UPSKILL MATH PLUS

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### Theory:

Hemisphere
A section of the sphere cut by a plane through any of its great circles is a hemisphere. In another way, we can say, one half of a sphere is called a hemisphere.
Curved surface area:
C. S. A. $$=$$ $$\frac{\text{Surface area of a sphere}}{2}$$

$$=$$ $\frac{4\mathrm{\pi }{r}^{2}}{2}$

$$=$$ $$2\pi r^2$$
Curved surface area of a hemisphere $$=$$ $$2\pi r^2$$ sq. units.
Total surface area:
T. S. A. $$=$$ Curved surface area of a hemisphere $$+$$ Area of the top region

$$=$$ $$2\pi r^2$$ $$+$$ $$\pi r^2$$

$$=$$ $$3 \pi r^2$$
Total surface area of a hemisphere $$=$$ $$3 \pi r^2$$ sq. units.
Hollow hemisphere
A hemisphere emptied from the inner side and has a difference in the outer and inner radius of a hemisphere is called a hollow hemisphere.

Curved surface area:
Let $$r$$ be the inner radius and $$R$$ be the outer radius of the hollow hemisphere.

The thickness of the hemisphere, $$t$$ $$=$$ $$R - r$$

C. S. A. $$=$$ Area of an internal hemisphere $$+$$ Area of an external hemisphere

$$=$$ $$2 \pi R^2 + 2 \pi r^2$$

$$=$$ $$2 \pi (R^2 + r^2)$$
Curved surface area of a hollow hemisphere $$=$$ $$2 \pi (R^2 + r^2)$$ sq. units
T. S. A. $$=$$ Curved surface area of a hollow hemisphere $$+$$ Area of the ring formed

$$=$$  $$2 \pi (R^2 + r^2)$$ $$+$$ $$\pi R^2$$ $$-$$ $$\pi r^2$$

$$=$$ $$2 \pi R^2 + 2 \pi r^2$$ $$+$$ $$\pi R^2$$ $$-$$ $$\pi r^2$$

$$=$$ $$2 \pi R^2$$ $$+$$ $$\pi R^2$$ $$+$$ $$2 \pi r^2$$$$-$$ $$\pi r^2$$

$$=$$ $$3 \pi R^2$$ $$+$$ $$\pi r^2$$

$$=$$ $$\pi(3R^2 + r^2)$$
Total surface area of a hollow hemisphere $$=$$ $$\pi(3R^2 + r^2)$$ sq. units
Important!
The properties of a hemisphere are same as the properties of a sphere.