Scale of Areas and Volumes
If you double the side of a square, you don't double its area: you quadruple it. The scale factor $k$ doesn't affect lengths, areas, and volumes in the same way, and mixing them up is one of the costliest mistakes in real life, literally in the price of things.
Lengths, areas, and volumes: three different ratios
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…
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…
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…
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?
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…
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 centimeter on the plan represents $50\ \text{cm}^2$ in reality.
Calculating with the ratio of areas
If you know how the area changes, to find $k$ you have to take the square root, not divide by $2$.
Percentages also give the scale factor: enlarging a photo or a figure to $150\,\%$ means every length is multiplied by $1.5$, that is, $k = 1.5$. Reducing to $50\,\%$ is $k = 0.5$. Careful: that percentage refers to the lengths, not the area.
Scales on plans and maps
A $1:75$ scale means $k = \dfrac{1}{75}$ for lengths. To go from the plan to real life, areas are multiplied by $75^2$.
On a map at scale $1:100$, a room occupies $12\ \text{cm}^2$. What is its actual area? Express it in square meters.
On a map at scale $1:100$, a room occupies $12\ \text{cm}^2$. What is its actual area? Express it in square meters.
A photo measuring $40\ \text{cm}^2$ is enlarged on a photocopier to $150\%$. What is the area of the enlargement?
A photo measuring $40\ \text{cm}^2$ is enlarged on a photocopier to $150\%$. What is the area of the enlargement?
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…
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?
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…
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 development, drawn to scale $1:75$, a rectangular garden occupies $96\ \text{cm}^2$. What is the actual area of the garden in square meters?
On the map of a housing development, drawn to scale $1:75$, a rectangular garden occupies $96\ \text{cm}^2$. What is the actual area of the garden in square meters?
On a map at a scale of $1:2\,000$, a soccer field occupies $17.5\ \text{cm}^2$. a) What is its actual area in square meters? b) Is it close to the official area of a…
On a map at a scale of $1:2\,000$, a soccer field occupies $17.5\ \text{cm}^2$.
a) What is its actual area in square meters?
b) Is it close to the official area of a soccer field, which is around $7\,000\ \text{m}^2$?
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…
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?
Calculating with the ratio of volumes
Same as before, but with the cube root: if you know how the volume changes, $k$ comes from $\sqrt[3]{\ }$.
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 meters.
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 meters.
For a neighborhood 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…
For a neighborhood 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 $6.4$ liters.
a) How much cardboard will be needed for the large house?
b) What will its interior volume be?
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…
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?
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…
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 meter 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?
Models, maps, and how much it costs to scale up
These ratios show up in everyday things: how much glass a fish tank needs, how much it costs to fill a bigger water tank, or how much real land a lot drawn on a plan takes up.
Back to the beginning: doubling the side quadruples the area, and doubling the edge of a box multiplies its capacity by $8$.
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…
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 $\$600$, 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 $\$100$, how much will it cost to fill the large one?
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…
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 metric tons.
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 liters will be needed to paint…
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 liters 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?
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…
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 $\$300$ and painting its outer surface costs $\$800$. a)…
Two cylindrical tanks are similar and the ratio of their heights is $2$. Filling the small tank with water costs $\$300$ and painting its outer surface costs $\$800$.
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.