The figure shows a quarter circle in a square. Its centre is at one vertex, and it passes through two adjacent vertices. There are two semicircles on two adjacent sides as diameters. They create the shaded regions A and B . Show that A and B have equal area.
Step-by-step solution
Idea: Count the quarter disc in two ways. The two semicircles together have exactly the quarter disc’s area, but they overlap in A and leave out B, so A and B must balance.
- Let the side of the square be s. Quarter disc (radius s): area = 14πs2.1 mark
- Each semicircle has radius s2: area = 12π(s2)2 = πs28. Together: πs24, the same as the quarter disc.1 mark
- Both semicircles lie inside the quarter disc (no point of a semicircle on a side from the centre corner is further than s from that corner). They overlap exactly in A, and the part of the quarter disc they miss is B. So quarter disc = (semicircle 1 + semicircle 2 − A) + B.1 mark
- πs24 = πs24 − A + B ⇒ A = B.1 mark
Check: Numbers, s = 2: A = lens of two unit circles meeting at right angles = 2(π4 − 12) ≈ 0.571; B = π − (2 × π2 − 0.571) ≈ 0.571 ✓.
Answer to write in the exam
Quarter disc = 14πs2
Each semicircle = 12π(s2)2 = πs28; two = πs24
Quarter disc = (semicircle 1 + semicircle 2 − A) + B
πs24 = πs24 − A + B
∴ A = B
Common mistakes that cost marks
- Adding the two semicircles without subtracting their overlap A.
- Taking the semicircles’ radius as s; it is s2.
- Trying to compute A and B separately and getting lost; the counting argument is enough.
How this can come in the exam
A semicircle is drawn on each of two radii OA and OB of a quarter circle OAB of radius 10 cm. The total area of the two semicircles is (use π = 3.14)
- 39.25 cm2
- 157 cm2
- 314 cm2
- 78.5 cm2
Show answer
(D) 78.5 cm2
2 × 12 × 3.14 × 25 = 78.5 cm2, equal to the quarter circle 14 × 314.
Try one yourself
In the figure of this question the square has side 4 cm. Find the area of region A. (Use π = 3.14.)
Show answer
A = 2 × (14π × 22 − 12 × 2 × 2) = 2 × (3.14 − 2) = 2.28 cm2, and B is the same.
More questions like this
- In the figure, four semicircles have been drawn within the given square whose side is 2 units. The centres of these semicircles are the midpoints of the sides. They create a 4-petalled flower (shown in blue). Find the perimeter and the area of this flower.
- In the figure we see two concentric circles with a common centre O. A chord BC of the larger circle is drawn, touching the smaller circle at A. The length of BC is l. Show that the area of the green region enclosed between the two circles is 14πl2.
- In the figure, semicircles have been drawn on all the sides of a right-angled triangle as shown. Show that Area (A) + Area (B) = Area (C).
- The figure shows two circles passing through each other’s centres. Find the area of the region enclosed by the two circles in terms of the common radius r.
- In the figure, we see three triangles within a rectangle. The areas of the triangles are A, B, C, as marked. Show that the area of the rectangle is 2(A + C)(B + C)C.