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.
Step-by-step solution
Idea: A petal is the overlap of two semicircles of radius 1. Its boundary is two quarter-circle arcs, and its area is two circular segments (quarter disc minus right triangle).
- Each semicircle has radius 1 (half the side). Look at the petal at the bottom-left corner: it lies between the semicircle on the bottom side (centre (1, 0)) and the one on the left side (centre (0, 1)); they meet at the corner (0, 0) and the centre (1, 1).1 mark
- Each arc of the petal turns through 90° about its centre (e.g. from (0, 0) to (1, 1) about (1, 0)), so it is a quarter circle of length π2. Perimeter = 4 petals × 2 arcs × π2 = 4π ≈ 887 ≈ 12.57 units.1 mark
- Each half of a petal (between the chord from (0, 0) to (1, 1) and one arc) is a segment = quarter disc − right triangle = π4 − 12. One petal = 2(π4 − 12) = π2 − 1.1 mark
- Flower = 4(π2 − 1) = 2π − 4 = 447 − 4 = 167 ≈ 2.29 sq. units.1 mark
Check: The four petals are the overlaps of the four semicircles: 4 semicircles (total 2π) cover the square (4) and count the petals twice, so petals = 2π − 4 ✓.
Answer to write in the exam
Radius of each semicircle = 1
Each petal = 2 quarter arcs ⇒ perimeter = 8 × π2 = 4π = 887 ≈ 12.57 units
Segment = π4 − 12; petal = 2(π4 − 12) = π2 − 1
Area = 4(π2 − 1) = 2π − 4
∴ Area = 167 ≈ 2.29 sq. units
Common mistakes that cost marks
- Using radius 2 (the side) instead of 1.
- Counting only one arc per petal for the perimeter.
- Taking a petal’s area as a full quarter disc (π4) instead of two segments.
How this can come in the exam
Two circles of radius 7 cm cross so that each arc of their common region is a quarter circle. The perimeter of the common region is (use π = 227)
- 11 cm
- 22 cm
- 44 cm
- 14 cm
Show answer
(B) 22 cm
Two quarter arcs: 2 × 14 × 2 × 227 × 7 = 22 cm.
Try one yourself
Repeat this question for a square of side 4 units. (Use π = 3.14.)
Show answer
Radius 2: perimeter 8 × π = 25.12 units; area 4 × 2(π − 2) = 8π − 16 = 9.12 sq. units.
More questions like this
- 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.
- In the figure we see two shaded regions formed by a quarter circle, a semicircle, and a triangle. Show that the areas of the two shaded regions are equal.