Scale factors for 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?

State whether the following statements are true or false, giving reasons:

a) If I double the side lengths of a rectangle, I double its area.

b) The ratio of the areas of two similar shapes is the square of the scale factor.

c) If the scale factor of two 3D solids is $2$, the volume of the large one is $8$ times that of the small one.

d) On a plan at scale $1:50$, one square centimetre on the plan represents $50\ \text{cm}^2$ in reality.

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?

Inverse problem. A sustainable parcel delivery company wants to redesign its standard cardboard box. They are going to order a new model with exactly the same shape (similar), but reducing the dimensions so that the total surface area of cardboard needed is $64\,\%$ of the surface area of the original box (that is, the area of the new box is $0.64$ times the area of the old one).\n\na) Set up the relationship between the areas and obtain the linear similarity ratio $k$ between the new box and the original one.\nb) If the height of the original box was $35\text{ cm}$, how much should the height of the new model measure?\nc) What percentage of the original volume will the new box keep? Is the volume reduced in the same proportion as the cardboard? Justify your answer by interpreting the volume scale.

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?

In a scale model at $1:50$, a tank has a volume of $80\ \text{cm}^3$. What is the volume of the real tank? Express it in cubic metres.

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?

Model. A rapeseed oil cooperative from Kilkenny sells its gourmet extra virgin rapeseed oil in glass bottles of similar design in two formats: an individual bottle of $100\text{ ml}$ and a family bottle of $800\text{ ml}$.\n\na) Both bottles are similar geometric solids. From their capacities, determine the linear similarity ratio $k$ between the family bottle and the individual one.\nb) If the individual bottle has a height of $14\text{ cm}$, what height will the family bottle have?\nc) If the adhesive label wrapped around the individual bottle has a surface area of $25\text{ cm}^2$, how many square centimeters will the label of the family bottle measure?\nd) To manufacture the body of the individual bottle, $120\text{ g}$ of molten glass are used (which exclusively forms the outer wall of the container, whose thickness is the same in both formats). A worker claims that for the family bottle, $960\text{ g}$ of glass will be needed ($8$ times more). Reason whether the worker is right or not, indicating the approximate actual amount of glass required.

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?

On the map of a housing estate, drawn to scale $1:75$, a rectangular garden occupies $96\ \text{cm}^2$. What is the actual area of the garden in square metres?

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.