Arcs and sectors: the garden sprinkler
Set how far a sprinkler turns and how far it sprays, and work out the area it waters and the length of its curved edge.
Core concept
A sector with angle θ is θ/360 of the circle: area = θ/360 × πr², arc length = θ/360 × 2πr
Topic: Arcs and sectors
A garden sprinkler sprays water r metres and turns back and forth through an angle: The amount of turn between two lines that meet, measured in degrees (°). All key words θ. The patch it waters is a sector, a slice of a circle. A full turn is 360°, so a sector with angle θ is the fraction: A part of a whole, written like ¾: the bottom number says how many equal parts, the top how many you take. All key words θ/360 of the whole circle. Its area: How much flat space a shape covers, measured in square units like cm². All key words is θ/360 of πr², and its curved edge, the arc, is θ/360 of the circumference 2πr. A 90° sprinkler waters a quarter of a circle; a 180° sprinkler waters a semicircle.
What you will learn
- Express a sector as a fraction θ/360 of a circle.
- Find the area of a sector and the length of an arc.
- Find the perimeter of a sector.
Before you start
- Area and circumference of a circle; fractions.
Materials
- A paper circle, scissors and string
Equipment
- A protractor and a measuring tape
Procedure
- Drag the end of the spray round to change the angle; tap − or + to change how far it sprays.
- Read the fraction of the circle, the sector area, the arc length and the perimeter.
- Record a quarter-circle sprinkler and a semicircle sprinkler.
- Record a full circle, and a 60° sprinkler reaching 7 m.
How this simulation is simplified
- The spray reaches the same distance in every direction.