Question

What is the relationship between height, radius, and slant height of a cone?

Answer 1

Hint: Cone is a three-dimensional geometric shape that tapers smoothly from a circular base to a vertex. Apply Pythagoras theorem as there is a right angle triangle including height, radius, and slant height.

Complete step-by-step solution -
Apply Pythagoras theorem,
\[{{\text{(perpendicular)}}^{2}}+{{\text{(base)}}^{2}}={{\text{(hypotenuse)}}^{2}}\]
\[{{P}^{2}}+{{B}^{2}}={{H}^{2}}\]

Here,
\[\begin{align}
  & H=\text{Slantheight} \\
 & B=\text{Radius} \\
 & P=\text{Height} \\
\end{align}\]
\[\begin{align}
& {{\text{(Height)}}^{2}}+{{\text{(Radius)}}^{2}}={{\text{(slantheight)}}^{2}} \\
\Rightarrow & {{H}^{2}}+{{R}^{2}}={{L}^{2}} \\
\Rightarrow & L=\sqrt{{{H}^{2}}+{{R}^{2}}} \\
 \Rightarrow & R=\sqrt{{{L}^{2}}-{{H}^{2}}} \\
 \Rightarrow & H=\sqrt{{{L}^{2}}-{{R}^{2}}} \\
\end{align}\].

Note: The key concept to solve this problem is the Pythagoras theorem. By using the Pythagoras theorem we can find out the relation between slant height, radius, and height from which we can use to find slant height, radius, and height.