Exercises: Area

For an art project, Lucas needs a piece of card that measures exactly $2.400 \text{ cm}^2$. When he goes to the stationer's, the shop assistant tells him that measurements are given in square decimetres. How many $\text{dm}^2$ should Lucas ask for for his project?

Calculate the area of the following shaded figure by counting the grid squares. Note that each small square represents an area of $1 \text{ m}^2$. Express the final result in square decametres ($\text{dam}^2$).

In farming, special units are used. Match each International System measurement with its corresponding land area unit:

1. a) $1 \text{ hm}^2$
2. b) $1 \text{ dam}^2$
3. c) $1 \text{ m}^2$

I. Are ($a$)

II. Hectare ($ha$)

III. Centiare ($ca$)

A local council has decided to create a central park of $5 \text{ hectares}$. If they want to install a play area occupying $1.500 \text{ m}^2$, how many square metres will be left for trees and paths in the rest of the park?

Select the most appropriate unit of measurement to express the area of the following objects:

1. a) The screen of a smartphone.
2. b) A top-division football pitch.
3. c) The surface area of Spain.
4. d) The floor of your bedroom.

(Options: $km^2$, $hm^2$, $m^2$, $cm^2$)

A farmer has three plots with the following areas:

• Plot A: $0,45 \text{ hm}^2$

• Plot B: $5.200 \text{ m}^2$

• Plot C: $48 \text{ dam}^2$

Order the plots from smallest to largest area by comparing their measurements in $\text{m}^2$.

Express in simple form (using a single unit) in square metres the following area measurement given in compound form:

$3 \text{ dam}^2$, $25 \text{ m}^2$ and $75 \text{ dm}^2$

In a housing estate, a square communal area measuring $10 \text{ metres}$ on each side is going to be paved. Square tiles with a surface area of $1 \text{ dm}^2$ each will be used. How many tiles will be needed to cover the entire area?

Mateo wants to change the flooring in his room and needs to find out how many square decimetres a $3,5\text{ m}^2$ rug measures. In his notebook, he writes the following resolution: "Since going from metres to decimetres means going down one step on the ladder, we multiply by $10$; therefore, $3,5\text{ m}^2 = 35\text{ dm}^2$".\na) Identify the mistake Mateo made and explain in your own words (supporting your explanation with the geometric concept of a square with a side length of $1\text{ m}$) why with surface units you do not multiply or divide by $10$.\nb) Correct the calculation and find the actual measurement of the rug in square decimetres.

A town council is going to renew the flooring of a $60\text{ m}^2$ exhibition hall using square tiles with a side length of $50\text{ cm}$.\na) Before operating with formulas, make a reasoned estimate: if each tile has a side length of half a metre, how many tiles do you need to cover a single square metre ($1\text{ m}^2$)? From that fact, estimate mentally how many tiles will be needed in total for the hall.\nb) Calculate the surface area of a tile in square metres ($m^2$) and obtain the exact number of tiles needed.\nc) Compare the exact result with your initial estimate and justify whether your estimate was consistent and useful.

A local council is planning the creation of a new suburban park. They have bought three adjacent plots whose areas are expressed in different units:

• Plot A: $0,052\text{ km}^2$

• Plot B: $140\text{ dam}^2$

• Plot C: $8.500\text{ m}^2$

Express the total area of the future park in hectares ($ha$).

The screen of a new tablet model has dimensions of $250\text{ mm}$ high by $160\text{ mm}$ wide. The manufacturer claims that the usable area of the screen is exactly $4\text{ dm}^2$. Is the manufacturer's claim true? Justify your answer by carrying out the necessary calculations in $\text{cm}^2$.

A forest fire has affected an area of the mountains. News reports indicate that $12\text{ km}^2$ and $45\text{ hm}^2$ of woodland have burned. To organise the reforestation, forestry technicians need to know the affected area in ares ($a$) and centiares ($ca$). Convert to both units.

