Here we will discuss about the area and circumference (Perimeter) of a circle and some solved example problems.
The area (A) of a circle or circular region is given by
A = πr\(^{2}\)
where r is the radius and, by definition,
π = \(\frac{\textrm{circumference}}{\textrm{diameter}}\) = \(\frac{22}{7}\) (approximately).
The circumference (P) of a circle with radius r is given by, P = 2πr
or,
The perimeter (circumference) of a circular region, with radius r is given by, P = 2πr
Solved example problems on finding the area and circumference (Perimeter) of a circle:
1. The radius of a circular field is 21 m, find its perimeter and area. (Use π = \(\frac{22}{7}\))
Solution:
According to the question, given r = 21 m.
Then, perimeter of a circular field = 2πr
= 2 × \(\frac{22}{7}\) × 21 m
= 2 × 22 × 3 m
= 132 m
Area of a circular field = πr\(^{2}\)
= \(\frac{22}{7}\) × 21\(^{2}\) m\(^{2}\)
= \(\frac{22}{7}\) × 21 × 21 m\(^{2}\)
= 22 × 3 × 21 m\(^{2}\)
= 1386 m\(^{2}\)
2. The perimeter of a circular plate is 132 cm, find its area. (Use π = \(\frac{22}{7}\))
Solution:
Let the radius of the plate be r.
Then, perimeter of a circular plate = 2πr
or, 132 cm = 2 × \(\frac{22}{7}\) × r
or, r = \(\frac{132 \times 7}{2 \times 22}\) cm
= \(\frac{6 \times 7}{2}\)
= 21 cm
Therefore, area of a circular plate = πr\(^{2}\)
= \(\frac{22}{7}\) × 21\(^{2}\) cm\(^{2}\)
= \(\frac{22}{7}\) × 21 × 21 cm\(^{2}\)
= 22 × 3 × 21 cm\(^{2}\)
= 1386 cm\(^{2}\)
3. If the area of a circle is 616 cm\(^{2}\) then, find its circumference. (Use π = \(\frac{22}{7}\))
Solution:
Let the radius of the circle be r cm.
Area of the circle = πr\(^{2}\)
or, 616 cm\(^{2}\) = \(\frac{22}{7}\) × r\(^{2}\)
or, r\(^{2}\) = \(\frac{616 \times 7}{22}\) cm\(^{2}\)
or, r = \(\sqrt{\frac{616 \times 7}{22}}\) cm
= \(\sqrt{28 \times 7}\) cm
= \(\sqrt{2 \times 7 \times 2 \times 7}\) cm
= \(\sqrt{14 \times 14}\) cm
= 14 cm
Therefore, radius of the circle = 14 cm.
Therefore, circumference of the circle = 2πr
= 2 × \(\frac{22}{7}\) × 14
= 2 × 22 × 2 cm
= 88 cm
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