Grade 10 · Core Mathematics · Lesson 6 of 11

Determining Area and Volume Scale Factors

A Grade 10 Core Mathematics CBE lesson from the sub-strand Similarity and Enlargement. In it, learners determine the volume scale factor of similar solids by computing and comparing their volumes.

Strand Measurements and GeometrySub-strand Similarity and EnlargementLesson code 10CM-21-L640 minutes · 2 slides

KICD learning outcome (c)

Determine the area and volume scale factor of different figures and solids.

By the end of this lesson, learners should be able to

Determine the volume scale factor of similar solids by computing and comparing their volumes
Calculate the area scale factor of similar plane figures by comparing their surface areas
Apply area and volume scale factors to solve mathematical and real-life scaling problems

Key vocabulary

Volume scale factor: The ratio obtained by dividing the volume of one solid by that of a similar solid.; The volume scale factor of a $2 cm$ cube to a $6 cm$ cube is $\frac{1}{27}$ .
Area scale factor: The ratio obtained by dividing the area of one figure by that of a similar figure.; A $20 {cm}^{2}$ figure enlarged to $80 {cm}^{2}$ has an area scale factor of $4$ .

Lesson preview

Slide 1 · Volume Scale Factor of Similar SolidsModel · 18 min

Figure 1: Two similar spheres with radii of 3 cm and 6 cm, showing their calculated volumes and the resulting volume scale factor.

When comparing two similar three-dimensional solids, the ratio of their volumes defines the volume scale factor. To calculate this ratio, we compute the volume of both solids using their respective linear dimensions, such as radius, height, or side length, and then divide the volume of one solid by the volume of the corresponding similar solid. The ratio of the volumes of two similar solids is the volume scale factor. Use Figure 1 to see how the volume scale factor is calculated from the volumes of similar solids.

Similar solids have corresponding linear dimensions that are enlarged or reduced by a constant ratio.
The ratio of the volumes of two similar solids is the volume scale factor.
Computing solid volumes directly allows us to determine the volume scale factor between two shapes.
The volume scale factor compares the 3D space occupied by an object and its similar image.

Worked example. 1. Calculate volume of smaller sphere:

V_{1} = \frac{4}{3} \times \frac{22}{7} \times 3^{3} = \frac{4}{3} \times \frac{22}{7} \times 27 = \frac{792}{7} = 113 \frac{1}{7} {cm}^{3}

. 2. Calculate volume of larger sphere:

V_{2} = \frac{4}{3} \times \frac{22}{7} \times 6^{3} = \frac{4}{3} \times \frac{22}{7} \times 216 = \frac{6336}{7} = 905 \frac{1}{7} {cm}^{3}

. 3. Divide smaller volume by larger volume to find the volume scale factor:

\frac{V_{1}}{V_{2}} = \frac{792 / 7}{6336 / 7} = \frac{792}{6336} = \frac{1}{8}

. Answer: The volume scale factor is $\frac{1}{8}$ ..

Check question: Two similar cylinders have volumes of $50 {cm}^{3}$ and $400 {cm}^{3}$ . What is the volume scale factor of the larger cylinder to the smaller cylinder?

$8$
$\frac{1}{8}$
$2$
$350$

Show answer

$8$ . To find the volume scale factor of the larger cylinder to the smaller cylinder, divide the larger volume by the smaller volume: $\frac{400 {cm}^{3}}{50 {cm}^{3}} = 8$ .

Slide 2 · Determining Area and Volume Scale Factors in PracticePractice · 22 min

Figure 2: A pair of rectangular tiles and a pair of cubes, demonstrating the calculation of area and volume scale factors from corresponding measurements.

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Homework and the other 10 lessons in this sub-strand

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Determining Area and Volume Scale Factors

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Similarity and Enlargement