Exercises: Perimeter, length and area of 2D shapes

A pentagon has a perimeter of $45\text{ cm}$. If another similar pentagon is constructed with a scale factor of $k = 3$, calculate the perimeter of the new pentagon.

The surface area of a triangular road sign is $1200\text{ cm}^2$. If a scale replica is manufactured with a scale factor of $k = \frac{1}{4}$, determine the area of the replica.

Two irregular hexagons are similar. The perimeter of the first is $60\text{ cm}$ and that of the second is $15\text{ cm}$. Find the scale factor to go from the large hexagon to the small one.

The area of a rectangle is $50\text{ cm}^2$ and the area of a rectangle similar to it is $200\text{ cm}^2$. What is the scale factor of its sides?

On a map, the distance between two cities is $8\text{ cm}$. If the scale of the map is $1:500\,000$, calculate the actual distance in kilometres.

A photograph measuring $10\text{ cm}$ wide by $15\text{ cm}$ high is enlarged by multiplying its dimensions by $2,5$. Calculate the area of the enlarged photograph.

Two isosceles triangles are similar. The non-equal side of the first measures $8\text{ cm}$ and its area is $24\text{ cm}^2$. If the non-equal side of the second measures $24\text{ cm}$, what is its area?

Knowing that the ratio of the perimeters of two similar polygons is $\frac{3}{5}$, what will the ratio of their areas be?

Lucía works on designing posters for her high school's recycling campaign in Madrid. She has created an initial digital poster measuring $30\text{ cm}$ wide by $40\text{ cm}$ high (with an area of $1200\text{ cm}^2$). For the main event in the auditorium, she is commissioned to print a similar canvas banner, but twice as wide—that is, with an enlargement factor of $k = 2$ ($60\text{ cm}$ wide by $80\text{ cm}$ high).

Lucía knows that printing the original poster costs $4{,}50\text{ €}$ and reasons as follows: "Since the dimensions of the sides are multiplied by $2$, the surface area will be double ($2400\text{ cm}^2$) and, therefore, the banner will cost me twice as much: $9\text{ €}$".

1. a) Point out the error in Lucía's reasoning and explain in your own words why the area does not double when we double the lengths of the sides of a plane figure.
2. b) Correct Lucía's calculations: determine the actual area of the large banner and the exact cost that the print shop will charge if the price is proportional to the printed square meters.

A sports facility construction company in Seville adapts children's courts based on the technical regulations of the Spanish Basketball Federation (FEB). An official regulation basketball court measures $28\text{ m}$ long by $15\text{ m}$ wide, with a playing area of $420\text{ m}^2$.

For a preschool playground, they design a mini-court similar to the regulation one by applying a scale factor of $k = 0{,}5$ to all its lengths.

1. a) Without calculating beforehand the measurements of the new sides of the children's court, use the properties of similarity to directly deduce the area that the school mini-court will have.
2. b) If the cost of painting and applying non-slip resin to the entire regulation court amounts to $3360\text{ €}$, how much will it cost to paint the mini-court if the same rate per square meter is applied?
3. c) What percentage of the cost of the official court does the expense of the children's court represent? Justify why that percentage matches exactly the value of $k^2$ expressed as a percentage.

On an architect's floor plan, drawn to a scale of $1:150$, the living room of a house occupies an area of $12\text{ cm}^2$. Calculate the actual area of the living room in square metres.

An artist paints a sketch on a canvas $20\text{ cm}$ wide by $30\text{ cm}$ high using $15\text{ ml}$ of blue paint. If he then reproduces exactly the same drawing on a similar mural $3\text{ m}$ high, how many litres of blue paint will he need?

There are two similar rhombuses. The sum of their perimeters is $140\text{ cm}$ and the scale factor between them is $k = 2,5$. Determine the perimeter of each rhombus.

A pizzeria sells a medium pizza with a diameter of $20\text{ cm}$ for $8\text{ €}$. If the price is strictly proportional to the amount of pizza (its area), how much should a family pizza with a diameter of $40\text{ cm}$ made with the same recipe cost?

Look at the figure:

The ratio of the areas of two regular octagons is $\frac{16}{9}$. If the apothem of the smaller octagon measures $6\text{ cm}$, calculate the length of the apothem of the larger octagon.

In a park there is a circular pond. On a plan of the park, the pond has an area of $15\text{ cm}^2$. If the plan is to a scale of $1:2000$, determine the actual area of the pond in hectares ($1\text{ ha} = 10\,000\text{ m}^2$).

A triangle $\triangle ABC$ of area $72\text{ cm}^2$ is cut by a segment $DE$ parallel to the base $BC$, forming a smaller triangle $\triangle ADE$ at the top. If the height of the small triangle is one-third of the height of the original triangle, calculate the area of the trapezium $DBCE$.

The swimming club of a coastal town in Cantabria is going to order similar protective thermal pool covers for its two open-air rectangular pools in order to prevent heat loss during the night.

The cover for the wading pool has a perimeter of $30\text{ m}$ and an area of $50\text{ m}^2$. The cover for the training pool is similar to the small one and its outer boundary has a perimeter of $60\text{ m}$.

1. a) Before performing operations with formulas, make a reasoned estimate: since the perimeter has doubled, will the area of the large cover be double ($100\text{ m}^2$), triple ($150\text{ m}^2$), or quadruple ($200\text{ m}^2$)? Write your justified prediction in one sentence.
2. b) Check your estimate by mathematical calculation: find the similarity ratio $k$ from the perimeters, calculate the exact area of the training pool, and determine how many total square meters of cover the club must purchase to cover both pools.
3. c) Compare the result obtained with your initial estimate and reflect: why does common sense often fail when estimating how the surface area of a figure increases when its lengths are doubled?

