Mathematics
Grade 10
15 min
Surface area and volume of similar solids
Surface area and volume of similar solids
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify if two solids are similar by comparing the ratios of their corresponding linear measures.
Determine the scale factor between two similar solids.
State and apply the relationship between the scale factor and the ratio of surface areas.
State and apply the relationship between the scale factor and the ratio of volumes.
Calculate a missing surface area or volume of a similar solid using the appropriate ratio.
Determine the scale factor of two similar solids given the ratio of their surface areas or volumes.
How can a tiny, detailed model of a skyscraper be created so perfectly? How much less 'surface area' (glass) and 'volume' (concrete) does the model have compared to the real thing? 🏙️
This tutorial explores the powerful relat...
2
Key Concepts & Vocabulary
TermDefinitionExample
Similar SolidsTwo solids of the same type are similar if the ratio of their corresponding linear measures (like height, radius, or side length) is constant.A rectangular prism with dimensions 2x3x4 is similar to one with dimensions 4x6x8 because every corresponding ratio is 1:2.
Scale Factor (k)The constant ratio of corresponding linear measures of two similar solids. It's often written as a fraction a/b.If a small cone has a radius of 3 cm and a similar large cone has a radius of 12 cm, the scale factor from small to large is 12/3 = 4.
Linear MeasureA one-dimensional measurement of a solid.Length, width, height, radius, slant height, or the perimeter of a base.
Ratio of Surface AreasThe ratio of the total surface areas of two similar solids. It is equal to the...
3
Core Formulas
Surface Area Ratio Theorem
If the scale factor of two similar solids is a:b, then the ratio of their surface areas is a²:b².
Use this rule to find a missing surface area when you know the scale factor and one of the surface areas. Square the scale factor to get the area ratio.
Volume Ratio Theorem
If the scale factor of two similar solids is a:b, then the ratio of their volumes is a³:b³.
Use this rule to find a missing volume when you know the scale factor and one of the volumes. Cube the scale factor to get the volume ratio.
4 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
Two similar solids have volumes of 54 cm³ and 128 cm³. The surface area of the smaller solid is A. What is the surface area of the larger solid in terms of A?
A.(4/3)A
B.(256/81)A
C.(64/27)A
D.(16/9)A
Challenging
A solid gold statue has a height of 10 cm and a mass of 1.5 kg. A similar, larger statue is to be made from a cheaper, lighter metal that has half the density of gold. If the larger statue has a height of 25 cm, what will its mass be?
A.11.72 kg
B.12.50 kg
C.23.44 kg
D.18.75 kg
Challenging
Two similar spheres have radii in the ratio 2:3. The sum of their volumes is 945π cm³. What is the volume of the smaller sphere?
A.216π cm³
B.108π cm³
C.315π cm³
D.729π cm³
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free