Volume of a Cylinder
Summary
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.
Examples
What is the volume of a cylinder with base radius 2 and height 3?
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?
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?
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?
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 \(\frac{50\pi}{0.25\pi}=200.\) \( _\square \)