Volume comparison!

Geometry Level 2

Which of the following shapes has a greater volume?

a) A cone of height $r$ and radius of base $r$
b) A cylinder of base radius $r$ and height $r$
c) A hemisphere of radius $r$

Note: The measurement in the picture is not included in the problem.

Cylinder and Cone have same greatest volume Hemisphere Cylinder Cone

2 solutions

Rishabh Jain
Feb 2, 2016

Volume of: $\color{#EC7300}{(a)Cone=V_a=\dfrac{1}{3}\pi(r^2)r}=\color{#20A900}{(\dfrac{1}{3})}\pi r^3$ $\color{#D61F06}{(b)Cyclinder=V_b=\pi(r^2)r}=\color{#20A900}{(1)}\pi r^3$ $\color{darkviolet}{(c)Hemisphere=V_c=\dfrac{2}{3}\pi(r^3)}=\color{#20A900}{\dfrac{2}{3}}\pi r^3$ And since $\color{#D61F06}{1}>\color{darkviolet}{\dfrac{2}{3}}>\color{#EC7300}{\dfrac{1}{3}}$ $\Large V_b>V_c>V_a$ Hence volume of $\color{#D61F06}{cyclinder}$ is greatest.

Colourful! :)

Mehul Arora - 5 years, 4 months ago

A Former Brilliant Member
Feb 2, 2016

Volume of $\color{#20A900}{\text{cone}}$ $\Rightarrow \frac{1}{3} \pi r^2\color{#D61F06}{h}=(\color{#20A900}{\frac{1}{3}}) \pi r^3$

Volume of $\color{magenta}{\text{cylinder}}$ $\Rightarrow \pi r^2\color{#D61F06}{h}=(\color{magenta}{1})\pi r^3$

Volume of $\color{#3D99F6}{\text{hemisphere}}$ $\Rightarrow (\color{#3D99F6}{\frac{2}{3}})\pi r^3$

Now, by comparing all three volumes.

$\color{#624F41}{\Rightarrow} \color{magenta}{1}>\color{#3D99F6}{\frac{2}{3}}>\color{#20A900}{\frac{1}{3}}$

$\therefore$ $\color{magenta}{\text{Cylinder}}$ have greatest volume.

Volume comparison!

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