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 kilometres 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 centimetres.

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.

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.

An artisan pottery workshop in La Bisbal manufactures similar traditional clay pitchers (botijos) in two sizes. The standard model has a capacity of $1\text{ litre}$ and painting it with a special waterproof varnish costs $3.50\text{ \euro}$. The artisan wishes to design a giant model with a capacity of exactly $8\text{ litres}$. If the varnishing cost is directly proportional to the outer surface area of the clay pitcher and the customer has set a spending limit of $15\text{ \euro}$ for varnishing this new model:

1. a) Find the linear similarity ratio $k$ that the dimensions of the giant clay pitcher must have with respect to the standard one.
2. b) Determine whether the varnishing cost of the giant model will respect the budget set by the customer.

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 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.

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 metres?

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 $\text{€}\ 18$.

The Town Hall of a municipality commissions the construction of a public swimming pool in the shape of a prism with an irregular trapezoidal base. In the technical project, a similar scale model was designed whose bottom and walls total an interior surface area of $1.2\text{ m}^2$ and whose water capacity is $0.3\text{ m}^3$. The actual swimming pool must have lengths $10$ times greater than those of the model (linear similarity ratio $k = 10$).

Two technicians discuss how to budget for the ceramic tiling and the filling:

• Technician A proposes first solving for each of the actual linear dimensions (length, widths, and depth) from the geometric formulas of the trapezoidal prism to then calculate the new surface area and the new volume.

• Technician B proposes directly applying the ratios between magnitudes of similar bodies from the overall data of the model.

Explain with reasons which of the two methods is the most appropriate and efficient in this case, and indicate what insurmountable difficulty the other one presents. Then, calculate the area to be tiled and the cubic metres of water needed to fill the actual pool.

A canning company from Howth sells albacore tuna in similar cylindrical tin cans. The single-portion can measures $4\text{ cm}$ in height, contains a net mass of $100\text{ g}$ of fish, and its container costs $0.25\text{ €}$ to manufacture. The company launches a similar family-size format with a height of $8\text{ cm}$. Assuming that the density of the content is identical in both formats and that the cost of the container depends exclusively on the total surface area of tinplate used:

1. a) Model the relationship between the masses and calculate how many grams of tuna the family-size format will contain.
2. b) Model the relationship between the surface areas and calculate the manufacturing cost of the large can.
3. c) Calculate the packaging cost rate per gram of product (in $\text{€}/\text{g}$) for both formats and explain why it is more economically efficient for the company to pack in large sizes.

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 $\text{€}\ 2.50$ (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 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 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 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 $k = 3$.

In biology, Bergmann's rule describes the thermal adaptation of animals through the relationship between their body surface area (through which they dissipate heat) and their volume (which generates heat). Consider two mammals of similar silhouette and identical density: a field mouse of length $6\text{ cm}$ and an otter of length $60\text{ cm}$. The metabolic energy they must burn per gram of body mass to keep their temperature constant is proportional to the surface-to-volume ratio ($S/V$).

1. a) Before carrying out any numerical calculation, estimate with reasoning: who do you think needs to burn proportionally more food per day relative to its own weight so as not to cool down, the mouse or the otter? Formulate a qualitative hypothesis.
2. b) Knowing that the linear similarity ratio of the otter with respect to the mouse is $k = 10$, algebraically deduce how the $S/V$ ratio of a body changes when its lengths are multiplied by $k$, and calculate the value of the ratio $\dfrac{(S/V)_{\text{mouse}}}{(S/V)_{\text{otter}}}$.
3. c) Check whether your initial estimate was correct and interpret the result: if the mouse needs to ingest each day the equivalent of $30\,\%$ of its body mass in food to compensate for its heat loss, what percentage of its own body mass would the otter need to ingest for the same purpose?

A sculptor has cast a solid bronze scale model $25\text{ cm}$ high for a monument to be installed in a public square. The model weighs $3.2\text{ kg}$ and applying a special protective patina over its entire surface cost $6\text{ \euro}$. The urban planning commission sets two strict conditions for commissioning the final sculpture, which will be a geometrically similar scale figure made of the same solid bronze:

1. For safety reasons during crane transport, the total mass of the sculpture cannot exceed $1\,350\text{ kg}$.

2. The maximum budget allocated for the protective patina of the sculpture is $216\text{ \euro}$ (the cost of the treatment per square metre is identical to that of the model).

Set up the inequalities corresponding to both constraints as a function of the similarity ratio $k$, determine the maximum permissible value for $k$, and calculate the maximum height that the final monument can have. Justify in detail which of the two constraints (the weight or the budget for the patina) is the one that imposes the actual limit on the size of the artwork.

Topic Challenge. How can you measure the inaccessible and reproduce it to scale? You will design a technical report calculating the actual dimensions of a nearby urban element (a tree, street lamp, or building) and those of its virtual model.

1. Real data collection: Choose a sunny day and measure with a tape measure the length of the shadow of the chosen element and, simultaneously, your own shadow and your height (or that of a $1\text{ m}$ slat).

2. Geometry and calculation: Model the similar right triangles formed by the sun's rays in GeoGebra or in a spreadsheet. Apply Thales's theorem to deduce the height of the object. Using the calculator and the Pythagorean theorem, determine the distance from the highest point of the element to the tip of its shadow.

3. Three-dimensional scale: Assimilate the object to a basic geometric solid (cylinder, cone, or prism) and estimate its real volume and lateral area. If you were to build a model with a similarity ratio $k = 0.05$ ($1:20$), calculate the surface area and volume it would occupy by applying the ratios $k^2$ and $k^3$.

Submit your report with the data obtained, the justified calculations, and a screenshot of the digital model.

Assessment criteria:

• Rigorous application of Thales and Pythagoras in similar right triangles.

• Correct use of the calculator and justification of similarity ratios in lengths ($k$), areas ($k^2$), and volumes ($k^3$).

• Clarity of the technical report and appropriate use of digital tools (GeoGebra or spreadsheet).