Mr Tan had a rectangular piece of card. He cut out 4 identical semicircles of diameter 28 cm from the card. Two semicircles were cut along the top edge and two along the bottom edge, with each diameter lying on an edge. The perimeter of the remaining piece of card is 300 cm. (Take π = 227)
- What is the total area of all the semicircles that Mr Tan cut out?
- The width of the original rectangle is 50 cm. Find the length of the original rectangle.
Need a hint?
For each semicircle, the straight diameter disappears from the outline but a curved edge appears. Work out the original perimeter first, then halve it to get length + width.
Worked answer
- (a) Radius of each semicircle = 28 ÷ 2 = 14 cm.
- (a) Area of one full circle = 227 × 14 × 14 = 616 cm², so one semicircle = 616 ÷ 2 = 308 cm².
- (a) Total area of 4 semicircles = 4 × 308 = 1232 cm².
- (b) Curved edge of one semicircle = 227 × 28 ÷ 2 = 44 cm, so 4 curved edges = 176 cm.
- (b) The 4 diameters removed from the edges = 4 × 28 = 112 cm.
- (b) Original perimeter = 300 − 176 + 112 = 236 cm.
- (b) Length + width = 236 ÷ 2 = 118 cm.
- (b) Length = 118 − 50 = 68 cm.
The mistake this question tests
Area and perimeter formulas
Pupils add the semicircle arcs to the perimeter without first removing the diameters, or they use the full circle area instead of half of it.
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