What is a Sphere?
A sphere is a 3D figure which is derived from the two-dimensional rotation of a circle. Therefore, a sphere is a three-dimensional solid figure, made up of all points in the space which lie at a constant distance from the centre of the sphere. Example - a ball. In other words, a sphere is a set of three-dimensional points on the surface which is equidistant from the centre. Our earth is one of the other examples of a sphere.
What is Hemisphere?
Now let's take a solid sphere and slice it into two equal halves with a plane that passes through its centre. The resultant half of the sphere is called a hemisphere. The word ‘Hemi’ means half and the sphere is a three-dimensional shape of a circle where a fixed distance is taken in all the directions from the centre in space. The fixed distance from the centre to the surface is called the radius. For example, cutting the globe into half will result in two hemispheres, the shape of our brain is also hemisphere, half-cut apple, grapefruit, guava, etc are also hemisphere.
The Surface Area of Hemisphere
The surface area of the hemisphere is also considered as the sum of the surfaces of all the structures covering the surface of the hemisphere. Suppose if you want to wrap a hemisphere then you need to cover the entire surface of the hemisphere. The curved surface area of a hemisphere is half the curved surface area of sphere which equals to 1/2 x 4πr2
The curved surface area of a hemisphere = 22πr2
The total surface area of a hemisphere is the sum of its curved surface area and the area of its base. The base of a hemisphere is circle of radius r same as the radius of the sphere.
We know, the radius of a circle = πr2
The Total Surface Area of Hemisphere = The curved surface area + The area of the base.
= 2πr2 + πr2
= 3πr2
Solved Problems:
Example1- If the radius of the hemisphere is 7 cm then find the curved surface area of the hemisphere.
Solution: According to the formula of the curved surface area of a hemisphere,
The curved surface area of hemisphere = 1/2 x 4πr2
= 2πr2
= 2π(7)2
= 308 cm2
Example 2 - Find (i) the curved surface area and (ii) the total surface area of a hemisphere of radius 21 cm.
Solution: The radius of the curved surface area of the hemisphere is 21 cm
= 2πr2 = 2 x 21x21 = 2772 cm2
The total curved surface area = 3πr2 = 3x 21x21 = 4158 cm2
Example 3 - Find the surface area of the hemispherical dome of a building whose base is of circumference 17.6 cm. If the cost of painting is Rs 5 per 100 cm2 then find the cost of painting the Dome.
Solution: Painting a Dome requires only the curved surface to be painted. Thus, the curved surface area of the hemisphere of the dome can be calculated if the radius of the dome is given.
If the circumference of a dome = 17.6 m
2πr = 17.6 m.
So, the radius of the dome = 17.6 x 7/ (2 x 22) m= 2.8 m
The curved surface area of the dome = 2πr2
= 2 x 2.8 x 2.8 m2
= 49.28m2
If the cost of painting 100 cm2 is Rs 5 then the cost of painting 1 m2 is Rs 500.
Therefore, the cost of painting the whole dome = Rs 500 x 49.28
= Rs 24,640.
The total price of painting the hemispherical dome is Rs 24,640.
