Exercises: Perimeter, Area, Circles and Sectors

A rectangular garden in Sligo is 12 m long and 7.5 m wide.

1. a) Find its perimeter.
2. b) Find its area.

A triangle has a base of 14 cm and a perpendicular height of 9 cm.

1. a) Find its area.
2. b) A second triangle has the same area and a base of 7 cm. Find its perpendicular height.

A circular table has a radius of 0.6 m. Use $\pi = 3.14$ or the $\pi$ key on your calculator.

1. a) Find the circumference, correct to two decimal places.
2. b) Find the area, correct to two decimal places.

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.

1. a) Find the area of the yard, in terms of $\pi$ and as a decimal to one decimal place.
2. b) Find the perimeter of the yard, correct to one decimal place.

A sector of a circle has radius 10 cm and angle $72^\circ$.

1. a) Find the length of the arc in terms of $\pi$.
2. b) Find the area of the sector in terms of $\pi$.
3. c) Find the perimeter of the sector, 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.

1. a) Find the area of the path.
2. 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.

1. a) Find the length of one lap, correct to the nearest metre.
2. b) Find the area enclosed by the track, correct to the nearest square metre.
3. c) An athlete runs 5 km. How many complete laps is this, and how far into the next lap does she finish?

A circle has a circumference of $31.4$ cm (use $\pi = 3.14$).

1. a) Find its radius.
2. b) A sector of this circle has an arc length of $7.85$ cm. Find the angle of the sector.
3. c) Hence find the area of the sector.
4. d) Show that the area of the sector equals $\frac{1}{2} \times \text{arc length} \times \text{radius}$.

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.

1. a) Find the area of the square.
2. b) Find the radius of each circle, giving the second in surd form.
3. c) Find the area of the region between the two circles, in terms of $\pi$.
4. d) Show that the area of the large circle is exactly twice the area of the small circle.