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With the vertices A, B and $C$ of a triangle A B C as centres, arcs are drawn with radii $6 cm$ each. If $A B = 20 cm, B C = 48 cm$ and $C A = 52 cm,$ then find the area of the shaded region. (Use $π = 3.14)$

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435 c m^{2}

423.48 c m^{2}

438 c m^{2}

None of these

answer is A.

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Detailed Solution

Given that, With the vertices A ,B & C of a triangle as centers and arcs are drawn with radii 6 cm,

A B = 20 cm, B C = 48 cm, C A = 52 cm

.
We know that, the area of sector is

θ 360 ° π r 2

.
According to Heron’s formula, the area is A

= s s − a 5 − b s − c

.
Calculating the area of arcs,

θ 360 ° π r 2 = 180 ° 360 ° × 3.14 × 62

⇒ 18 × 3.14

⇒ 56.52 c m 2

Calculating the value of semi-perimeter,

S = a − b − c 2

⇒ 20 + 48 + 52 2

⇒ 1202

⇒

60
Calculating the area of triangle by using Heron’s Formula

A = s s − a 5 − b s − c

A

60

( 60 - 20 ) (60 - 48 ) (60 - 52)

⇒ 60 × 40 × 12 × 8

⇒

480 × 480

⇒ 480 c m 2

Calculating the area of shaded region by subtracting the area of the three arcs from the area of triangle,

480 − 56.52 = 423.48

Therefore, the area of the shaded region is 423.48

m^{2}

.
Hence the correct option is 1.

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