DWG NO. 11.5 — Lesson 5 of 5

Similar Solids

Unit 11: Surface Area and Volume · ~15–30 min

Objective

Use the scale factor between two similar solids to find ratios of their surface areas and volumes.

What makes solids similar

Two solids are similar if one is an enlargement or reduction of the other — same shape, every corresponding linear measurement (edge, radius, height, slant height) in the same ratio. That ratio is the scale factor, k. Because a solid's surface is made of two–dimensional faces and its interior is three–dimensional space, area and volume don't scale the same way length does.

The scale factor rule

If two similar solids have scale factor k (ratio of corresponding lengths), then:

Lengths (edges, radii, heights, diagonals) are in ratio k.
Surface areas are in ratio k² — because area is a product of two lengths.
Volumes are in ratio k³ — because volume is a product of three lengths.

This is the same "squares for area, cubes for volume" pattern you may have seen with scale drawings and dilations — similar solids just extend it into three dimensions.

Scale factor k = 5/2 between two cubes means surface areas are in ratio (5/2)² and volumes in ratio (5/2)³.

Worked Example 1 · Finding both ratios

ProblemTwo similar cylinders have radii 3 cm and 9 cm. Find the ratio of their surface areas and the ratio of their volumes.

1.Scale factor: k = 9/3 = 3.

2.Surface area ratio: k² = 3² = 9, so the larger cylinder's surface area is 9 times the smaller's.

3.Volume ratio: k³ = 3³ = 27, so the larger cylinder's volume is 27 times the smaller's.

Surface area ratio = 9 : 1 • Volume ratio = 27 : 1

Worked Example 2 · Working backward from volume

ProblemTwo similar pyramids have volumes 64 in³ and 216 in³. Find the scale factor and the ratio of their surface areas.

1.Volume ratio: 216/64 = 27/8. Since k³ = 27/8, take the cube root: k = 3/2.

2.Surface area ratio: k² = (3/2)² = 9/4.

Scale factor = 3 : 2 • Surface area ratio = 9 : 4

Guided practice

1.Two similar spheres have radii 4 ft and 12 ft. Find the scale factor and the ratio of their surface areas.
Show answer
k = 12/4 = 3. Surface area ratio = k² = 9, so 9 : 1.
2.Using the spheres from Problem 1, find the ratio of their volumes.
Show answer
Volume ratio = k³ = 3³ = 27, so 27 : 1.
3.Two similar cones have a volume ratio of 8 : 1. Find their scale factor.
Show answer
k³ = 8, so k = ∛8 = 2. Scale factor is 2 : 1.
4.A small storage bin and a large geometrically similar bin have a surface area ratio of 25 : 4. If the large bin holds 500 ft³, roughly how much does the small bin hold?
Show answer
k² = 25/4, so k = 5/2 (large : small). Volume ratio = k³ = 125/8. Small bin's volume = 500 ÷ (125/8) = 500(8/125) = 32 ft³.
5.Explain why doubling every dimension of a solid more than doubles the amount of material needed to build it and more than doubles the space it holds.
Show answer
Doubling every length gives a scale factor k = 2. Surface area (and material needed) scales by k² = 4, and volume (space held) scales by k³ = 8 — both grow much faster than the linear factor of 2 because they depend on products of two or three lengths, not just one.

← Lesson 11.4 Unit 11 Overview Review & Practice Day →