What is a Hemisphere?
- A hemisphere is essentially half of a sphere.
- If you cut a sphere directly through its center, you get two equal halves, each known as a hemisphere.
- This shape retains many properties of the sphere but has a flat, circular base.
Formula for Area of a Hemisphere
The surface area of a hemisphere consists of two parts: the curved surface area and the area of the base.
The formula for the total surface area of a hemisphere with a radius r is:
Total\:Surface\:Area=curved\left(lateral\right)\:surface\:area+area\:of\:the\:circular\:base\\
Total\:Surface\:Area=2\pi r^2+\pi r^2\\
Total\:Surface\:Area=3\pi r^2
Solving the Equation for Radius r
To find the radius r of a hemisphere when given its surface area, we use the formula above. Let’s denote the total surface area as A:
A=3\pi r^2
Divide both sides by 3𝜋
\frac{A}{3\pi }=r^2\\
r^2=\frac{A}{3\pi }\\
Take the square root of both sides.
r=\sqrt{\frac{A}{3\pi }}
This gives us the radius in terms of the total surface area.
Properties of a Hemisphere
- Symmetry: A hemisphere is symmetric about its central axis.
- Curved Surface: The curved surface area is half that of a full sphere.
- Flat Base: It has a flat, circular base which is a significant feature distinguishing it from a full sphere.
- Constant Diameter: The diameter of the base is equal to the diameter of the original sphere.
Applications of Hemispheres
Hemispheres are commonly found in various applications:
- Architecture: Dome structures in buildings.
- Astronomy: Celestial hemispheres used in mapping the night sky.
- Engineering: Components of machinery and devices.
- Everyday Objects: Items like bowls and lenses often use a hemispherical shape.
Solved Examples
Example 1: Find the surface area of a hemisphere with a radius of 5 cm.
The total surface area of a hemisphere is given by,
A=3\pi r^2
Substitute r=5
A=3\pi (5)^2\\
A=3\pi 25\\
A=75\pi
So, the total surface area is 75𝜋 square centimetres.
Example 2: Find the curved(lateral) surface area of a hemisphere with a radius of 5 cm.
The curved(lateral) surface area of a hemisphere is given by,
A=2\pi r^2
Substitute r=5
A=2\pi (5)^2\\
A=2\pi 25\\
A=50\pi
So, the total surface area is 50𝜋 square centimetres.
Example 3: A hemisphere has a total surface area of 150𝜋 square meters. What is its radius?
The radius of the hemisphere is given by,
r=\sqrt{\frac{A}{3\pi }}
Substitute A=150𝜋
r=\sqrt{\frac{150\pi }{3\pi }}\\
r=\sqrt{50}\\
r=5\sqrt{2}\\
So, the radius of the hemisphere is 5\sqrt{2} meters.
FAQs on Area of Hemisphere.
1) What is the difference between a sphere and a hemisphere?
Answer: A sphere is a full 3D round shape, while a hemisphere is half of a sphere, having a flat circular base.
2) How do you Calculate the Area of a Hemisphere?
Answer: The formula to calculate the surface area A of a hemisphere is A=2\pi r^2, where r is the radius of the hemisphere.
3) What are the Units of Measurement for the Area of a Hemisphere?
Answer: The area of a hemisphere is measured in square units, like square meters (m²) or square centimeters (cm²), depending on the context.
