Lengths, Areas, and Volumes
If you enlarge a photo to twice its size, does it use twice as much paper? No: it uses four times as much. When you scale a figure by $k$, lengths change by $k$, area by $k^2$, and volume by $k^3$. Here you will use this in both directions: given $k$, calculate an area or a volume; and given an area or a volume, find $k$. It is the tool behind scale models, maps, pizza prices, and even the reason mice get cold faster than elephants.
Perimeter, area, and volume with ratio $k$
Why squared? A square with side $1$ enlarged by $k = 2$ has side $2$, and $2 \times 2 = 4$ squares like the original fit inside it. A cube enlarged by $2$ contains $2 \times 2 \times 2 = 8$ of the original cubes: area depends on two measurements and volume on three.
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, similar to the first, is drawn with sides that measure exactly half as long. Calculate the area of this…
The area of a triangle is $12 \text{ cm}^2$. A second triangle, similar to the first, is drawn with sides that measure exactly half as long. Calculate the area of this new triangle.
You can do it on paper and send a photo with your answer.
From the ratio of areas (or volumes) to $k$
When the given information is a ratio of areas or volumes, you have to work backward: a square root to go from areas to $k$, a cube root to go from volumes to $k$.
The ratio of the areas of two similar hexagons is $\frac{16}{25}$. Determine the scale factor between their corresponding sides.
The ratio of 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.
Two triangular fields are similar. The dimensions of the smaller field are $30 \text{ m}$, $40 \text{ m}$ and $50 \text{ m}$, and its area is $600 \text{ m}^2$. If the…
Two triangular fields are similar. The dimensions of the smaller field are $30 \text{ m}$, $40 \text{ m}$ and $50 \text{ m}$, and its area is $600 \text{ m}^2$. If the perimeter of the larger field is $360 \text{ m}$, calculate the area of this second field.
Scales on maps and plans: length and area
A scale $1:n$ is a similarity ratio $k = n$ between the drawing and reality, but that $n$ applies directly only to lengths. To find real areas from areas on the plan (or the other way around), you have to square it.
A road map is drawn to a scale of $1:50{,}000$. If a straight stretch of road measures $4 \text{ cm}$ on the map, calculate how many kilometers this stretch measures in…
A road map is drawn to a scale of $1:50{,}000$. If a straight stretch of road measures $4 \text{ cm}$ on the map, calculate how many kilometers this stretch measures in reality.
A rectangular plot of land has an actual area of $400 \text{ m}^2$. On a town plan, the same plot occupies an area of $4 \text{ cm}^2$. Calculate the numerical scale to…
A rectangular plot of land has an actual area of $400 \text{ m}^2$. On a town plan, the same plot occupies an area of $4 \text{ cm}^2$. Calculate the numerical scale to which the plan is drawn.
On a topographic map with a scale of $1:250\,000$, a natural park 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 natural park occupies an area of $12 \text{ cm}^2$. Knowing that $1 \text{ hectárea (ha)}$ is equivalent to $10\,000 \text{ m}^2$, calculate the actual area of the natural park expressed in hectares.
Mass, volume, and density
If two objects are made of the same material, their mass is proportional to their volume: it changes with $k^3$, just like volume. That is why a figure that is "a little bigger" can weigh much more.
A model of a sports car is manufactured to a scale of $1:24$. If the actual car has a length of $4.8 \text{ m}$, find the length of the model in centimeters.
A model of a sports car is manufactured to a scale of $1:24$. If the actual car has a length of $4.8 \text{ m}$, find the length of the model in centimeters.
The scale model of a historical monument has been built to a scale of $1:40$. If the volume of the model is $0.05 \text{ m}^3$ and its exterior surface area measures…
The scale model of a historical monument has been built to a scale of $1:40$. If the volume of the model is $0.05 \text{ m}^3$ and its exterior surface area measures $1.2 \text{ m}^2$, find the actual volume of the monument in $\text{m}^3$ and its actual surface area in $\text{m}^2$.
