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 \(\frac{50\pi}{0.25\pi}=200.\) \( _\square \)