Composite areas: a path round the lawn
Pave a path round a rectangular lawn and dig a round pond in it, then work out the area of path to pave and the grass left to mow.
Core concept
In short: Find the area of a lawn by taking the path away from the whole garden.
The area of a shaded region is the area of the whole shape minus the area cut out of it
Topic: Composite and shaded areas
A school lawn is a rectangle, and a paved path of the same width runs all the way round it. The path is the shaded region between two rectangles: the big one (lawn and path together) and the lawn. Its area: How much flat space a shape covers, measured in square units like cm². All key words is the big rectangle's area minus the lawn's area. If a path is w m wide, the big rectangle is (L + 2w) by (B + 2w), because the path is on both sides. A round pond dug in the lawn is taken away from the grass in the same way: grass = lawn − πr².
What you will learn
- Find the area of a region by subtracting one area from another.
- Find the outer dimensions of a rectangle with a border of width w.
- Combine rectangles and circles in area problems, including costs.
Before you start
- Area of a rectangle and of a circle.
Materials
- Squared paper
Equipment
- A measuring tape
Procedure
- Tap − or + to change the lawn's length and breadth, the path's width and the pond's radius.
- Read the path's area as big rectangle − lawn, and the grass as lawn − pond.
- Record a 1 m path round a 10 m by 6 m lawn, then add a pond of radius 2 m.
- Make a path whose area equals the lawn's area, and record a 2 m wide path.
How this simulation is simplified
- The path has the same width all round; paving costs Rs 500 per m².