Exercises: 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 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?

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

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

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.

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.

Look at the two similar rectangles in the diagram.

a) Check that the scale factor is $3$.

b) If the area of the small rectangle is $8\ \text{cm}^2$, what is the area of the large one?

Someone else's mistake. Marcos wants to print an advertising banner for the neighborhood festivities from a sketch measuring $20\text{ cm} \times 30\text{ cm}$ (area: $600\text{ cm}^2$). For the final banner, he decides to triple the size of all the sides, applying a similarity ratio $k = 3$. At the print shop, Marcos calculates the cost by saying: "Since I multiplied the sides by $3$, I will need exactly three times as much canvas and ink, that is, $3 \cdot 600 = 1800\text{ cm}^2$".\n\na) Without needing to calculate the new measurements of the sides, explain in your own words why Marcos's reasoning is wrong, relying on the relationship between lengths and areas.\nb) Correct the calculation: what will be the actual surface area of the banner in $\text{cm}^2$ and in $\text{m}^2$? By what number is the cost of canvas and ink actually multiplied?

Estimate and verify. An artisan ice cream shop sells similar cylindrical tubs in two sizes. The "Small" tub has a diameter of $6\text{ cm}$ and a capacity of $125\text{ ml}$. The "Giant" tub is exactly similar to the small one, but its diameter measures $12\text{ cm}$.\n\na) Estimate: Before doing operations with formulas, think: when the diameter doubles, will the giant tub contain twice as much ice cream ($250\text{ ml}$), four times as much ($500\text{ ml}$), or considerably more? Write your prediction, briefly explaining your reasoning.\nb) Verify: Find the linear similarity ratio $k$ and calculate the exact capacity of the giant tub in milliliters.\nc) Compare the result with your initial estimate and assess whether it is reasonable for the giant tub to cost three times as much as the small one, taking into account the amount of ice cream it contains.

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?

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 $30\text{ €}$, 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 $5\text{ €}$, how much will it cost to fill the large one?

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?

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?

A shipyard builds a scale model of a sailboat at a scale of $1:40$ weighing $3$ kg. If the real boat were built with the same materials and the same average density, how much would it weigh? Express the answer in tonnes.

A sculpture and its souvenir miniature are similar with scale factor $5$.

a) If $120$ ml of paint is used to paint the miniature, how many litres will be needed to paint the sculpture?

b) If the miniature weighs $2$ kg and both are made of the same material, how much does the sculpture weigh?

On a map at a scale of $1:2\,000$, a football pitch occupies $17,5\ \text{cm}^2$.

a) What is its actual area in square metres?

b) Is it close to the official area of a football pitch, which is around $7\,000\ \text{m}^2$?

Look at the two cubes in the figure: the small one has an edge of $2$ cm and the large one, $6$ cm. The volume of the small cube is $8\ \text{cm}^3$.

a) What is the similarity ratio?

b) Calculate the volume of the large cube without using the formula for the volume of a cube, starting from the volume of the small one.

c) If the total surface area of the small cube is $24\ \text{cm}^2$, what is the total surface area of the large one?

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.

Model. An olive oil cooperative from Jaén sells its gourmet extra virgin olive 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) 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 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?

A removals company wants to "double the capacity" of its cubic boxes and decides to make them with edges twice as long. The boss claims: "That way they will hold twice as much stuff".

a) Explain the manager's mistake.

b) How many times more capacity will the new box actually have?

c) If the small box has a volume of $27\ \text{dm}^3$, what will the volume of the new one be?

Two cylindrical tanks are similar and the ratio of their heights is $2$. Filling the small tank with water costs $15\text{ €}$ and painting its outer surface costs $40\text{ €}$.

a) How much will it cost to fill the large tank with water?

b) How much will it cost to paint its outer surface?

c) A worker claims that "everything will cost double, because the tank is twice as big". Correct their statement using the previous results and explain why they are wrong.

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- Alba'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- Bruno'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 $38\text{ \euro}$, what will be the material cost for the roof?

Model and transfer. A sculptor casts a commemorative statue in bronze. To check its balance and calculate the budget, he first makes a solid model $20\text{ cm}$ high, which weighs $1,5\text{ kg}$ and whose supporting base has an area of $75\text{ cm}^2$. The final monumental sculpture will be $3\text{ m}$ high and will be cast using the same bronze alloy.\n\na) Determine the linear similarity ratio $k$ between the actual sculpture and the model.\nb) If the monumental statue were made completely solid (just like the model), what would its mass be in kilograms and in tons?\nc) What surface area will the supporting base of the large sculpture occupy on the pedestal?\nd) The pressure supported by the pedestal under the sculpture is proportional to $\frac{\text{Mass}}{\text{Area}}$. Express how this fraction changes as a function of $k$ and demonstrate why the large sculpture exerts a much greater pressure per square centimeter than the model. What technical solution do sculptors adopt in reality to avoid this overweight problem?

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.