**Curved Surface Area of Hemisphere**

- Sphere is three-dimensional geometrical figure having surface area and volume like a ball. And if we cut the sphere from middle by a plane then the structure formed will be hemisphere or half sphere as shown in figure.
- If r is the radius of sphere, then the total surface area of sphere is given by,

Total surface area of sphere = 4*π*r^{2}

__Derivation of total surface area of sphere:__

- If we apply same force from all sides on the cylinder having circular bases of radius r, height h then it will be converted into sphere of radius r as shown in figure.
- And also, if we open sphere from all sides then it forms the cylinder again with circular base having radius r and height h.
- Hence, we can say that, the curved surface area of cylinder is equal to the total surface area of sphere.
- But, when we form the sphere from cylinder then height of the cylinder becomes equal to the diameter of the cylinder as shown in figure.

So, height of cylinder = diameter of sphere

Hence, h = d = 2*r

Now, we can write as,

Total surface area of sphere = curved surface area of cylinder

But we already know that,

Curved surface area of cylinder = 2πrh

Hence, total surface area of sphere = 2πrh

But, h = 2r

Total surface area of sphere = 2πrh = 2πr*2r

**Total surface area of sphere = = 4*π*r ^{2 }**(Hence proved.)

**Derivation of Curved Surface area of Hemisphere:**

- We know that hemisphere is formed when we cut sphere from its middle in two equal parts. So, hemisphere is the half part of sphere. And when we join to equal hemisphere then there will be formation of sphere as shown in figure.
- Hemisphere has outer curved surface and a circular upper face also.
- Let us consider the sphere of radius r is cut into two equal parts then hemisphere of radius r will be formed as shown in figure below.

- We already derived the total surface area of sphere which is given by,

Total surface area of sphere = 4πr^{2}

- Hence, the curved surface area of hemisphere will be half of the total surface area of sphere.

Curved surface area of hemisphere = ½*total surface area of sphere

= ½*4πr^{2}

= 2πr^{2}

Thus, the curved surface area of hemisphere is given by,

**Curved surface area of hemisphere = 2πr ^{2}**

Hence proved.