We want to replace the floor of a kitchen that is rectangular in shape and measures $420\text{ cm}$ long by $3\text{ m}$ wide. If we are going to use square tiles with an area of $20\text{ dm}^2$ each:

1. a) Calculate the total area of the kitchen in $\text{m}^2$.
2. b) How many tiles will we need to cover the entire floor?

Order the following areas of Spanish natural parks from largest to smallest, first converting them all to the same unit of your choice:

• Park A: $12.400.000\text{ m}^2$

• Park B: $1.350\text{ ha}$

• Park C: $128\text{ km}^2$

• Park D: $1.250.000\text{ dam}^2$

A farmer owns a farm of $3\text{ ha}$, $5\text{ a}$ and $25\text{ ca}$. He decides to sell part of his land for the construction of a road. The part sold measures exactly $4.200\text{ m}^2$. What area of the farm, expressed in compound form (hectares, ares and centiares), does the farmer have left?

On a scale plan of a shopping centre, the main corridor is represented by a rectangle with an area of $450\text{ cm}^2$. If we know that in reality this corridor occupies $0,18\text{ dam}^2$:

1. a) Express both quantities in $\text{dm}^2$.
2. b) How many times larger is the real area than the area on the plan?

Calculate the area of the following irregular figure by splitting it into squares and rectangles. Each square of the grid background represents an area of $100\text{ cm}^2$. Express the final result in square decimetres ($\text{dm}^2$).

In forest emergency reports in Spain, burnt area data is usually recorded by combining the metric decimal system and agrarian units. After a small wildfire in a region of Castilla y León, environmental technicians issued the following report with two detected hotspots:\n- Hotspot A: affected an area of $0,045\text{ km}^2$.\n- Hotspot B: affected an area of $380\text{ ares}$.\na) Convert the surface area of each hotspot to square metres ($m^2$) and calculate the total burnt area expressed in hectares ($ha$).\nb) On TV news, they often translate these disasters into "regulation football pitches" (whose average standard area is about $0,7\text{ ha}$). How many full football pitches is the total devastated area approximately equivalent to? Critically assess whether this comparison helps the population comprehend the scale of the environmental damage.

An agricultural cooperative has a rectangular plot with a total area of $1,2\text{ ha}$ and a width of $80\text{ m}$. The members decide to organise the plot into three sectors:\n- The olive grove sector will occupy exactly $50\,\%$ of the land.\n- The vegetable garden sector will occupy $45\text{ ares}$.\n- The remaining space will be allocated to an irrigation pond and a tool shed.\na) Calculate the total length of the plot in metres.\nb) Determine the surface area that the irrigation pond and the shed will occupy, expressing the result in both square metres ($m^2$) and centiares ($ca$).\nc) If the vegetable garden sector is designed taking advantage of the full width of the plot ($80\text{ m}$), how many metres long must that area be?

The council has allocated land for school urban allotments with an area of $0,12 \text{ km}^2$ and $5,4 \text{ hm}^2$. To make better use of the land, they divide it into plots of $60 \text{ áreas}$ each. If each of these plots produces an average of $850 \text{ kg}$ of vegetables per year and the average market price of the vegetables is $0,75 \text{ €/kg}$, calculate the total value of the production of the entire land in one harvest.

For the construction of a new shopping centre, a rectangular building plot measuring $2,4 \text{ hm}$ long and $150 \text{ m}$ wide has been fenced off. The council requires $25 \%$ of the total area to be allocated to green spaces and another $15 \%$ to car parks. If the remainder is divided into shop units with an average area of $120 \text{ m}^2$, how many shop units can be built in total?

An artist wants to cover the side facade of a building with square solar panels measuring $50 \text{ cm}$ on each side. The facade is a rectangle $12 \text{ m}$ high and $3,5 \text{ dam}$ long, but 8 symmetrical windows measuring $150 \text{ dm}^2$ each must be deducted. Calculate the total number of solar panels he needs to buy to cover all the available area of the facade.

