Ex.12.2 Q10 Areas Related to Circles Solution - NCERT Maths Class 10

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Question

An umbrella has \(8\) ribs which are equally spaced (see given figure). Assuming umbrella to be a flat circle of radius \(\text{45 cm,}\) find the area between the two consecutive ribs of the umbrella.

Text Solution

 

What is known?

An umbrella has \(8\) ribs which are equally spaced. Assume umbrella to be a flat circle of radius \(= 45\,\rm{ cm.}\)

What is unknown?

The area between the \(2\) consecutive ribs of the umbrella

Reasoning:

Since there are \(8\) equal spaced ribs in an umbrella and the umbrella is assumed to be a flat circle.

\(\therefore \;\)Angle between \(2 \) consecutive ribs at the centre \(\begin{align}= \frac{{{{360}^o}}}{8} = {45^o}\end{align}\)

Area between \(2\) consecutive ribs of the umbrella = Area of a sector with angle \(45^\circ\)

\[\begin{align}&= {\frac{\theta }{{{{360}^\circ }}} \times \pi {r^2}}\\& = {\frac{{{{45}^\circ }}}{{{{360}^\circ }}} \times \pi {r^2}}\\ &= {\frac{{\pi {r^2}}}{8}}\end{align}\]

Steps:

Since there are \(8\) equally spaced ribs in the umbrella

\(\therefore\) Angle between \( 2\) consecutive ribs \(\left( \theta \right)\)

\[\begin{align}&= \frac{{{{360}^\circ }}}{8}\\& = {45^\circ }\end{align}\]

Area between \(2\) consecutive ribs of umbrella

\[\begin{align} &= {\frac{\theta }{{{{360}^\circ }}} \times \pi {r^2}}\\ &= {\frac{{{{45}^\circ }}}{{{{360}^\circ }}} \times \frac{{22}}{7} \times 45 \times 45}\\ &= {\frac{1}{8} \times \frac{{22}}{7} \times 45 \times 45}\\& = {\frac{{22275}}{{28}}{\text{c}}{{\text{m}}^2}}\\&{ = 795.535\,\,{\text{c}}{{\text{m}}^2}}\end{align}\]