Grade 10 · Core Mathematics · Lesson 5 of 11

Calculating Area and Volume Scale Factors for Similar Figures

A Grade 10 Core Mathematics CBE lesson from the sub-strand Similarity and Enlargement. In it, learners work out the area scale factor between matching plane shapes by comparing their surface areas.

Strand Measurements and GeometrySub-strand Similarity and EnlargementLesson code 10CM-21-L540 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

Work out the area scale factor between matching plane shapes by comparing their surface areas.
Determine the volume scale factor between matching solids by comparing their calculated volumes.

Key vocabulary

Area scale factor: The ratio comparing the area of an enlarged image figure to the original figure.; Dividing an area of $180 {cm}^{2}$ by $20 {cm}^{2}$ gives an area scale factor of $9$ .
Volume scale factor: The ratio comparing the volume of an enlarged solid to the original solid.; Dividing a volume of $12320 {cm}^{3}$ by $1540 {cm}^{3}$ yields a volume scale factor of $8$ .

Lesson preview

Slide 1 · Area Scale Factor in Similar Plane FiguresOpening · 18 min

Figure 1: Two similar right-angled triangles, P and Q, with their base lengths, heights, and areas labelled.

When plane figures are geometrically similar, their corresponding linear dimensions share a common enlargement ratio. The area scale factor is defined as the ratio of the area of the enlarged image figure to the area of the original object figure. By calculating the actual surface areas of two similar figures and taking their ratio, we determine how many times the surface area expands or shrinks under an enlargement. Use Figure 1 to calculate the ratio of the enlarged triangle's area to the original's.

Similar plane figures have identical shape but proportional side lengths and distinct surface areas.
The area scale factor measures the exact ratio of the enlarged area to the original area.
Area of a right triangle is computed using the standard formula half times base times height.
Dividing the area of the image figure by the original figure yields the area scale factor.

Worked example. Calculate area of triangle P:

{Area}_{P} = \frac{1}{2} \times 8 \times 5 = 20 {cm}^{2}

. Calculate area of triangle Q:

{Area}_{Q} = \frac{1}{2} \times 24 \times 15 = 180 {cm}^{2}

. Divide the area of Q by the area of P:

ASF = \frac{{Area}_{Q}}{{Area}_{P}} = \frac{180}{20} = 9

. Answer: $Area scale factor = 9$ .

Check question: Rectangle A has dimensions $5 cm$ by $2 cm$ . Similar rectangle B has dimensions $10 cm$ by $4 cm$ . Determine the area scale factor relating rectangle B to rectangle A.

$2$
$4$
$10$
$40$

Show answer

$4$ . The area of rectangle A is $5 \times 2 = 10 {cm}^{2}$ . The area of rectangle B is $10 \times 4 = 40 {cm}^{2}$ . The area scale factor is calculated by dividing the area of B by the area of A: $40 \div 10 = 4$ .

Slide 2 · Volume Scale Factor of Similar SolidsModel · 22 min

Figure 2: Two similar cylindrical containers, C1 and C2, displaying their respective radii, heights, and volumes.

Teach the full lesson on the projector

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

See the school projector WhatsApp us

Calculating Area and Volume Scale Factors for Similar Figures

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