Composite Figures 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?

A giant letter 'T' for a photo backdrop 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 centered.

1. a) Calculate the total area of the sign.
2. b) If the material costs $\$170$ per square meter, how much does the sign cost? First, express the area in $\text{m}^2$.

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 $\$300$ per square meter, how much does the material for the sign cost? First express the area in $\text{m}^2$.

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.

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 $8 \times 5 + 8 \times 3 = 64$ 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?

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.

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 $\$498$. How many tins are needed to apply two coats of paint? How much will they cost?

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 $\$658$ per square meter installed. How much will it cost to pave it?
4. d) For safety, the entire outer edge needs to be fenced. If fencing costs $\$290$ per meter, what is the total cost of the work (paving plus fencing)?

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 $\$448$ per bag. How many bags need to be bought and how much will they cost?

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 meters of fencing are needed to enclose the entire garden?

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.

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 centimeters of tape are needed?

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.

A rectangular room measuring $4.5$ m by $3.2$ m has two doors measuring $0.9$ m wide and a built-in closet occupying $1.8$ m of one of the walls.

1. a) Calculate the perimeter of the room.
2. b) How many meters of baseboard are needed? Remember that baseboard is not placed across doorways or where the closet is.
3. c) Baseboard costs $\$135$ per meter and is only sold in strips of $2$ m. How many strips need to be bought and how much will they cost?

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.

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?