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

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.

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 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 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.

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 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.

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 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 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.

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.

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 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. 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 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 $\$8$. 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 $\$18$.

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.

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 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.

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 $2 \text{ kg}$. 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-shaped town plaza has a perimeter of $160 \text{ m}$. The municipality decides to remodel it to build a new, similar square plaza, such that the paved area of the new plaza is exactly triple 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 polygon.

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 $k = 3$.