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

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 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?

Aoife wants to calculate the area of a cardboard rhomboid for an Art project. She measures with her ruler and writes down: base $b = 12\text{ cm}$, slanted side $c = 7\text{ cm}$, and height perpendicular to the base $h = 5\text{ cm}$. In her notebook she writes:

"$A = b \times c = 12 \times 7 = 84\text{ cm}^2$".

1. a) Clearly explain the mistake made by Aoife and why, geometrically, it makes no sense to multiply the base by the slanted side to find the surface area.
2. b) Calculate the correct area of the rhomboid and its perimeter.
3. c) Which geometric shape with sides of $12\text{ cm}$ and $7\text{ cm}$ would have an area of exactly $84\text{ cm}^2$? Justify your answer.

To fence off a rectangular play area in the high school courtyard, a full $24\text{ m}$ roll of barrier tape is available, without cutting or wasting any of it (that is, the perimeter of the area must be exactly $24\text{ m}$). The dimensions of the sides must be whole numbers of meters.

1. a) Write down all possible combinations of base and height (integer values) that this rectangle can have.
2. b) Calculate the area of each of the rectangles obtained in the previous part.
3. c) If the goal is for the students to have the greatest possible space to play inside the delimited enclosure, which of the options should you choose? What specific name does that rectangle have?

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

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 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 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 football pitch 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 football pitches like this one would fit inside a nature reserve of $1\ \text{km}^2$?

The basin of a children's pool at a municipal sports center is shaped like an isosceles trapezoid. The longer base measures $14.8\text{ m}$, the shorter base measures $9.2\text{ m}$, and the perpendicular distance between both bases (the height) is $6.1\text{ m}$. The city council is going to order a custom-made protective tarp to cover it in winter.

1. a) Before working with decimals, round the three measurements to the nearest meter and make a mental estimate of the tarp's area. Write down the operation you thought of and the estimated result.
2. b) Calculate the exact area the tarp must have from the actual measurements of the trapezoid.
3. c) Check whether your estimate was reasonable by calculating the difference between the exact and estimated area. If the tarp costs $8.50\text{ €/m}^2$, how many euros of difference would budgeting the purchase with the estimate instead of the exact figure have entailed?

A neighborhood association in Galway is going to renovate a small square shaped like a regular hexagon with a side length of $6\text{ m}$. The city council's blueprints indicate that the apothem of this polygon measures $5.2\text{ m}$. The project consists of placing a wooden edging around the entire perimeter, at a price of $€4.20$ per linear meter, and covering the entire interior ground with rustic turf that costs $€2.50$ per square meter. For the work, they have a municipal grant of $€350$.

1. a) Model the situation by calculating the linear meters of edging needed and the surface area in square meters that the turf will cover.
2. b) Calculate the total budget required for the work (edging plus turf).
3. c) Interpret the result obtained: is the municipal grant of $€350$ enough to fund the project? If money is lacking or left over, indicate the exact amount.

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 metres of fencing are needed if a $3$ m entrance is left unfenced?
2. b) If the fencing costs $\text{€}\ 12.50$ per metre installed, what is the budget to enclose the plot?
3. c) The grass turf is sold in rolls of $5\ \text{m}^2$ at $\text{€}\ 18.95$ per roll. How many rolls need to be bought and how much will they cost?

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 $84\ \text{m}^2$.

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.

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 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.

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 $\text{€}\ 18.95$ 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 $\text{m}^2$).

In the courtyard of a school, a rhombus-shaped logo is going to be painted with diagonals measuring $D = 8\text{ m}$ and $d = 6\text{ m}$. Two 1st Year students argue about the best way to calculate its surface area:

• Cian proposes: "We draw a rectangle with a base of $8\text{ m}$ and a height of $6\text{ m}$ enclosing the rhombus. Its area is $8 \times 6 = 48\text{ m}^2$. In the corners, there are four equal right-angled triangles with a base of $4\text{ m}$ and a height of $3\text{ m}$; if we subtract those four triangles from the outer rectangle, we get the area of the rhombus."

• Aoife proposes: "We can directly apply the formula $A = \dfrac{D \times d}{2}$, because the rhombus takes up exactly half of the rectangle that contains it."

1. a) Fully develop Cian's procedure: calculate the area of one of the corner triangles, the sum of the four, and the final subtraction.
2. b) Apply Aoife's direct method and check whether both results match.
3. c) Compare both methods: which one is more efficient in this problem and why? In what cases of irregular polygonal shapes drawn on a grid is it essential to use Cian's strategy (enclosing in a rectangle and subtracting the excess corners)?

A livestock farmer in Kilkenny has $120\text{ m}$ of metal fencing to enclose a resting pen for his sheep and wants it to have the maximum possible area. He is hesitating between three designs, all with an exact perimeter of $120\text{ m}$:

1. An equilateral triangle with a side length of $40\text{ m}$ and whose perpendicular height measures $34.64\text{ m}$.

2. A square with a side length of $30\text{ m}$.

3. A regular hexagon with a side length of $20\text{ m}$ and whose apothem measures $17.32\text{ m}$.

1. a) Calculate the area enclosed by each of the three proposed options.
2. b) Compare the three results: with the same perimeter ($120\text{ m}$), which of the three regular polygons encloses the greatest area? What geometric conclusion can you deduce about how the area changes as the number of sides of a regular polygon with a constant perimeter increases?
3. c) Animal welfare regulations require that the pen have at least $1000\text{ m}^2$ for the size of his flock. Does any of the three options meet this? If a regular polygon with many more sides were built (for example, a regular decagon or an almost circular shape with those same $120\text{ m}$ of fence), do you think it would be possible to exceed that $1000\text{ m}^2$ threshold? Justify your answer based on the observed trend.