Lengths, Areas, and Volumes in Similar Figures
A corner store sells the same juice in two bottles of the same shape, one small and one a bit taller. Does the big one hold twice as much juice if it is twice as tall? You already know that when you scale a figure by $k$, the area changes by $k^2$ and the volume by $k^3$. Here you will use that in both directions: given $k$, find the area or the volume; and given an area or a volume, find $k$. It is the tool behind scale models, maps, prices, and even why a mouse cools off faster than an elephant.
Review: perimeter, area, and volume with ratio k
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 data is a ratio of areas or volumes, you have to undo the process: 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.
Map and plan scales: length and area
A scale $1:n$ says that each unit in the drawing represents $n$ real units. That $n$ applies directly only to lengths. To convert areas from the plan to reality (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 $k^3$. That is why a figure that is "a little bigger" can weigh a lot 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.
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.
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 different magnitudes: cost (proportional to area, like painting or manufacturing a can) and capacity (proportional to volume). First identify which of the two scales in each part of the problem.
A brand of soup sells its product in two similar cylindrical cans. The amount of tinplate required to manufacture the large can (its surface area) is $4$ times greater…
A brand of soup sells its product in two similar cylindrical cans. The amount of tinplate 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 $\text{RD\$}\ 150…
A toy factory produces solid wooden pyramids. Painting the surface of a small pyramid costs $\text{RD\$}\ 150$ (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?
When the solid is only partly full
If a tank in the shape of an inverted cone is filled to a certain height, the water forms a smaller cone similar to the tank. Its similarity ratio is the ratio of the heights, and the filled volume is $k^3$ of the total.
A water tank is shaped like an inverted cone. The liquid inside reaches a depth equivalent to $\frac{2}{3}$ of the total height of the tank. Determine what fraction of…
A water tank is shaped like an inverted cone. The liquid inside reaches a depth equivalent to $\frac{2}{3}$ of the total height of the tank. Determine what fraction of the total volume of the tank is currently filled with water.
The Great Pyramid of Giza originally had an approximate height of $146 \text{ m}$. Imagine a similar replica were built nearby, but designed to have exactly half the…
The Great Pyramid of Giza originally had an approximate height of $146 \text{ m}$. Imagine a similar replica were built nearby, but designed to have exactly half the volume of the original pyramid. Calculate the exact height of this replica, expressing the result using surds.
A square town square has a perimeter of $160 \text{ m}$. The municipality decides to remodel it to build a new similar square town square, such that the paved area of…
A square town square has a perimeter of $160 \text{ m}$. The municipality decides to remodel it to build a new similar square town square, such that the paved area of the new square is exactly triple the area of the old one. Calculate the perimeter of the new square, giving the exact value 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 polygon.
The surface-to-volume ratio
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 $k = 3$.
Go back to the beginning: the bottle that is twice as tall has $k = 2$, so its capacity is $2^3 = 8$ times that of the small one, not double.