# Find the angle subtended at the center of a circle of radius ‘a’ by an arc of length \[\left( \dfrac{a\pi }{4} \right)\]cm.

Last updated date: 21st Mar 2023

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Hint:.The arc is any portion of the circumference of a circle. Arc length is the distance from one endpoint of the arc to the other point. Use an equation to find arc length \[=2\pi r\left( \dfrac{\theta }{360} \right)\].

Complete step-by-step answer:

Given that the radius of the circle = a

We need to find the angle subtended by arc length \[\left( \dfrac{a\pi }{4} \right)\]

From the figure it is clear that the length of arc is\[\Rightarrow \dfrac{a\pi }{4}\]

We need to find \[\theta \].

The formula for finding the arc length is given by

\[\Rightarrow \]arc length\[=2\pi r\left( \dfrac{\theta }{360} \right)\]

We can find the arc length or portion of the arc in the circumference, if we know at what portion of 360 degrees the arc’s central angle is.

Arc length\[=2\pi r\left( \dfrac{\theta }{360} \right)\], where r is the radius of the circle.

\[\therefore 2\pi r\left( \dfrac{\theta }{360} \right)=\dfrac{a\pi }{4}\]

We have been given arc length\[=\dfrac{a\pi }{4}\]

Put radius, r = a

\[\Rightarrow 2\pi a\left( \dfrac{\theta }{360} \right)=\dfrac{a\pi }{4}\]

Simplifying the above equation,

\[\begin{align}

& 2\pi a\left( \dfrac{\theta }{360} \right)=\dfrac{a}{4}\pi \\

& \Rightarrow \dfrac{\theta }{360}=\dfrac{1}{8}\Rightarrow \theta =\dfrac{360}{8}={{45}^{\circ }} \\

\end{align}\]

\[\therefore \]We get the angle subtended at the center of the circle\[={{45}^{\circ }}\]

Note: Here, arc length \[=\dfrac{a\pi }{4}\]

If we assume value of \[\theta ={{45}^{\circ }}\]and applying we get

\[\begin{align}

& =2\pi a\left( \dfrac{45}{360} \right) \\

& =2\pi a\left( \dfrac{1}{8} \right)=\dfrac{a\pi }{4} \\

\end{align}\]

Complete step-by-step answer:

Given that the radius of the circle = a

We need to find the angle subtended by arc length \[\left( \dfrac{a\pi }{4} \right)\]

From the figure it is clear that the length of arc is\[\Rightarrow \dfrac{a\pi }{4}\]

We need to find \[\theta \].

The formula for finding the arc length is given by

\[\Rightarrow \]arc length\[=2\pi r\left( \dfrac{\theta }{360} \right)\]

We can find the arc length or portion of the arc in the circumference, if we know at what portion of 360 degrees the arc’s central angle is.

Arc length\[=2\pi r\left( \dfrac{\theta }{360} \right)\], where r is the radius of the circle.

\[\therefore 2\pi r\left( \dfrac{\theta }{360} \right)=\dfrac{a\pi }{4}\]

We have been given arc length\[=\dfrac{a\pi }{4}\]

Put radius, r = a

\[\Rightarrow 2\pi a\left( \dfrac{\theta }{360} \right)=\dfrac{a\pi }{4}\]

Simplifying the above equation,

\[\begin{align}

& 2\pi a\left( \dfrac{\theta }{360} \right)=\dfrac{a}{4}\pi \\

& \Rightarrow \dfrac{\theta }{360}=\dfrac{1}{8}\Rightarrow \theta =\dfrac{360}{8}={{45}^{\circ }} \\

\end{align}\]

\[\therefore \]We get the angle subtended at the center of the circle\[={{45}^{\circ }}\]

Note: Here, arc length \[=\dfrac{a\pi }{4}\]

If we assume value of \[\theta ={{45}^{\circ }}\]and applying we get

\[\begin{align}

& =2\pi a\left( \dfrac{45}{360} \right) \\

& =2\pi a\left( \dfrac{1}{8} \right)=\dfrac{a\pi }{4} \\

\end{align}\]

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