Perimeters and Areas of Polygons

Calculate the perimetre and area of the rectangle in the diagram, with a base of $12$ cm and a height of $7$ cm.

Calculate mentally:

1. a) The perimetre and area of a square with a side length of $9$ cm.
2. b) The perimetre of a rhombus whose side length is $6.5$ m.
3. c) The area of a rectangle with a base of $8$ m and a height of $6$ m.

A rectangular billboard has an area of $48\ \text{m}^2$ and its base measures $8$ m.

1. a) What is its height?
2. b) What is its perimetre?
3. c) If the billboard had the same area but a square shape, would that be possible? Reason your answer.

A soccer field measures $105$ m long and $68$ m wide.

1. a) Calculate its area in $\text{m}^2$.
2. b) Express the area in hectares. Remember that $1\ \text{ha} = 10,000\ \text{m}^2$.
3. c) Approximately how many soccer fields like this one would fit inside a nature reserve of $1\ \text{km}^2$?

We want to lay a new floor on a rectangular terrace $6$ m long and $4$ m wide using square tiles of side length $40$ cm, without cutting any.

1. a) How many tiles fit along the length and along the width? How many tiles are needed in total?
2. b) The tiles are sold in boxes of $12$ units at $\text{RD\$}\ 1\,137$ per box. How many boxes need to be bought? How much will the flooring cost?
3. c) What is the area of one tile, in $\text{cm}^2$? Check that the total area of the tiles used matches the area of the terrace (express both in $\text{m}^2$).

Calculate the area of the triangle in the diagram, with a base of $10$ m and a height of $6$ m.

A triangular billboard has a base of $4$ m and a height of $3$ m.

1. a) What is its area?
2. b) If the poster is printed on both sides, what total surface area needs to be printed?

To calculate the area of a triangle with base $12$ cm and height $7$ cm, a student calculated $12 \times 7 = 84$ and answered: "the area is $84\ \text{cm}^2$".

1. a) Explain what mistake they made.
2. b) Calculate the correct area.
3. c) The same student calculates the area of a rhombus with diagonals $10$ cm and $6$ cm and gets $60\ \text{cm}^2$. Do they make the same mistake again? Correct it.

The rhomboid in the diagram has a base of $9$ cm and a height of $5$ cm.

1. a) Calculate its area.
2. b) If the sloping side of the rhomboid measures $6$ cm, calculate its perimetre as well.

Calculate the area of the trapezium in the diagram, whose bases measure $14$ m and $8$ m and its height $6$ m.

The area of a trapezium is $84\ \text{m}^2$, its height is $7$ m and the longer base is $16$ m.

1. a) Substitute the data into the area formula of a trapezium, $A = \frac{(B + b) \times h}{2}$, and set up the corresponding equation.
2. b) Solve the equation and calculate the length of the shorter base.
3. c) Check your result by verifying that the area comes out to $84\ \text{m}^2$.

Calculate the area of the rhombus in the diagram, whose diagonals measure $16$ cm and $10$ cm.

The diagonals of a rhombus measure $20$ cm and $14$ cm.

1. a) Calculate its area.
2. b) If we cut the rhombus along its two diagonals, it is divided into $4$ identical right-angled triangles. What are the base and height of each of these triangles?
3. c) Calculate the area of one of these triangles and check that the four areas sum to the area of the rhombus.

The regular hexagon in the diagram has a side length of $6$ cm and its apothem measures $5.2$ cm.

1. a) Calculate its perimetre.
2. b) Calculate its area using the formula $A = \frac{P \times a}{2}$, where $P$ is the perimetre and $a$ is the apothem.

A housing development wants to fence and sow with grass a rectangular plot measuring $45$ m long and $28$ m wide.

1. a) How many meters of fencing are needed if a $3$ m entrance is left unfenced?
2. b) If the fencing costs $\text{RD\$}\ 750$ per meter installed, what is the budget to enclose the plot?
3. c) The grass turf is sold in rolls of $5\ \text{m}^2$ at $\text{RD\$}\ 1\,137$ per roll. How many rolls need to be bought and how much will they cost?

State whether the following statements are true or false, reasoning your answer or providing a counterexample:

1. a) If we double the side lengths of a rectangle, its area also doubles.
2. b) A rhomboid and a rectangle with the same base and the same height have the same area.
3. c) A triangle has half the area of a rectangle with the same base and height.
4. d) Two shapes with the same perimetre always have the same area.