As part of a reforestation plan following a wildfire, work will be carried out over an area of $14 \text{ ha}, 5 \text{ a}$ and $20 \text{ ca}$. The forestry engineers have decided that one tree must be planted for every $12 \text{ m}^2$ of land. If the cost of each sapling is $4,80 \text{ €}$ and labour costs $250 \text{ €}$ per square decametre worked, what will be the total budget required for the reforestation?

A graphic designer is creating a logo on a metal sheet. The logo consists of a central square with an area of $4 \text{ dm}^2$. Around this square, he adds four smaller squares whose sides are half the length of the central square's side. Express the total area of the logo in square millimetres and in square decimetres using only simple form.

A map of a nature reserve has a scale such that $1 \text{ cm}^2$ on paper represents $2.500 \text{ m}^2$ in reality. If the park has an irregular shape that occupies $480 \text{ cm}^2$ on the map, calculate its real area in hectares, ares and centiares (compound form). If a fire destroys $0,4 \text{ km}^2$ of the park, what percentage of the total area has been saved?

The floor of a sports centre with an area of $12,5 \text{ dam}^2$ is going to be renovated. Boxes of heavy-duty tiles have been bought, where each box contains enough material to cover $2,5 \text{ m}^2$ and costs $45 \text{ €}$. In addition to the cost of the material, the installation company charges $18 \text{ €}$ per square metre laid. How much is the total bill if $5 \%$ more material than needed is bought to make up for what is wasted in cuts and breakage?

Calculate the total area of the following composite figure made of metal parts. The main part is a rectangle $0,8 \text{ m}$ by $15 \text{ cm}$, to which square tabs measuring $25 \text{ mm}$ on each side have been welded at each of its four vertices. In addition, a circular hole with an area of $450 \text{ mm}^2$ has been drilled in the centre. Express the final result in square centimetres rounded to the nearest whole number.

To pave an "L"-shaped small square, the public works department has a sketch with perpendicular corners and the following dimensions: a total base of $60\text{ m}$, a total height of $40\text{ m}$, and an empty recessed corner measuring $25\text{ m}$ wide by $15\text{ m}$ deep.\nTwo quantity surveyors from the town council debate how to calculate its area in hectares ($ha$):\n- Laura's method (additive): Break down the shape into two non-overlapping interior rectangles, calculate the area of each one in $\text{m}^2$, add them together, and convert the total to hectares.\n- Carlos's method (subtractive): Calculate the area of the full rectangle that would enclose the entire figure ($60\text{ m} \times 40\text{ m}$), subtract the area of the rectangular cutout that is not part of the square, and convert the final result to hectares.\na) Apply both methods in detail and check whether they lead to exactly the same result.\nb) Analyse and justify which of the two methods requires fewer operations and poses less risk of error when interpreting the missing dimensions.\nc) A third technician proposes converting all side measurements to hectometres ($hm$) before multiplying to obtain the hectares directly. Justify whether this third approach is more or less advantageous when performing calculations with decimals.

A company plans to install a photovoltaic solar power plant in Andalusia. Regional environmental regulations require that, to protect steppe fauna, at least one-third ($1/3$) of the estate's total area must be kept intact as an ecological corridor, without any construction.\nThe project's technical specifications state that the solar panels will occupy $180\text{ dam}^2$, while the access roads, inverters, and the transformer substation will occupy an area of $60\text{ ares}$.\na) Model the situation mathematically and calculate the minimum total surface area that the estate must have (expressed in hectares, $ha$) for the project to strictly comply with environmental regulations.\nb) If the market price of rural land in that area is $12\,500\text{ €}$ per hectare, what will be the minimum outlay that the developer company must make to acquire the estate?\nc) If a local farmer offers to sell them an adjoining estate of $0,032\text{ km}^2$, determine with reasoning whether that estate is sufficient to carry out the project or if it must be rejected.