Lengths, areas and volumes
You already know that when a figure is scaled by $k$, its area changes by $k^2$ and its volume by $k^3$. Now you are going to use this in both directions: given $k$, find the area or volume; and given an area or a volume, find $k$. It is the tool behind scale models, maps, product prices and even why mice cool down faster than elephants.
Revision: 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 information is a ratio of areas or of volumes, you have to reverse the process: take the square root to go from areas to $k$, and the 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.
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 local council 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 local council 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 surds.
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.
Map and plan scales: length and area
A scale $1:n$ is a scale factor $k = n$ 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 round), it has to be squared.
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 kilometres 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 kilometres 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.
Aoife checks the $1:500$ scale plan on the online portal of Tailte Éireann for a rectangular plot that her family wants to fence and sow with grass in Kilkenny. On the…
Aoife checks the $1:500$ scale plan on the online portal of Tailte Éireann for a rectangular plot that her family wants to fence and sow with grass in Kilkenny. On the plan, the plot measures $4\text{ cm}$ wide by $6\text{ cm}$ long. To calculate the actual area, Aoife carries out the following calculation: “Since the area on paper is $4 \times 6 = 24\text{ cm}^2$, the actual area of the plot will be $24 \times 500 = 12\,000\text{ cm}^2 = 1.2\text{ m}^2$”. Identify and explain in detail the double error (geometric and common-sense) that Aoife makes, and correctly calculate the actual area of the plot in square metres.
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 by $k^3$, just like the 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 centimetres.
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 centimetres.
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.
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 $\text{m}^2$.
Area, cost and capacity
Many real-life problems link two different quantities: cost (proportional to area, such as painting or manufacturing a tin) and capacity (proportional to volume). You first have to identify which of the two scales in each part of the problem.
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 brand of soup sells its product in two similar cylindrical tins. The amount of tinplate required to manufacture the large tin (its surface area) is $4$ times greater…
A brand of soup sells its product in two similar cylindrical tins. The amount of tinplate required to manufacture the large tin (its surface area) is $4$ times greater than that needed for the small tin. If the small tin contains $250 \text{ ml}$ of soup, find how many litres the large tin 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 pizzeria offers its circular pizzas in two similar sizes: medium and family-size. The medium pizza has a diameter of $28 \text{ cm}$ and costs $\text{€}\ 8$. If the…
A pizzeria offers its circular pizzas in two similar sizes: medium and family-size. The medium pizza has a diameter of $28 \text{ cm}$ and costs $\text{€}\ 8$. If the price of the pizza is directly proportional to its area, find what diameter the family-size pizza must have if its menu price is $\text{€}\ 18$.
A toy factory produces solid wooden pyramids. Painting the surface of a small pyramid costs $\text{€}\ 2.50$ (the cost is directly proportional to the surface area) and…
A toy factory produces solid wooden pyramids. Painting the surface of a small pyramid costs $\text{€}\ 2.50$ (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-area-to-volume ratio
When a body grows while keeping its shape, its volume grows faster than its surface area ($k^3$ compared with $k^2$). That is why the surface-area-to-volume ratio decreases as size increases: it is the key to why small animals lose heat faster and why cells cannot grow indefinitely.
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 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?
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 $k = 3$.