Exercises: Lengths, areas and volumes

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 ratio between the areas of two similar hexagons is $\frac{16}{25}$. Determine the scale factor between their corresponding sides.

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.

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 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.

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.

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 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.

Two triangular crop 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 model of a historical monument has been built at a scale of $1:40$. If the volume of the model is $0,05 \text{ m}^3$ and its outer surface measures $1,2 \text{ m}^2$, find the real volume of the monument in $\text{m}^3$ and its real surface area in $\text{m}^2$.

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.

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.

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 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.

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?

A pizzeria offers its circular pizzas in two similar sizes: medium and large. The medium pizza has a diameter of $28 \text{ cm}$ and costs $8 \text{ \$}$. If the price of the pizza is strictly proportional to its area, find what diameter the large pizza must have if its menu price is $18 \text{ \$}$.

A water tank has the shape of an inverted cone. It is known that 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.

A toy factory produces solid wooden pyramids. Painting the surface of a small pyramid costs $2,50 \text{ \$}$ (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 is manufactured, similar to the first one, which requires $800 \text{ cm}^3$ of wood, what will be the cost of painting this large pyramid?

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.

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?

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 $\text{m}^2$.

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.

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.

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 $k = 3$.