Grade 10 · Core Mathematics · Lesson 7 of 11

Connecting Length, Surface Area, and Volume Ratios

A Grade 10 Core Mathematics CBE lesson from the sub-strand Similarity and Enlargement. In it, learners deduce that for similar figures, area scale factor equals the square of the linear scale factor.

Strand Measurements and GeometrySub-strand Similarity and EnlargementLesson code 10CM-21-L740 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

Deduce that for similar figures, area scale factor equals the square of the linear scale factor.
Deduce that for similar solids, volume scale factor equals the cube of the linear scale factor.
Determine surface area and volume ratios using length measurements of similar shapes.

Key vocabulary

Area scale factor: The constant ratio of the surface areas of two mathematically similar figures.; Squaring the linear scale factor of three gives an area scale factor of nine.
Volume scale factor: The constant ratio comparing the volumes of two geometrically similar solid figures.; Cubing the linear scale factor of two gives a volume scale factor of eight.
Linear scale factor: The ratio of any linear measurement of an enlarged figure to the corresponding measurement.; Dividing the new height by the original height yields the linear scale factor.

Lesson preview

Slide 1 · Establishing the Relationship Between Scale FactorsOpening · 18 min

Figure 1: Two pairs of similar 3D solids and a completed table showing the calculation of their linear, area, and volume scale factors.

When a solid shape is enlarged, every linear dimension changes by a linear scale factor, denoted as $k$ . Comparing corresponding surface areas shows that area expands by $k^{2}$ , known as the area scale factor. Similarly, comparing volumes reveals that volume increases by $k^{3}$ , termed the volume scale factor. This fundamental relationship allows us to determine surface areas and volumes directly from length ratios without recalculating individual dimensions.

The linear scale factor $k$ is the ratio of any two corresponding lengths on similar shapes.
The area scale factor equals the square of the linear scale factor, so $ASF = k^{2}$ .
The volume scale factor equals the cube of the linear scale factor, so $VSF = k^{3}$ .
Comparing ratios in a structured table confirms that area depends on $k^{2}$ and volume on $k^{3}$ .

Worked example. Calculate the linear scale factor by simplifying the ratio of corresponding lengths:

LSF = \frac{15}{6} = \frac{5}{2}

. Use the relationship that the area scale factor is the square of the linear scale factor:

ASF = {LSF}^{2}

. Square the fraction:

ASF = {(\frac{5}{2})}^{2} = \frac{25}{4}

. Answer: Linear scale factor = $\frac{5}{2}$ and Area scale factor = $\frac{25}{4}$ .

Check question: Two similar triangular wooden blocks have corresponding base lengths of $4 cm$ and $12 cm$ . What is the area scale factor?

$3$
$6$
$9$
$27$

Show answer

$9$ . The linear scale factor is $\frac{12}{4} = 3$ . Squaring this linear ratio gives the area scale factor as $3^{2} = 9$ .

Slide 2 · Calculating Scale Factors for Similar 2D and 3D ShapesPractice · 22 min

Figure 2: A diagram of two similar spherical tanks demonstrating how to calculate linear, area, and volume scale factors.

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This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

Both slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Connecting Length, Surface Area, and Volume Ratios

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