Grade 10 · Core Mathematics · Lesson 8 of 11

Connecting Linear, Surface Area, and Volumetric Scale Factors in Similar Solids

A Grade 10 Core Mathematics CBE lesson from the sub-strand Similarity and Enlargement. In it, learners calculate volume scale factor from linear measurements of similar solids.

Strand Measurements and GeometrySub-strand Similarity and EnlargementLesson code 10CM-21-L840 minutes · 2 slides

KICD learning outcome (d)

Relate linear scale factor, area scale factor, and volume scale factor in enlargements.

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

Calculate volume scale factor from linear measurements of similar solids.
Determine linear and area scale factors from volume ratios using cube roots and squaring.
Solve multi-step contextual problems involving lengths, surface areas, and volumes of similar containers and figures.

Key vocabulary

Volume Scale Factor: The constant numerical ratio obtained by comparing capacities of two similar 3D shapes.; Cubing a linear multiplier of 2 yields a volume multiplier of 8.
Area Scale Factor: The ratio comparing corresponding surface areas or flat regions of similar geometric solids.; Squaring a linear multiplier of 3 yields an area multiplier of 9.

Lesson preview

Slide 1 · Deriving Scale Factors from Dimensions and Volume RatiosModel · 18 min

Figure 1: A flowchart demonstrating how to calculate the linear scale factor and area scale factor using the volumes of two similar pots.

When working with similar three-dimensional figures, the linear scale factor (LSF) relates corresponding lengths. The area scale factor (ASF) equals ${LSF}^{2}$ , whereas the volume scale factor (VSF) equals ${LSF}^{3}$ . Given a cubic ratio, taking its cube root determines the linear multiplier. Squaring that linear multiplier then provides the surface ratio, allowing computation of unknown surface areas or dimensions across matching solids.

The linear scale factor is derived from the ratio of any two matching linear measurements.
Cubing the linear scale factor yields the volume scale factor relating capacities of matching solids.
Taking the cube root of the volume multiplier restores the linear multiplier between matching figures.
Squaring the restored linear scale factor gives the area scale factor needed for surface area calculations.

Worked example. Express the volume ratio as

VSF = \frac{216}{125}

. Take the cube root to determine the linear ratio:

LSF = \sqrt[3]{\frac{216}{125}} = \frac{6}{5}

. Square the linear ratio to determine the area scale factor:

ASF = {(\frac{6}{5})}^{2} = \frac{36}{25}

. Multiply smaller surface area by area scale factor:

{Surface Area}_{larger} = 175 \times \frac{36}{25} = 252 {cm}^{2}

. Answer: Linear scale factor is $\frac{6}{5}$ , area scale factor is $\frac{36}{25}$ , and larger surface area is $252 {cm}^{2}$ ..

Check question: Two similar spherical balloons have radii of $7.5 cm$ and $10 cm$ . Compute the linear scale factor of enlargement from the smaller balloon to the larger balloon, and the corresponding volume scale factor.

Linear scale factor = $\frac{4}{3}$ , Volume scale factor = $\frac{64}{27}$
Linear scale factor = $\frac{4}{3}$ , Volume scale factor = $\frac{16}{9}$
Linear scale factor = $\frac{3}{4}$ , Volume scale factor = $\frac{27}{64}$
Linear scale factor = $\frac{4}{3}$ , Volume scale factor = $4$

Show answer

Linear scale factor = $\frac{4}{3}$ , Volume scale factor = $\frac{64}{27}$ . The linear scale factor (LSF) for the enlargement is the larger radius divided by the smaller radius: $\frac{10}{7.5} = \frac{100}{75} = \frac{4}{3}$ . Cubing this linear multiplier yields the volume scale factor (VSF): $VSF = {(\frac{4}{3})}^{3} = \frac{64}{27}$ .

Slide 2 · Solving Multi-Step Area and Volume Scaling ProblemsPractice · 22 min

Figure 2: A step-by-step diagram showing how to find an unknown surface area using the volumes of two similar cylindrical drums.

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Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Connecting Linear, Surface Area, and Volumetric Scale Factors in Similar Solids

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