Mathematics
Grade 8
15 min
Volume and surface area of similar solids
Volume and surface area of similar solids
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify similar three-dimensional solids.
Determine the scale factor between two similar solids.
Explain the relationship between the scale factor and the ratio of surface areas of similar solids.
Explain the relationship between the scale factor and the ratio of volumes of similar solids.
Calculate the surface area of a similar solid given the scale factor and the original surface area.
Calculate the volume of a similar solid given the scale factor and the original volume.
Have you ever wondered how architects scale up a small model of a building to its actual size, or how toy manufacturers shrink a real object into a miniature replica? 📏🏠
In this lesson, you'll discover the fascinating mathematical relationships between the dimensions, surface...
2
Key Concepts & Vocabulary
TermDefinitionExample
Similar SolidsThree-dimensional figures that have the same shape but different sizes. All corresponding linear dimensions (like side lengths, radii, heights) are proportional.A small cube and a large cube are similar solids. A small sphere and a large sphere are similar solids. A rectangular prism and a rectangular pyramid are NOT similar solids.
Corresponding Sides/DimensionsSides, edges, radii, or heights that are in the same relative position on two different similar solids.If you have two similar rectangular prisms, the length of the first prism corresponds to the length of the second prism, the width to the width, and the height to the height.
Scale Factor (k)The ratio of any corresponding linear dimension of two similar solids. It tells you how much larger or s...
3
Core Formulas
Ratio of Corresponding Linear Dimensions
$$\frac{\text{Dimension}_1}{\text{Dimension}_2} = k$$
The ratio of any corresponding linear measurements (like height, radius, side length) of two similar solids is the scale factor, 'k'.
Ratio of Surface Areas of Similar Solids
$$\frac{\text{Surface Area}_1}{\text{Surface Area}_2} = k^2$$
The ratio of the surface areas of two similar solids is equal to the square of their scale factor, 'k'.
Ratio of Volumes of Similar Solids
$$\frac{\text{Volume}_1}{\text{Volume}_2} = k^3$$
The ratio of the volumes of two similar solids is equal to the cube of their scale factor, 'k'.
5 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
A large block of cheese in the shape of a cube with side length 10 cm is cut into smaller, similar cubes, each with a side length of 2 cm. How many small cubes can be made?
A.25
B.125
C.50
D.5
Challenging
Two similar solids have a volume ratio of 8:27. The surface area of the larger solid is 225 cm². What is the surface area of the smaller solid?
A.67.5 cm²
B.150 cm²
C.100 cm²
D.50 cm²
Challenging
The weight of a solid object is directly proportional to its volume. A small, bronze pyramid weighs 4 lbs and has a height of 5 inches. A similar, larger bronze pyramid has a height of 15 inches. What is the weight of the larger pyramid?
A.108 lbs
B.12 lbs
C.36 lbs
D.60 lbs
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free