What is a Hemisphere?
The word hemisphere can be separated into hemi, which denotes half, and sphere, which refers to the geometrical 3D shape in mathematics. Consequently, a hemisphere is a 3D geometric object that is made up of half of a sphere, with one side being flat and the other being a bowl-like shape. The most common examples of hemispheres are the northern and the southern hemispheres, a bowl, mushrooms and so on.
A hemisphere is created by precisely cutting a sphere along its diameter, leaving behind two identical hemispheres. The base or face of the hemisphere is the flat side of the hemisphere. Hemispheres might either be hollow or of solid form.
Read about volume of spheres : Click here.
Properties of a Hemisphere
- A hemisphere has one curved surface and one flat circular base
- Just like a sphere a hemisphere has no edges or vertices
- A line segment that connects two opposite points on the circumference of the circular base of a hemisphere and passes through its center is said to be the diameter of the hemisphere
- A line segment from the center of the circular base to any point on its curved surface is the radius of the hemisphere
Formula for the Surface Area of a Hemisphere
The total surface area of a hemisphere is the sum of its curved surface area and base surface area.
Curved surface area of a hemisphere = \( 2\pi~r^2\) [Half of the surface area of a sphere with radius r]
Area of base = \( \pi~r^2\) [Circle shaped base with radius \(r\)]
Total Surface Area (TSA) = Curved Surface Area + Area of Base
TSA = \( 2\pi~r^2~+~\pi~r^2\)
TSA = \( 3\pi~r^2\)
where ‘r’ is the radius of the hemisphere and ‘\( \pi\)’ is a constant, which is taken to be 3.14 or \( \frac{22}{7}\). Surface area is measured in square units.
Formula for the Volume of a Hemisphere
The volume of a hemisphere is half the volume of a sphere.
Volume of hemisphere, \( V~=~\frac{1}{2}~\times~\frac{4}{3}~\pi~r^3\)
\( ~~~~=~\frac{2}{3}~\pi~r^3\)
Volume is measured in cubic units.
Solved Hemisphere Examples
Example 1: Find the total surface area of the given hemisphere.
Solution:
Apply the formula for the total surface area of a hemisphere:
Total Surface Area (TSA) = \( 3\pi~r^2\)
= \( 3~\times~\frac{22}{7}~\times~(15)^2\) [Substituting the given values]
= \( 3~\times~\frac{22}{7}~\times~225\) [Square of number]
= \( 2121.43~yd^2\) [Simplified]
Therefore, the total surface area of the given hemisphere is 2121.43 square yards.
Example 2: Julie made a hemispherical bowl using her pottery skills. How much water can she fill in the bowl if its radius is 5 centimeters?
Solution:
Find the volume of the bowl to find the amount of water that can be filled in it.
Apply the formula for the volume of a hemisphere:
Volume of the hemisphere = \( \frac{2}{3}~\pi~r^3\)
Volume of the hemisphere = \( \frac{2}{3}~\pi~(5)^3\) [Substituting the value of radius ‘\( r\)’]
= \( \frac{2}{3}~\times~\frac{22}{7}~\times~(5)^3\) [Substituting \( \frac{22}{7}\) for \( \pi\)]
= \( \frac{2}{3}~\times~\frac{22}{7}~\times~125\) [Cube of number]
= \( 216.9~cm^3\) [Simplify]
Therefore, Julie can fill 261.9 cubic centimeters of water in the bowl.
Example 3: The total surface area of a hemisphere is 8 square meters. What is the radius of the hemisphere?
Solution:
Use the formula for the total surface area of a hemisphere to find the radius.
Total Surface Area (TSA) = \( 3~\pi~r^2\)
\(8~=~3~\pi~(r)^2\) [Substitute the given value]
\(\frac{8}{3~\pi}~=~r^2\) [Divide both sides by \(3~\pi\)]
\(\sqrt{\left ( \frac{8}{3~\pi} \right )}~=~r\) [Square root on both sides]
\(0.921~=~r\) [Substitute 3.14 for \(\pi\) and simplify]
Radius, \(r~=~0.921\) meters
Hence, the radius of the hemisphere is 0.921 meters.
Bowls, mushrooms, igloos, northern and southern hemispheres of the planet earth are a few real life examples of hemispherical shapes.
A hemisphere is said to be hollow if the interior space is hollow. Two radii make up a hollow hemisphere: an internal radius for the hollow inner circle and an external radius for the hollow outer circle.
A hemisphere features a circle-shaped flat face and a curved surface.