A designer 3D prints a small plastic figurine that weighs $30 \text{ g}$. Seeing the result, she decides to print a similar version scaled up to $150\%$ of its original…
A designer 3D prints a small plastic figurine that weighs $30 \text{ g}$. Seeing the result, she decides to print a similar version scaled up 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 figurine.
Area, cost, and capacity
Many real-life problems link two different quantities: cost (proportional to area, as in painting or manufacturing a can) and capacity (proportional to volume). First you have to identify which of the two is the one that scales in each part of the problem.
A brand of soup sells its product in two similar cylindrical cans. The amount of sheet metal required to manufacture the large can (its surface area) is $4$ times…
A brand of soup sells its product in two similar cylindrical cans. The amount of sheet metal required to manufacture the large can (its surface area) is $4$ times greater than that needed for the small can. If the small can contains $250 \text{ ml}$ of soup, find how many liters the large can contains.
A regulation basketball has a diameter of $24 \text{ cm}$, whereas 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 $24 \text{ cm}$, whereas a tennis ball has a diameter of $6 \text{ cm}$. Assuming both are perfect spheres, determine how many times larger the surface area of the basketball is compared to the tennis ball, and how many times larger its volume is.
A painter wants to create a mural on a wall by enlarging a preliminary sketch. The sketch is drawn on a rectangular canvas $20 \text{ cm}$ wide by $30 \text{ cm}$ high.…
A painter wants to create a mural on a wall by enlarging a preliminary sketch. The sketch is drawn on a rectangular canvas $20 \text{ cm}$ wide by $30 \text{ cm}$ high. The wall intended for the mural has a total area of $24 \text{ m}^2$ and the final mural will be exactly similar to the sketch, covering the entire wall. What will the dimensions of the mural be in meters?
A toy factory produces solid wooden pyramids. Painting the surface of a small pyramid costs $\$50$ (the cost is directly proportional to the surface area) and $100…
A toy factory produces solid wooden pyramids. Painting the surface of a small pyramid costs $\$50$ (the cost is directly proportional to the surface area) and $100 \text{ cm}^3$ of wood is used to manufacture it. If a larger pyramid, similar to the first, requiring $800 \text{ cm}^3$ of wood is manufactured, what will be the cost of painting this large pyramid?
The surface-to-volume ratio
When a body grows while keeping its shape, its volume grows faster than its surface area ($k^3$ versus $k^2$). That is why the surface-to-volume ratio decreases as size increases: it is the key to why small animals lose heat faster and why cells cannot grow indefinitely.
The screen of a smartphone has an area of $A \text{ cm}^2$. The manufacturer launches a new "Pro" model with exactly the same proportions (that is, a similar screen),…
The screen of a smartphone has an area of $A \text{ cm}^2$. The manufacturer launches a new "Pro" model with exactly the same proportions (that is, a similar screen), but whose diagonal is $20\%$ larger than that of the standard model. By what exact percentage does the area of the screen increase? Additionally, if all three-dimensional dimensions of the phone increase by $20\%$, by what percentage will its total volume increase?
A bronze statue displayed in a square has a mass of $432 \text{ kg}$. The museum shop sells a scale replica made from the exact same bronze alloy, which has a mass of…
A bronze statue displayed in a square has a mass of $432 \text{ kg}$. The museum shop sells a scale replica made from the exact same bronze alloy, which has a mass of $2 \text{ kg}$. If the surface area of the replica is $450 \text{ cm}^2$, find the surface area of the original statue expressed in $\text{m}^2$.
In biology, the surface area to volume ratio of a cell determines its ability to exchange nutrients. Consider a spherical cell of radius $r$. If the cell grows so that…
In biology, the surface area to volume ratio of a cell determines its ability to exchange nutrients. Consider a spherical cell of radius $r$. If the cell grows so that its new radius is $k \times r$ (with $k > 1$), prove mathematically that the surface area to volume ratio decreases by a factor of $\frac{1}{k}$. Next, 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 $k = 3$.