Scale factors for areas and volumes
If you double the side of a square, you don't double its area: you quadruple it. The scale factor $k$ does not affect lengths, areas and volumes in the same way, and mixing them up is one of the most expensive mistakes in real life (literally, in the price of things).
Lengths, areas and volumes: three different factors
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?
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 $2$.
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?
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
Just as before, but with the cube root: if you know how the volume changes, $k$ comes from $\sqrt[3]{\,}$.
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…
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…
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?
Models, maps and the cost of enlarging
These ratios appear in very everyday situations: how much glass a fish tank needs, how much it costs to fill a larger tank with water, or how much real land a plot drawn on a map covers.
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 $30\text{ €}$, 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?
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?