Scaling areas and volumes

Two 2D shapes are similar with a scale factor of $k = 3$.

a) By what number are the lengths multiplied?

b) By what number is the area multiplied?

c) If they were 3D solids, by what number would the volume be multiplied?

Two squares are similar with scale factor $k = 2$. If the small one has a side length of $5$ cm:

a) What is the side length of the large one?

b) What are the areas of the two squares?

A photo measuring $40\ \text{cm}^2$ is enlarged on a photocopier to $150\%$. What is the area of the enlargement?

An artist paints a mural from a $30\ \text{cm} \times 20\ \text{cm}$ sketch applying a scale factor of $k = 8$.

a) What dimensions will the mural have?

b) What area will it have?

c) If a tin of paint covers $2\ \text{m}^2$, what is the minimum number of tins needed?

a) The ratio of the areas of two similar 2D shapes is $9$. What is the scale factor of their sides? And the ratio of their perimeters?

b) The volumes of two similar 3D solids are in the ratio $\frac{27}{8}$. What is the scale factor?

c) The areas of two similar plots of land are in the ratio $\frac{1}{4}$. What is the ratio of their sides?

A cube has an edge length of $4$ cm and another cube, similar to it, has edges twice as long.

a) What is the volume of each cube?

b) How many times larger is the volume of the large cube compared to the small one?

For a neighbourhood model, two similar small houses have been built with scale factor $1.5$. The small house required $7.2\ \text{dm}^2$ of cardboard and its interior volume is $6.4$ litres.

a) How much cardboard will be needed for the large house?

b) What will its interior volume be?

A projector forms a square with sides of $1.5$ m on a screen placed $3$ m away. When the screen is moved back to $6$ m, the image is still a square, as shown in the figure.

a) What is the similarity ratio between the two images? How long is the side of the square at $6$ m?

b) How many times larger is the area of the far image than that of the near one?

c) Since the same amount of light is spread over the whole surface, what fraction of the brightness per square metre does the image at $6$ m have compared with the one at $3$ m?

d) At what distance would the screen have to be placed so that the area of the image is $9$ times the initial one?

On a map at scale $1:100$, a room occupies $12\ \text{cm}^2$. What is its actual area? Express it in square metres.

Two fish tanks are similar: the small one is $40$ cm long and the large one is $80$ cm long.

a) What is the scale factor?

b) If the glass for the small fish tank cost $\text{€}\ 30$, how much will the glass for the large one cost, at the same price per unit area?

c) If filling the small one with water costs $\text{€}\ 5$, how much will it cost to fill the large one?

Choose the method. An architecture studio designs the roof of a municipal sports center made up of $150$ triangular panels, all different from each other, joined like a polyhedral mosaic. In the studio, they build a scale model at $1:20$ ($k = 20$). After measuring the model, they know that the total plastic surface area of the $150$ panels adds up to exactly $1.2\text{ m}^2$. To order the polycarbonate panels for the actual construction, two architects discuss how to proceed:\n\n- Aoife's method: Measure the three sides of each of the $150$ triangles on the model using calipers, multiply each length by $20$, calculate the area of each actual triangle, and add up the $150$ areas obtained.\n- Cian's method: Take the total surface area of the model ($1.2\text{ m}^2$) and multiply it directly by $k^2 = 20^2 = 400$.\n\na) Compare both methods: which of the two is preferable from the point of view of efficiency and the accumulation of rounding errors? Justify your choice.\nb) Apply the most convenient method to calculate the total surface area of polycarbonate that must be purchased for the actual roof.\nc) If a square meter of polycarbonate costs $38\text{ €}$, what will be the material cost for the roof?

Unit Challenge. Have you ever wanted to calculate the height of an object in your surroundings without climbing it, and plan its replica? Become a surveyor and technical designer from home.

1. Data collection and Thales: On a sunny day, measure your height ($h_1$) and the length of your shadow ($s_1$) with a tape measure. At the same time, measure the shadow cast by a tall object nearby ($s_2$, such as a lamppost or a tree). Applying triangle similarity and Thales's theorem, deduce its real height ($h_2$).

2. Application of Pythagoras: Taking $s_2$ and $h_2$ as the legs of a right triangle, apply the Pythagorean theorem to find the real distance from the top of the object to the tip of its shadow.

3. Scales in a spreadsheet: Plan a scale model at $1:20$ ($k = 0.05$). In a spreadsheet (or GeoGebra), tabulate the real measurements and those of the scale model. Calculate the area scale factor ($k^2$) to estimate the surface area of material you would use, and the volume scale factor ($k^3$) to compare the capacity or mass of the replica to the original.

Prepare a brief report with your data, justified calculations, and a screenshot of your spreadsheet.

Assessment criteria:

• Geometric rigor when applying Thales and Pythagoras with measurements taken by you.

• Correct deduction and use of the scale factors $k$, $k^2$, and $k^3$ in the spreadsheet.

• Clarity, order, and mathematical justification in the final report.