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Pythagorean Theorem Applications | Lesson 2 of 2

The Theorem in Three Dimensions

Lesson 2 of 2: Space Diagonals and Cones

In this lesson:

  • Find the space diagonal of a box
  • Find a cone's height from its slant height
Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

What You Will Be Able to Do

By the end of this lesson, you should be able to:

  1. Find the space diagonal of a rectangular box
  2. Find a cone's height using a cross-section
  3. Apply the theorem twice in sequence for one answer
Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

The Longest Rod in a Box

Rectangular box 3 by 4 by 12 in perspective, dashed space diagonal corner to corner

What is the longest straight rod that fits inside this 3-by-4-by-12 box?

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Start Flat: The Base Diagonal

Bottom 3 by 4 face shown flat, diagonal of length 5 marked as a right triangle

The bottom face is a 3-by-4 rectangle: .

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Stand a Triangle Up Inside the Box

Box showing the standing right triangle: base diagonal 5, vertical height 12, space diagonal 13

The base diagonal (5), the height (12), and the space diagonal form a right triangle.

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Finish Off the Space Diagonal

The standing triangle has legs 5 and 12:

The longest rod is 13 ft.

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

One Formula for the Whole Job

The two steps combine into a single shortcut:

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

A Room With an Irrational Diagonal

A room is 10 m by 8 m by 3 m:

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Your Turn: A Smaller Box

Find the space diagonal of a box that is 2 by 3 by 6.

Use the formula or the two-step method. Try it before advancing.

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Moving From Boxes to Cones

A box's triangle was hidden but flat-based. A cone has no such face.

To find the right triangle, we have to slice the cone open.

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Slicing Reveals the Right Triangle

Cone with a vertical cross-section through the apex exposing a right triangle: slant height as hypotenuse, radius and height as legs

The slant height is the hypotenuse; the radius and the height are the legs.

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Find the Height of the Cone

Slant height 15, radius 9:

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

What If: A Straw in a Cup

Cylindrical cup with a straw resting diagonally, radius and height as the two legs

A straw rests corner-to-corner in a cup. Which lengths are the legs?

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Find a Space Diagonal on Your Own

A crate is 6 by 6 by 2 feet.

Find the longest rod that fits inside. Show both steps — no help.

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Two 3-D Traps to Avoid

⚠️ Don't stop at step one — the base diagonal isn't the final answer
⚠️ In a cone, use the radius, not the diameter, as the leg

Grade 8 Math | 8.G.B.7
Pythagorean Theorem Applications | Lesson 2 of 2

Three Dimensions Are Two, Stacked

✓ Find one length, then use it as a leg of the next triangle
✓ A box uses two triangles; a cone uses a cross-section
⚠️ Don't stop at the intermediate length; mind radius versus diameter

Next: the theorem becomes distance on the coordinate plane.

Grade 8 Math | 8.G.B.7