On the Electronic Portal of the Spanish Cadastre, a farmer checks the details of a traditional olive grove plot in the region of La Loma (Jaén). On the official map downloaded at a scale of $1:2000$, the plot is represented as a polygonal shape enclosing an area of $18{,}5\text{ cm}^2$ on the map, with a drawn perimeter of $19\text{ cm}$.

To be eligible for an agricultural subsidy from the Regional Government of Andalusia, regulations require the holding to have a minimum actual area of $0{,}7\text{ hectares}$ ($1\text{ ha} = 10\,000\text{ m}^2$).

1. a) Identify the similarity ratio $k$ that relates the length measurements of the actual land to those on the cadastral map.
2. b) Deduce the actual area of the plot in square meters using the relationship between areas of similar figures (remember that $1\text{ m}^2 = 10\,000\text{ cm}^2$) and convert the result to hectares.
3. c) Justify whether the farm meets the area requirement to apply for the regional aid.
4. d) If the farmer decides to order a wire fence to surround the entire perimeter of the farm, how many meters of fencing must he buy at the hardware store?

An agricultural plot has the shape of a triangle with an area of $4500\text{ m}^2$. It is divided into two plots by a straight fence parallel to one of its sides, such that the resulting plot sharing a vertex with the original one is a triangle similar to the entire plot. If the length of the fence is $\frac{2}{3}$ of the length of the side to which it is parallel, calculate the area of each of the two resulting plots.

If the area of a regular polygon is doubled while maintaining its shape (that is, creating a similar one), by approximately what percentage does its perimeter increase?

Two regular hexagons are arranged concentrically as shown in the figure. The smaller hexagon has its vertices at the midpoints of the sides of an intermediate hexagon, which in turn has its vertices at the midpoints of the sides of the larger hexagon. If the area of the smaller hexagon is $12\text{ cm}^2$, find the area of the larger hexagon.

A cinema projector projects a rectangular image onto a screen. At a distance $d$, the image has an area of $2,4\text{ m}^2$. If we move the projector $1,5\text{ m}$ further from the screen, the area of the projected image becomes $5,4\text{ m}^2$. Knowing that the scale factor of the dimensions of the image is proportional to the distance to the screen, calculate the initial distance $d$.

The difference between the areas of two similar polygons is $105\text{ cm}^2$. Knowing that the ratio of their perimeters is $\frac{4}{3}$, determine the area of each polygon.

In a right-angled triangle $\triangle ABC$, right-angled at $A$, the altitude $AH$ to the hypotenuse $BC$ is drawn. This divides the original triangle into two similar triangles $\triangle HBA$ and $\triangle HAC$. If side $AB$ measures $15\text{ cm}$ and side $AC$ measures $20\text{ cm}$, calculate the area of the two small triangles using the scale factor between them.

You can do it on paper and send a photo with your answer.

The cover of an old book measures $15\text{ cm} \times 22\text{ cm}$. A pocket edition similar to the original is to be made. To save costs, the cover area of the pocket edition must be exactly $60\%$ of the original area. Calculate the dimensions of the pocket edition cover, rounding to two decimal places.

A rhombus has diagonals measuring $16\text{ cm}$ and $12\text{ cm}$. A rectangle is inscribed inside the rhombus such that its sides are parallel to the diagonals and its vertices lie on the sides of the rhombus. If the inscribed rectangle is similar to another rectangle of dimensions $4\text{ cm}$ and $3\text{ cm}$, determine the area of the inscribed rectangle.

The figure shows the construction described:

A team of window dressers at a department store in Barcelona is designing two similar polygonal decorative panels. The budget for special paint for the large panel is exactly $5{,}76$ times the budget for the small panel, since paint consumption depends exclusively on the area of each shape. Furthermore, due to assembly constraints, it is known that the perimeter of the large panel must exceed the perimeter of the small panel by $11{,}2\text{ m}$.

1. a) Deduce the linear similarity ratio $k$ between both panels from the area increase data.
2. b) Write a linear equation whose variable is the perimeter $P$ of the small panel and whose solution allows you to calculate that perimeter so that the project conditions are met.
3. c) Solve the proposed equation and find the value of the perimeter of the small panel and the large panel.
4. d) Knowing that the small panel is shaped like a rhomboid, propose the lengths of its adjacent sides so that it satisfies the obtained perimeter, and justify why your proposal is geometrically viable.

The Urban Planning Department of the Zaragoza City Council is putting out to tender the works to pave a pedestrian square shaped like a trapezoid, similar to another existing square in the city. The original square has a major base of $40\text{ m}$, a minor base of $24\text{ m}$, and a height of $15\text{ m}$. The design of the new square reduces all dimensions with a scale factor of $k = 0{,}75$.

To calculate the square meters of paving that must be ordered, two municipal technicians propose different procedures:

• Method 1 (Dimension scaling): First multiply each of the three dimensions (bases and height) by $0{,}75$ to obtain the reduced measurements of the new square, and then apply the trapezoid area formula with these new values.

• Method 2 (Area scaling): Directly calculate the area of the original square with its measurements and multiply the result by the area ratio ($k^2 = 0{,}75^2$).

1. a) Work out both methods step by step and check whether you arrive at exactly the same area in square meters.
2. b) If the tender price for the permeable pavement is $32\text{ €/m}^2$, calculate the total project budget for the new square.
3. c) If the City Council had to budget $15$ squares of different sizes, all similar to the original, which of the two methods is faster and more efficient for the technician's work? Which method would you choose if the scale factor were a repeating decimal such as $k = \frac{2}{3}$ to avoid rounding errors? Justify both answers.