Surface area of a hemisphere and hollow hemisphere — lesson. Mathematics State Board, Class 10.

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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.

$=$

$\frac{\text{Surface area of a sphere}}{2}$

$=$

$\frac{4 π r^{2}}{2}$

$=$

$2\pi r^2$

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.

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2. Surface area of a hemisphere and hollow hemisphere

Theory: