Exercises: Compound shapes and measurement problems

The figure shows a house formed by a rectangle $8$ m wide and $5$ m high, and a triangular roof $3$ m high.

1. a) Calculate the area of the rectangular part.
2. b) Calculate the area of the triangular roof.
3. c) What is the total area of the facade?

In the grid shown in the figure, each square represents $1\ \text{cm}^2$. The blue outline is the edge of a small lake.

1. a) Count the squares that lie entirely inside the lake.
2. b) Count those that the lake touches only in part.
3. c) Give an estimate of the area of the lake, explaining the criterion you used.

The frame in the figure is a rectangle $30$ cm by $24$ cm, and the photo inside measures $24$ cm by $18$ cm.

1. a) Calculate the area of the outer rectangle and that of the photo.
2. b) What is the area of the frame (the orange area)?
3. c) How wide is the frame at the top? Explain how you calculated it.

A giant letter 'T' for a photocall is formed by two rectangles: the top crossbar, $120$ cm long and $30$ cm high, and the vertical stem, $90$ cm high and $30$ cm wide, attached below and centred.

1. a) Calculate the total area of the sign.
2. b) If the material costs $8,50\ \text{€}$ per square metre, how much does the sign cost? First, express the area in $\text{m}^2$.

To mark a hiking trail, Lucía designs a wooden sign by joining two pieces without overlapping them: a square with a side length of $10\text{ cm}$ and, on its top side, an equilateral triangle also with a side length of $10\text{ cm}$.

To order a metal strip to protect the entire outer edge, Lucía makes the following calculation:

"Perimeter of the square = $4 \cdot 10 = 40\text{ cm}$. Perimeter of the triangle = $3 \cdot 10 = 30\text{ cm}$. Therefore, I need $40 + 30 = 70\text{ cm}$ of metal strip".

1. a) Explain with reasoning what conceptual error Lucía has made and which part of the figures she has overcounted.
2. b) How many centimetres of metal strip are actually needed to surround the outer perimeter of the sign?
3. c) If the height of the equilateral triangle is approximately $8,7\text{ cm}$, calculate the total area of wood of the sign.

In the Biology and Geology subject, students have designed on a grid plan a small amphibian pond in the school garden. On the plan, each grid square represents $1\text{ m}^2$.

The pond has an irregular shape and is completely contained within a rectangular area $5\text{ m}$ long by $4\text{ m}$ wide. When counting on the plan, the pond completely covers $9$ full squares and touches or partially covers another $8$ squares.

1. a) Before doing any calculations, make an estimate: is it possible for the surface area of the pond to be $22\text{ m}^2$? Justify your answer looking solely at the rectangle that contains it.
2. b) State between which two values (minimum and maximum) the actual area of the pond certainly lies.
3. c) Apply the criterion of adding half of the partial squares to the complete squares to obtain an estimate of the area. Assess whether the result obtained is consistent with your analysis in section a).

The garden in the diagram is a rectangle measuring $12$ m by $8$ m with a circular swimming pool of radius $2$ m inside it.

1. a) Calculate the area of the whole garden.
2. b) Calculate the area of the pool (take $\pi \approx 3,14$).
3. c) What area of lawn remains?
4. d) How many metres of fencing are needed to enclose the entire garden?

We want to paint a rectangular wall $5$ m wide and $2,5$ m high that has a window measuring $1,5$ m by $1,2$ m.

1. a) What surface area needs to be painted?
2. b) A tin of paint covers $10\ \text{m}^2$ and costs $24,90\ \text{€}$. How many tins are needed to apply two coats of paint? How much will they cost?

The diagram shows a running track made up of two straight sections of $80$ m and two semicircles of radius $20$ m at the ends.

1. a) How long is each of the two curved sections? Remember that two identical semicircles form a complete circle (take $\pi \approx 3,14$).
2. b) Calculate the total length of one full lap of the track along its inside edge.
3. c) The lanes of the track are $1,22$ m wide. Does a runner cover the same distance running in the inside lane as in the outside lane? Reason your answer without doing calculations.

An arrow-shaped sign is formed by a rectangle $30$ cm long and $12$ cm high (the body) and a triangle with base $12$ cm and height $9$ cm (the tip).

1. a) Draw a sketch of the sign in your notebook.
2. b) Calculate the total area of the sign.
3. c) If the vinyl to print it costs $15\ \text{€}$ per square metre, how much does the material for the sign cost? First express the area in $\text{m}^2$.

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

A wooden shield is pentagonal in shape: it is a rectangle $20$ cm wide and $24$ cm high, topped at the bottom by a triangle with a base of $20$ cm (the tip of the shield) and a height of $10$ cm.

1. a) Calculate the area of the shield.
2. b) We want to edge the entire outline with a metallic tape. The sloping sides at the tip measure $14,1$ cm each. How many centimetres of tape are needed?

We want to construct an L-shaped mural on a wall, formed by two rectangles: a vertical one $3$ m high and $0,8$ m wide, and a horizontal one $1,6$ m long and $0,8$ m high sticking out to the right from the bottom of the first, without overlapping it.

1. a) Draw a sketch and calculate the total area of the mural.
2. b) The paint for the mural is bought in tins that cover $2\ \text{m}^2$ each. How many tins are needed?
3. c) What is the outer perimetre of the mural?

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

The communal courtyard of a building is "L"-shaped. It can be described geometrically as a large rectangle $12\text{ m}$ long by $8\text{ m}$ wide that is missing, in one of its corners, a rectangular gap $5\text{ m}$ long by $3\text{ m}$ wide occupied by the meter shed.

1. a) Method 1 (by subtraction): calculate the area of the courtyard by subtracting the area of the gap from the area of the large rectangle.
2. b) Method 2 (by decomposition): mentally divide the courtyard into two rectangles using a straight cut. Write down the dimensions of the two rectangles you have chosen, calculate their areas separately, and add them together to verify that you obtain the same result.
3. c) If you were asked to calculate the exterior perimeter of the entire courtyard, does any of the previous area calculations help you directly, or must you proceed differently? Give your reasoning on which method seemed more convenient to you for the area and why.

A municipality's town hall is going to resurface the floor of the sports centre's skating rink. The rink consists of a central rectangle $25\text{ m}$ long by $12\text{ m}$ wide, and two semicircular areas attached to its two shorter ends (each semicircle has a diameter of $12\text{ m}$, that is, a radius of $6\text{ m}$).

To protect the floor, a synthetic resin will be applied with the following technical specifications:

• $0,4\text{ kg}$ of resin are needed for each square metre of the rink.

• The product is only sold in $10\text{ kg}$ drums.

• Each drum costs $45,80\text{ €}$.

(Take $\pi \approx 3,14$).

1. a) What complete geometric figure do the two semicircular ends form together? Calculate the total area of the skating rink.
2. b) How many kilograms of resin are needed at a minimum to cover the entire surface?
3. c) How many whole drums must the town hall purchase and what will be the total cost of the paint?

A rectangular swimming pool measuring $10$ m long and $4$ m wide is surrounded by a path $1,5$ m wide.

1. a) What are the total length and width of the overall area (pool plus path)? Make a sketch in your notebook.
2. b) Calculate the area of the path.
3. c) The paving slabs for the path cost $32,90\ \text{€}$ per square metre installed. How much will it cost to pave it?
4. d) For safety, the entire outer edge needs to be fenced. If fencing costs $14,50\ \text{€}$ per metre, what is the total cost of the work (paving plus fencing)?

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

A small square of side length $2$ cm is cut out from a square piece of card of side length $8$ cm.

1. a) Calculate the area of the resulting shape. Does it depend on where the cut-out is made?
2. b) Calculate the perimetre of the resulting shape in each of these three cases: the cut-out is made at a corner, in the middle of a side and in the interior of the card (a hole).
3. c) A student claims: "If removing a piece from a shape decreases its area, the perimetre always decreases too." Is this true? Use your results from part b) to justify your answer.

To find the area of the frontage of a house consisting of a rectangle $8$ m wide and $5$ m high with a triangular roof of height $3$ m, a student calculated $8 \cdot 5 + 8 \cdot 3 = 64$ and answered $64\ \text{m}^2$.

1. a) Explain what mistake she made in calculating the area of the roof.
2. b) Calculate the correct area of the frontage.
3. c) The frontage also has a door measuring $2$ m by $1$ m and a square window of side length $1,5$ m. What surface area needs to be painted?

A rectangular room measuring $4,5$ m by $3,2$ m has two doors measuring $0,9$ m wide and a built-in wardrobe occupying $1,8$ m of one of the walls.

1. a) Calculate the perimetre of the room.
2. b) How many metres of skirting board are needed? Remember that skirting board is not placed across doorways or where the wardrobe is.
3. c) Skirting board costs $6,75\ \text{€}$ per metre and is only sold in strips of $2$ m. How many strips need to be bought and how much will they cost?

A trapezium-shaped park has a longer base of $50$ m, a shorter base of $30$ m and a height of $24$ m. Inside it, there is a triangular play area with a base of $18$ m and a height of $12$ m; the rest is grass.

1. a) Calculate the total area of the park.
2. b) Calculate the area covered by grass.
3. c) To sow the grass, $1$ kg of seed is needed for every $40\ \text{m}^2$, and the seed is sold in bags of $5$ kg at $22,40\ \text{€}$ per bag. How many bags need to be bought and how much will they cost?

An ink blot has fallen onto grid paper with grid squares of $1\ \text{cm}^2$. Counting carefully, the blot completely covers $14$ squares and partially touches another $18$.

1. a) Between which two values is the area of the blot guaranteed to lie? Reason your answer.
2. b) Give a reasonable estimate of the area and explain the criteria you used.
3. c) What is the maximum possible error of your estimate?

A renovation company is going to paint the façade of an agricultural warehouse. The façade consists of a rectangular wall $10\text{ m}$ wide by $4\text{ m}$ high, topped at its upper part by a triangular pediment with a base of $10\text{ m}$ and an unknown height $h$.

Located in the rectangular wall are an access door $3\text{ m}$ wide by $3\text{ m}$ high and two square ventilation windows with a side length of $1,5\text{ m}$ each. Neither the door nor the windows are going to be painted.

1. a) Write a simplified algebraic expression representing the net area to be painted in terms of the height $h$ of the triangular roof.
2. b) The final invoice states that exactly $46,5\text{ m}^2$ of façade were painted. Set up an equation using your previous expression and calculate the height $h$ of the triangular pediment.
3. c) To finish the work, a metal gutter is placed on the two sloping sides of the triangular roof. If each of those sides measures $6,4\text{ m}$, what is the complete exterior perimeter of the finished façade?

In a secondary school, a rectangular dirt courtyard $24\text{ m}$ long by $16\text{ m}$ wide is going to be remodelled into a green area with natural grass. Inside this area, three zones must be kept free of grass:

• A circular space around a centuries-old olive tree, with a radius of $3\text{ m}$.

• Two identical square areas for chess tables, each with a side length of $4\text{ m}$.

(Take $\pi \approx 3,14$).

1. a) Model the situation and calculate the net area of ground that will be covered with grass.
2. b) Around the border of the olive tree circle and the two square areas, a protective rubber edging will be installed to separate the soil from the grass. How many linear metres of edging are needed at a minimum? Round up to the nearest whole metre.
3. c) Two quotes are received from local companies:

• VerdeSur Company: charges a flat fee of $750\text{ €}$ for transport and ground levelling, plus $8,50\text{ €}$ for each square metre of installed grass.

• EcoJardín Company: does not charge a flat fee, but bills $11,00\text{ €}$ for each square metre of installed grass.

Which of the two companies is more economical for the school? Justify your decision by comparing the total amounts of both options.

Unit challenge. Design your ideal nook! You are going to plan the renovation of your bedroom or a room in your house by applying plane geometry to a real situation.

1. Real measurement: Using a tape measure, measure the length and width of the room (assume it is rectangular) and accurately obtain its perimeter and total area in $\text{m}^2$.

2. Scale design: In GeoGebra or on grid paper, draw the new layout. You must include: a triangular table, a quadrilateral rug, and a beanbag or lamp with a circular or semicircular base.

3. Geometric calculations: Determine the perimeter and area of each 2D element. Then, find the remaining walkable area by solving the composite figure problem (subtracting the furniture from the total area).

4. Budget and report: Check the price per $\text{m}^2$ of vinyl flooring on a DIY store's website (for example, $8,95\text{ €/m}^2$) and calculate the cost of covering the free area. Write a brief one-page report with the floor plan and justified calculations.

Assessment criteria:

• Correctness in taking real measurements and in applying perimeter and area formulas with their corresponding units.

• Accurate and justified geometric drawing of triangles, quadrilaterals, and circles.

• Correct resolution of the composite area calculation and the final budget estimation.