Grade 10 · Core Mathematics · Lesson 9 of 11

Applications of Similarity and Enlargement in Real Life

A Grade 10 Core Mathematics CBE lesson from the sub-strand Similarity and Enlargement. In it, learners determine missing linear dimensions, surface areas, and volumes of real-life objects using scale models and scale factors.

Strand Measurements and GeometrySub-strand Similarity and EnlargementLesson code 10CM-21-L940 minutes · 2 slides

KICD learning outcome (e)

Apply similarity and enlargement to real-life situations.

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

Determine missing linear dimensions, surface areas, and volumes of real-life objects using scale models and scale factors.
Apply the relationships between linear ratio $k$ , area scale factor $k^{2}$ , and volume factor $k^{3}$ to practical contexts.

Key vocabulary

Linear Scale Factor: The ratio of any linear dimension in a model to the corresponding actual dimension.; A scale of 1 : 50 gives a linear scale factor of 0.02 for the classroom model.
Area Scale Factor: The ratio of the area of a model surface to its corresponding actual area.; The area scale factor for a 1 : 50 model is equal to 1 over 2500.

Lesson preview

Slide 1 · Modelling Real Structures with Scale NetsModel · 18 min

Figure 1: A three-dimensional rectangular hall with its dimensions alongside an unfolded two-dimensional scale net, showing the scale factors.

Applying geometric similarity to real structures like school halls or grain stores involves taking their physical measurements. Learners construct two-dimensional unfolded patterns on manila paper to build accurate miniature structures. The linear scale factor $k$ relates dimensions of the miniature pattern to the real structure. Squaring $k$ determines the area scale factor $k^{2}$ for covering surfaces, while cubing $k$ yields the volume scale factor $k^{3}$ for internal spatial capacity. Use Figure 1 to visualise the relationship between a full-sized building and its unfolded miniature pattern.

Measuring physical dimensions allows us to create accurate reduced or enlarged scale models of real structures.
The linear scale factor $k$ relates corresponding measured linear dimensions of models and actual objects.
Total surface area of a miniature pattern scales by the area factor $k^{2}$ relative to actual surfaces.
Volume of space inside a scaled structure changes by the volume factor $k^{3}$ relative to actual volume.

Worked example. Actual surface area = Floor & Ceiling + 4 Walls =

2 (12 \times 10) + 2 (12 \times 4 + 10 \times 4) = 240 + 176 = 416 m^{2}

. Linear scale factor

k = \frac{1}{50}

. Area scale factor

ASF = k^{2} = {(\frac{1}{50})}^{2} = \frac{1}{2500}

. Replica surface area =

416 \times \frac{1}{2500} = 0.1664 m^{2}

. In

{cm}^{2}

0.1664 \times 10 000 = 1664 {cm}^{2}

. Actual volume =

12 \times 10 \times 4 = 480 m^{3}

. Volume scale factor

VSF = k^{3} = {(\frac{1}{50})}^{3} = \frac{1}{125 000}

. Model volume =

480 \times \frac{1}{125 000} = 0.00384 m^{3}

. In

{cm}^{3}

0.00384 \times 1 000 000 = 3840 {cm}^{3}

. Answer: (a) $1664 {cm}^{2}$ , (b) $3840 {cm}^{3}$ .

Check question: A miniature grain store is constructed using a linear ratio of 1 : 50. If the internal volume of the actual grain store is $250 m^{3}$ , calculate the capacity of the miniature store in ${cm}^{3}$ .

$20 {cm}^{3}$
$2000 {cm}^{3}$
$100 000 {cm}^{3}$
$5 000 000 {cm}^{3}$

Show answer

$2000 {cm}^{3}$ . The linear ratio $k = \frac{1}{50}$ , so the volume scale factor is ${(\frac{1}{50})}^{3} = \frac{1}{125 000}$ . Converting $250 m^{3}$ to ${cm}^{3}$ gives $250 \times 1 000 000 = 250 000 000 {cm}^{3}$ . Dividing this actual volume by $125 000$ yields $2000 {cm}^{3}$ .

Slide 2 · Solving Scale Model Problems Using Scale FactorsPractice · 22 min

Figure 2: A standard oil container and a giant marketing display model demonstrating how base area and capacity scale.

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

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