Volume of a Cylinder

A cylinder is a right circular prism. It is a solid object with 2 identical flat circular ends, and a curved rectangular side. Like other prisms, the volume can be obtained by multiplying height by base area.

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Consider a cylinder with base radius $r$ and height $h$, as shown in the figure above. Since the area of the base is $\pi r^2$, the volume is equal to $\pi r^2 h$.

Note: Sometimes, the definition of cylinders may not require having a circular base. In such cases, the base shape will need to be given. The above definition is then called a circular cylinder.

What is the volume of a cylinder with base radius 2 and height 3?

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The volume is $\pi \times 2^2 \times 3 = 12 \pi$. $_\square$

A cylinder has a volume of $100\pi$ and a height of $4$. What is the radius of its base?

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Let $r$ denote the radius of the base. Then we have

$$\pi r^2\times4=100\pi\Rightarrow r^2=25\Rightarrow r=5.\ _\square$$

If we make a cylinder two times taller, and lengthen its base radius by three-fold, how many times larger would it become in volume?

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Let $r$ denote the initial base radius and $h$ denote the initial height of the cylinder. Then its initial volume is $V=\pi r^2h$. The problem states that the cylinder's new base radius is $r'=3r$, and new height is $h'=2h$. Hence the final volume is $V'=\pi r'^2h'=\pi\times(3r)^2\times2h=18\pi r^2h$. Therefore the answer is 18 times. $_\square$

A container contains a total $50\pi\text{ m}^3$ of water. If we are to distribute this water into cylindrical cups of base radius $0.5\text{ m}$ and height $1\text{ m}$, how many cups do we need?

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The volume of each cylindrical cup is $\pi \times 0.5^2 \times 1 = 0.25 \pi\text{ m}^3$. Therefore the number of cups we need is $\frac{50\pi}{0.25\pi}=200.$ $_\square$