So far you have calculated perimeters and areas of similar figures separately. In practice, real problems mix both at once: a plan with an area in $\text{cm}^2$ that corresponds to square metres in real life, a photo that is enlarged and you need to know how much ink it uses...
Before you start, make sure you know: Surface area (1st ESO)
Before you start, make sure you know: Plans and scale (2nd ESO)
From length to area: the scale factor squared
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A pentagon has a perimeter of $45\text{ cm}$. If another similar pentagon is constructed with a scale factor of $k = 3$, calculate the perimeter of the new pentagon.
A pentagon has a perimeter of $45\text{ cm}$. If another similar pentagon is constructed with a scale factor of $k = 3$, calculate the perimeter of the new pentagon.
Two isosceles triangles are similar. The non-equal side of the first measures $8\text{ cm}$ and its area is $24\text{ cm}^2$. If the non-equal side of the second…
Two isosceles triangles are similar. The non-equal side of the first measures $8\text{ cm}$ and its area is $24\text{ cm}^2$. If the non-equal side of the second measures $24\text{ cm}$, what is its area?
On a plan with a scale of $1:n$, lengths are multiplied by $n$ when converting to real life, but areas are multiplied by $n^2$: it is easy to forget this and multiply by $n$ only.
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On a map, the distance between two cities is $8\text{ cm}$. If the scale of the map is $1:500\,000$, calculate the actual distance in kilometres.
On a map, the distance between two cities is $8\text{ cm}$. If the scale of the map is $1:500\,000$, calculate the actual distance in kilometres.
On an architect's floor plan, drawn to a scale of $1:150$, the living room of a house occupies an area of $12\text{ cm}^2$. Calculate the actual area of the living room…
On an architect's floor plan, drawn to a scale of $1:150$, the living room of a house occupies an area of $12\text{ cm}^2$. Calculate the actual area of the living room in square metres.
Enlarging a photo into a poster is, mathematically, constructing a similar figure. The area grows much faster than the dimensions: that is why an enlargement that seems modest in terms of side lengths can need far more ink or canvas.
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A photograph measuring $10\text{ cm}$ wide by $15\text{ cm}$ high is enlarged by multiplying its dimensions by $2,5$. Calculate the area of the enlarged photograph.
A photograph measuring $10\text{ cm}$ wide by $15\text{ cm}$ high is enlarged by multiplying its dimensions by $2,5$. Calculate the area of the enlarged photograph.
An artist paints a sketch on a canvas $20\text{ cm}$ wide by $30\text{ cm}$ high using $15\text{ ml}$ of blue paint. If he then reproduces exactly the same drawing on a…
An artist paints a sketch on a canvas $20\text{ cm}$ wide by $30\text{ cm}$ high using $15\text{ ml}$ of blue paint. If he then reproduces exactly the same drawing on a similar mural $3\text{ m}$ high, how many litres of blue paint will he need?
Real-life proportionality: price according to area
When a price or an amount of material depends on the area (not on the side), two similar figures with scale factor $k$ make that price multiply by $k^2$, not by $k$: this is the usual trap of "the family-size pizza should cost double".
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A pizzeria sells a medium pizza with a diameter of $20\text{ cm}$ for $8\text{ €}$. If the price is strictly proportional to the amount of pizza (its area), how much…
A pizzeria sells a medium pizza with a diameter of $20\text{ cm}$ for $8\text{ €}$. If the price is strictly proportional to the amount of pizza (its area), how much should a family pizza with a diameter of $40\text{ cm}$ made with the same recipe cost?
Reasoning backwards: from area to the scale factor $k$
Some problems directly give the ratio of areas or of perimeters and ask for $k$ or for some length: you have to undo the process with a square root.
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Two irregular hexagons are similar. The perimeter of the first is $60\text{ cm}$ and that of the second is $15\text{ cm}$. Find the scale factor to go from the large…
Two irregular hexagons are similar. The perimeter of the first is $60\text{ cm}$ and that of the second is $15\text{ cm}$. Find the scale factor to go from the large hexagon to the small one.
There are two similar rhombuses. The sum of their perimeters is $140\text{ cm}$ and the scale factor between them is $k = 2,5$. Determine the perimeter of each rhombus.
There are two similar rhombuses. The sum of their perimeters is $140\text{ cm}$ and the scale factor between them is $k = 2,5$. Determine the perimeter of each rhombus.