Question

A cone made up of a cardboard has a vertical height of 8 cm and a radius of 6 cm. If this cone is cut along the slanted height to make a sector, what is the central angle in degrees of the sector?
(a) $ {{216}^{\circ }} $
(b) $ {{180}^{\circ }} $
(c) $ {{90}^{\circ }} $
(d) $ {{36}^{\circ }} $

Answer 1

Hint: First of all find the slant height of the given cone by using the formula of slant height which is equal to $ l=\sqrt{{{\left( r \right)}^{2}}+{{\left( h \right)}^{2}}} $ where r and h is the radius and height of the cone respectively. Then the cut is made through slant height then the sector’s radius is equal to slant height and arc of the sector is equal to circumference of the base of the cone which is equal to $ 2\pi r $ where r is the radius of the cone.

Complete step-by-step answer:
The radius and height of the cone is given as 6 cm and 8 cm respectively.
In the below diagram, we have shown a cone with radius and height as r and h respectively.

In the above figure, slant height is given as CG.
As you can see that triangle CAG is forming a right angled triangle and CG is the hypotenuse so applying Pythagoras theorem in this triangle we get,
 $ {{\left( CG \right)}^{2}}={{\left( CA \right)}^{2}}+{{\left( AG \right)}^{2}} $
Substituting CA as h and AG as r in the above equation we get,
 $ {{\left( CG \right)}^{2}}={{\left( h \right)}^{2}}+{{\left( r \right)}^{2}} $
Taking square root on both the sides we get,
 $ \begin{align}
  & \sqrt{{{\left( CG \right)}^{2}}}=\sqrt{{{\left( h \right)}^{2}}+{{\left( r \right)}^{2}}} \\
 & \Rightarrow CG=\sqrt{{{\left( h \right)}^{2}}+{{\left( r \right)}^{2}}} \\
\end{align} $
Substituting the value of h and r as 8 cm and 6 cm respectively we get,
 $ \begin{align}
  & CG=\sqrt{{{\left( 8 \right)}^{2}}+{{\left( 6 \right)}^{2}}} \\
 & \Rightarrow CG=\sqrt{64+36} \\
 & \Rightarrow CG=\sqrt{100} \\
 & \Rightarrow CG=10cm \\
\end{align} $
From the above, we have got the slant height as 10 cm.
Now, look carefully towards the below figure when we are cutting the cone along the slant height (i.e. CG) to make a sector then the radius of the sector is slant height and length of the arc is equal to the circumference of the base of the cone.

Circumference of the cone is equal to:
 $ 2\pi r $
Substituting the value of r as 6 cm in the above formula we get,
 $ 2\pi \left( 6 \right) $
 $ =12\pi cm $
After making a cut along the slant height the newly formed sector will look like:

The central angle that we have to find in the above problem is the angle $ \theta $ in the above figure.
We know that $ \theta $ in the sector of a circle is equal to the division of length of arc and radius of the sector.
 $ \theta =\dfrac{\text{Arc length}}{\text{Radius}} $
Substituting arc length as $ 12\pi $ and radius as 10 in the above equation we get,
 $ \theta =\dfrac{12\pi }{10} $
Substituting $ \pi $ as $ {{180}^{\circ }} $ in the above equation we get,
 $ \begin{align}
  & \theta =\dfrac{12\left( {{180}^{\circ }} \right)}{10} \\
 & \Rightarrow \theta =12\left( {{18}^{\circ }} \right) \\
 & \Rightarrow \theta ={{216}^{\circ }} \\
\end{align} $
From the above calculations, we have found the value of the central angle is equal to $ {{216}^{\circ }} $ .
Hence, the correct option is (a).

Note: The point where you go wrong in this problem is in writing the value of radius and arc length for the sector because it demands a clear visualization of how the cutting of cone along slant height will give you radius of the sector as slant height and arc length as the circumference of the base.
You can visualize it by imagining the given cone in the form of a birthday cap so when we make a cut along the slant height then the cone shape will roll out into an arc. For instance, let us assume that the below figure is in the birthday cap so when we make a cut alone CG then the folded cone will open into an arc whose radius is CG.