Catholic High School 2006 PSLE Math Prelim Question

The figure below, not shown to scale, is made up of semi-circles and circles. Given that the diameter of the large semicircle is 42 cm, find the area of the shaded region and express your answer as a fraction in the lowest term.

Solution

Area of large semi-circle

1/2 x (22/7) x 21cm x 21cm = 693 square cm


Area of 1 circle (within large semi-circle)

(22/7) x 7cm x 7cm = 154 square cm

Note -
Radius of circle is 7cm because diameter of large semi-circle is 42cm. This diameter is equivalent to the total diameters of the 3 smaller semi-circles at the base of the figure above. Diameter of 1 small semi-circle is therefore 42cm divided by 3 = 14cm. Radius of smaller semi-circle (and circle) is hence, 7 cm.


There are 2 circles and 3 smaller semi-circles in the large semi-circle. This is equivalent to an area of 3.5 circles within the large semi-circle. Therefore,

Area of 3.5 circles (within large semi-circle) is

154 square cm x 3.5 = 539 square cm.


Area of the portion that is inside large semi-circle, but outside the two circles and three small semi-circles is

693 square cm - 539 square cm = 153 square cm


Area of shaded portion is 1/3 the area above, therefore,

154 square cm divided by 3

= 51 and 1/3 square cm (Answer)