Question

Find the total surface area of a solid hemisphere of radius r.

Answer 1

Hint: The total surface area of a solid hemisphere is the sum of the area of the curved surface and the flat circular face of the hemisphere. Solid hemisphere is a three dimensional shape. The shape of the solid hemisphere is round at the curved end and flat at the circular end. It has three axes such as x-axis, y-axis and z-axis which defines its shape. We first calculate the area of the curved surface area of the hemisphere and then add the area of the circular flat end in order to get the final answer for total surface area of solid hemisphere of radius r.

Complete step-by-step answer:
We know that the curved surface area of a complete sphere of radius r units is $ 4\pi {r^2} $ . We know that the hemisphere is formed by cutting a sphere into two equal halves.
So, the curved surface area of a solid hemisphere of radius r units is $ \dfrac{{4\pi {r^2}}}{2} = 2\pi {r^2} $ .
Now, we have to find the area of the flat circular base of the solid hemisphere and then add both the areas to get the total surface area of a solid hemisphere.
We know that the area of a circle with radius r units is $ \pi {r^2} $ square units.
So, the area of circular base of radius r units $ = \pi {r^2} $
Now, the total surface area of the solid hemisphere $ = 2\pi {r^2} + \pi {r^2} $
Taking $ \pi {r^2} $ common from both the terms, we get,
 $ = \pi {r^2}\left( {2 + 1} \right) $
Simplifying the expression, we get,
 $ = 3\pi {r^2} $
So, the total surface area of a solid hemisphere of radius r units is $ 3\pi {r^2} $ square units.
So, the correct answer is “ $ 3\pi {r^2} $ square units”.

Note: The sphere is defined as the three-dimensional round solid figure in which every point on its surface is equidistant from its centre. The fixed distance is called the radius of the sphere and the fixed point is called the centre of the sphere. When we cut a sphere into two equal halves, we get two hemispheres. Hence, the curved surface area of a hemisphere is exactly half of that of a complete sphere.