Before you start, make sure you know: Similarity (Math 8 (Pre-Algebra))
Before you start, make sure you know: Plans and scale (Math 7)
A pizza shop sells a 10-inch pizza and a 20-inch pizza. The big one is twice as wide, so do you get twice as much pizza? You will answer that by the end of this page, and you will see the same idea in map scales, 3D-printed models, soup cans, and even living cells.
Lengths, Areas, and Volumes
When you enlarge or shrink a figure by a scale factor $k$, every length gets multiplied by $k$. Areas and volumes do not follow the same rule.
Try different values of $k$ and watch how the sides, the area, and the volume respond.
Percent changes work the same way. A length that grows by a percent does not make the area grow by that same percent.
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Two pentagons are similar with a scale factor of $k = 3$. If the perimeter of the smaller pentagon is $15 \text{ cm}$ and its area is $20 \text{ cm}^2$, calculate the…
Two pentagons are similar with a scale factor of $k = 3$. If the perimeter of the smaller pentagon is $15 \text{ cm}$ and its area is $20 \text{ cm}^2$, calculate the perimeter and area of the larger pentagon.
The area of a triangle is $12 \text{ cm}^2$. A second triangle is drawn similar to the first, but with sides measuring exactly half. Calculate the area of this new…
The area of a triangle is $12 \text{ cm}^2$. A second triangle is drawn similar to the first, but with sides measuring exactly half. Calculate the area of this new triangle.
A smartphone screen has an area of $A \text{ cm}^2$. The manufacturer launches a new "Pro" model with exactly the same proportion (i.e. a similar screen), but whose…
A smartphone screen has an area of $A \text{ cm}^2$. The manufacturer launches a new "Pro" model with exactly the same proportion (i.e. a similar screen), but whose diagonal is $20\%$ larger than that of the standard model. By what exact percentage does the screen area increase? Additionally, if all the three-dimensional dimensions of the phone increase by $20\%$, by what percentage will its total volume increase?
Sometimes you are given a ratio of areas or volumes and need the scale factor. Then you work backward: take a square root to go from an area ratio to $k$, and a cube root to go from a volume ratio to $k$.
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The ratio between the areas of two similar hexagons is $\frac{16}{25}$. Determine the scale factor between their corresponding sides.
The ratio between the areas of two similar hexagons is $\frac{16}{25}$. Determine the scale factor between their corresponding sides.
There are two similar cubes. The volume of the smaller cube is $8 \text{ cm}^3$ and the volume of the larger cube is $64 \text{ cm}^3$. Find the scale factor $k$ between…
There are two similar cubes. The volume of the smaller cube is $8 \text{ cm}^3$ and the volume of the larger cube is $64 \text{ cm}^3$. Find the scale factor $k$ between the edges of both cubes.
A painter wants to paint a mural on a wall by enlarging a preliminary sketch. The sketch is drawn on a rectangular canvas $20 \text{ in}$ wide by $30 \text{ in}$ high.…
A painter wants to paint a mural on a wall by enlarging a preliminary sketch. The sketch is drawn on a rectangular canvas $20 \text{ in}$ wide by $30 \text{ in}$ high. The wall intended for the mural has a total area of $24 \text{ ft}^2$ and the final mural will be exactly similar to the sketch, occupying the entire wall. What will be the dimensions of the mural in feet?
Most scale factors you have seen so far are "nice" numbers like $3$ or $\tfrac{2}{7}$. When the ratio is not a perfect square or cube, leave $k$ as a radical to keep the answer exact, and use a calculator only for an approximation at the end.
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A square plaza has a perimeter of $160 \text{ m}$. The city council decides to remodel it to build a new similar square plaza, such that the paved area of the new plaza…
A square plaza has a perimeter of $160 \text{ m}$. The city council decides to remodel it to build a new similar square plaza, such that the paved area of the new plaza is exactly three times the area of the old one. Calculate the perimeter of the new plaza, giving the exact value using radicals.
The Great Pyramid of Giza originally had an approximate height of $146 \text{ m}$. Imagine that a similar replica were built nearby, but designed to have exactly half…
The Great Pyramid of Giza originally had an approximate height of $146 \text{ m}$. Imagine that a similar replica were built nearby, but designed to have exactly half the volume of the original pyramid. Calculate what the exact height of this replica would be, expressing the result using radicals.
There are two similar polygons. The area of the first polygon is $S_1 = 45 \text{ cm}^2$ and the area of the second is $S_2 = 80 \text{ cm}^2$. If the sum of the…
There are two similar polygons. The area of the first polygon is $S_1 = 45 \text{ cm}^2$ and the area of the second is $S_2 = 80 \text{ cm}^2$. If the sum of the perimeters of both polygons is $70 \text{ cm}$, determine the perimeter of each of them.
A scale of $1:n$ means that $1$ unit on the map stands for $n$ of the same units in real life. So $n$ is the scale factor between the drawing and reality, but it multiplies lengths only. To go from an area on the map to a real area, you must multiply by $n^2$.
Unit conversions come up constantly in these problems. Keep this toolbox handy, and remember that squared units need squared conversion factors.
Length
Area
$1$ ft $= 12$ in.
$1$ ft$^2 = 144$ in.$^2$
$1$ yd $= 3$ ft
$1$ acre $= 43{,}560$ ft$^2$
$1$ mi $= 5{,}280$ ft $= 63{,}360$ in.
