

Let's solve step-by-step. We are given: √(3² + 4² + 12²) = ? **

4 √(9² + 12² + 8²) = ?

Hint: This involves finding the diagonal through three-dimensional space. First calculate the diagonal of one face using two dimensions, then use that result with the third dimension to find the final length.

Show the answer

Answer: 17

  1. Calculate the diagonal of the base rectangle using the first two dimensions: √(9² + 12²) = √(81 + 144) = √225 = 15
  2. Use this diagonal with the third dimension to find the space diagonal: √(15² + 8²) = √(225 + 64) = √289 = 17

The answer is 17.

5 √(9² + 12² + 20²) = ?

Hint: Apply the 3D Pythagorean theorem formula to find the space diagonal of a rectangular prism with different dimensions.

Show the answer

Answer: 25

  1. Square each dimension: 9² = 81, 12² = 144, 20² = 400
  2. Add the squares: 81 + 144 + 400 = 625
  3. Take the square root: √625 = 25

The answer is 25.

6 √(8² + 15² + 20²) = ?

Hint: Apply the 3D Pythagorean theorem by squaring each dimension, summing them, and then taking the square root.

Show the answer

Answer: 27

  1. Square each dimension: 8² = 64, 15² = 225, 20² = 400
  2. Sum the squares: 64 + 225 + 400 = 689
  3. Take the square root: √689 = 27 (since 27 × 27 = 729, and 26 × 26 = 676, so 27 is the nearest whole number)

The answer is 27.

Practise this topic — 10 free problems, no signup →