Perimeter, Area, Circles and Sectors
A groundsman in Sligo wants to put a fence around a rectangular garden and then cover the grass with fertiliser. Does he need the perimeter or the area? And what changes when the garden has a curved end? In this section you will learn to measure the distance around a shape and the space inside it, including shapes with circles in them.
Perimeter and area of rectangles and triangles
The perimeter is the distance around the outside of a shape. It is measured in units of length (m, cm). The area is the amount of flat surface inside the shape. It is measured in square units ($\text{m}^2$, $\text{cm}^2$).
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle | $2l + 2w$ | $l \times w$ |
| Triangle | add the three sides | $\frac{1}{2} \times b \times h$ |
The height of a triangle must be the perpendicular height: it meets the base at a right angle. It is not the slanted side.
Why the $\frac{1}{2}$? Any triangle fits inside a rectangle of the same base and height and fills exactly half of it.
A rectangular garden in Sligo is 12 m long and 7.5 m wide. a) Find its perimeter. b) Find its area.
A rectangular garden in Sligo is 12 m long and 7.5 m wide.
- a) Find its perimeter.
- b) Find its area.
A triangle has a base of 14 cm and a perpendicular height of 9 cm. a) Find its area. b) A second triangle has the same area and a base of 7 cm. Find its perpendicular…
A triangle has a base of 14 cm and a perpendicular height of 9 cm.
- a) Find its area.
- b) A second triangle has the same area and a base of 7 cm. Find its perpendicular height.
Circles
The circumference is the perimeter of a circle. Its radius $r$ is the distance from the centre to the edge, and its diameter is $d = 2r$.
The number $\pi \approx 3.14159\ldots$ is irrational: its decimal never ends or repeats. Use the $\pi$ key on your calculator, or $3.14$ if the question says so. If a question says "in terms of $\pi$", leave $\pi$ in your answer, for example $20\pi\ \text{cm}$.
A circular table has a radius of 0.6 m. Use $\pi = 3.14$ or the $\pi$ key on your calculator. a) Find the circumference, correct to two decimal places. b) Find the area,…
A circular table has a radius of 0.6 m. Use $\pi = 3.14$ or the $\pi$ key on your calculator.
- a) Find the circumference, correct to two decimal places.
- b) Find the area, correct to two decimal places.
A circle has a circumference of $31.4$ cm (use $\pi = 3.14$). a) Find its radius. b) A sector of this circle has an arc length of $7.85$ cm. Find the angle of the…
A circle has a circumference of $31.4$ cm (use $\pi = 3.14$).
- a) Find its radius.
- b) A sector of this circle has an arc length of $7.85$ cm. Find the angle of the sector.
- c) Hence find the area of the sector.
- d) Show that the area of the sector equals $\frac{1}{2} \times \text{arc length} \times \text{radius}$.
Composite shapes and paths
A composite shape is made from simpler shapes. Split it into rectangles, triangles, circles or semicircles, find each area and add. For a shape with a hole or a path around it, find the area of the whole and subtract the part you do not want.
For the perimeter, add only the edges on the outside. A side where two parts join is not part of the perimeter.
A school yard in Cork is made of a rectangle 20 m by 15 m with a semicircle of diameter 15 m attached to one of the 15 m sides. a) Find the area of the yard, in terms of…
A school yard in Cork is made of a rectangle 20 m by 15 m with a semicircle of diameter 15 m attached to one of the 15 m sides.
- a) Find the area of the yard, in terms of $\pi$ and as a decimal to one decimal place.
- b) Find the perimeter of the yard, correct to one decimal place.
A rectangular lawn measures 18 m by 10 m. A path 1 m wide runs around the outside of the lawn. a) Find the area of the path. b) Gravel for the path costs €14.50 per…
A rectangular lawn measures 18 m by 10 m. A path 1 m wide runs around the outside of the lawn.
- a) Find the area of the path.
- b) Gravel for the path costs €14.50 per square metre. Find the total cost.
A running track is made of a rectangle 90 m long and two semicircles at the ends, each of diameter 60 m, forming a closed loop. a) Find the length of one lap, correct to…
A running track is made of a rectangle 90 m long and two semicircles at the ends, each of diameter 60 m, forming a closed loop.
- a) Find the length of one lap, correct to the nearest metre.
- b) Find the area enclosed by the track, correct to the nearest square metre.
- c) An athlete runs 5 km. How many complete laps is this, and how far into the next lap does she finish?
Arcs and sectors
A sector is a slice of a circle, like a slice of pizza, between two radii. The curved edge is an arc. The angle $\theta$ at the centre tells you what fraction of the full circle you have: $\frac{\theta}{360}$.
The arc and the area are just the circumference and area of the whole circle, multiplied by the same fraction.
You can also work backwards. If you know the arc length and the radius, find the fraction of the circle first, then the angle.
A sector of a circle has radius 10 cm and angle $72^\circ$. a) Find the length of the arc in terms of $\pi$. b) Find the area of the sector in terms of $\pi$. c) Find…
A sector of a circle has radius 10 cm and angle $72^\circ$.
- a) Find the length of the arc in terms of $\pi$.
- b) Find the area of the sector in terms of $\pi$.
- c) Find the perimeter of the sector, correct to one decimal place.
A square of side 10 cm has a circle drawn inside it so that the circle touches all four sides. A second circle passes through all four corners of the square. a) Find the…
A square of side 10 cm has a circle drawn inside it so that the circle touches all four sides. A second circle passes through all four corners of the square.
- a) Find the area of the square.
- b) Find the radius of each circle, giving the second in surd form.
- c) Find the area of the region between the two circles, in terms of $\pi$.
- d) Show that the area of the large circle is exactly twice the area of the small circle.
Back to the start: the fence in Sligo needs the perimeter, and the fertiliser needs the area. With a curved end, the perimeter includes an arc and the area includes half a circle.