$1$ acre $\approx 4{,}047$ m$^2$
$1$ in. $= 2.54$ cm
$1$ m$^2 = 10{,}000$ cm$^2$
$1$ m $= 100$ cm
$1$ hectare (ha) $= 10{,}000$ m$^2$
For volume and capacity: $1$ ft$^3 = 1{,}728$ in.$^3$ and $1$ gal $= 128$ fl oz.
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A road map is designed at a scale of $1:50\,000$. If a straight stretch of road measures $4 \text{ in}$ on the map, calculate how many miles that stretch measures in…
A road map is designed at a scale of $1:50\,000$. If a straight stretch of road measures $4 \text{ in}$ on the map, calculate how many miles that stretch measures in reality.
A rectangular plot of land has a real area of $400 \text{ m}^2$. On an urban plan, that same plot occupies an area of $4 \text{ cm}^2$. Calculate the numerical scale at…
A rectangular plot of land has a real area of $400 \text{ m}^2$. On an urban plan, that same plot occupies an area of $4 \text{ cm}^2$. Calculate the numerical scale at which the plan is drawn.
On a topographic map with a scale of $1:250\,000$, a nature reserve occupies an area of $12 \text{ cm}^2$. Knowing that $1 \text{ hectárea (ha)}$ is equivalent to…
On a topographic map with a scale of $1:250\,000$, a nature reserve occupies an area of $12 \text{ cm}^2$. Knowing that $1 \text{ hectárea (ha)}$ is equivalent to $10\,000 \text{ m}^2$, calculate the real area of the nature reserve expressed in acres.
If two objects are made of the same material, their mass is proportional to their volume. So mass scales with $k^3$, just like volume. This is why an object that is "only a bit bigger" can weigh a lot more.
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A model of a sports car is made at a scale of $1:24$. If the real car has a length of $4,8 \text{ ft}$, find out the length of the model expressed in inches.
A model of a sports car is made at a scale of $1:24$. If the real car has a length of $4,8 \text{ ft}$, find out the length of the model expressed in inches.
A designer 3D prints a small plastic figure weighing $30 \text{ g}$. Upon seeing the result, she decides to print a similar version but enlarged to $150\%$ of its…
A designer 3D prints a small plastic figure weighing $30 \text{ g}$. Upon seeing the result, she decides to print a similar version but enlarged to $150\%$ of its original size. Assuming that the density of the plastic infill is identical in both prints, calculate the weight of the new figure.
A bronze statue exhibited in a square has a mass of $432 \text{ kg}$. The museum shop sells a scale replica made of exactly the same bronze alloy, whose mass is…
A bronze statue exhibited in a square has a mass of $432 \text{ kg}$. The museum shop sells a scale replica made of exactly the same bronze alloy, whose mass is $2 \text{ kg}$. If the surface area of the replica is $450 \text{ cm}^2$, find the surface area of the original statue expressed in $\text{m}^2$.
Many real problems connect two different quantities. Cost often depends on area (painting, covering, making a can), while capacity depends on volume. First decide which one scales in each part of the problem.
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Two water bottles have similar shapes. The larger bottle is $1,5$ times taller than the smaller bottle. If the smaller bottle has a capacity of $500 \text{ ml}$,…
Two water bottles have similar shapes. The larger bottle is $1,5$ times taller than the smaller bottle. If the smaller bottle has a capacity of $500 \text{ ml}$, determine the capacity of the larger bottle.
A soup brand sells its product in two similar cylindrical cans. The amount of metal needed to make the large can (its surface area) is $4$ times greater than that needed…
A soup brand sells its product in two similar cylindrical cans. The amount of metal needed to make the large can (its surface area) is $4$ times greater than that needed for the small can. If the small can contains $250 \text{ fl oz}$ of soup, find out how many gallons the large can contains.
A regulation basketball has a diameter of $24 \text{ cm}$, while a tennis ball has a diameter of $6 \text{ cm}$. Assuming both are perfect spheres, determine how many…
A regulation basketball has a diameter of $24 \text{ cm}$, while a tennis ball has a diameter of $6 \text{ cm}$. Assuming both are perfect spheres, determine how many times greater the surface area of the basketball is compared to the tennis ball, and how many times greater its volume is.
When a solid grows and keeps its shape, its volume grows faster than its surface area ($k^3$ versus $k^2$). So the ratio of surface area to volume shrinks as the object gets bigger. This is why small animals lose heat faster than big ones and why cells cannot grow forever.
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In biology, the surface-area-to-volume ratio of a cell determines its capacity to exchange nutrients. Consider a spherical cell of radius $r$. If the cell grows such…
In biology, the surface-area-to-volume ratio of a cell determines its capacity to exchange nutrients. Consider a spherical cell of radius $r$. If the cell grows such that its new radius is $k \cdot r$ (with $k > 1$), show mathematically that the surface-area-to-volume ratio is reduced by a factor of $\frac{1}{k}$. Then, calculate the exact value of the new surface-area-to-volume ratio if the original radius was $2 \,\mu\text{m}$ and the growth factor is $k = 3$.
Back to the start: the $20$-inch pizza has $k = 2$ compared with the $10$-inch one, so it has $2^2 = 4$ times as much pizza, not $2$ times. At the same price per square inch, it should cost four times as much.
Summary
Let's practice! 8 medium exercises in this section