Surface Area of Hemisphere

Hemisphere - A hemisphere is half of a sphere, typically split along a plane passing through its center and perpendicular to its base.

Curved Surface Area of Hemisphere

The curved surface area of a hemisphere refers to the total surface area of the curved or rounded portion of a hemisphere. A hemisphere is a three-dimensional geometric shape that resembles half of a sphere. It has a curved surface that extends from the base to the rounded top.

Formula of Curved Surface Area of a Hemisphere

The formula for the Curved surface area of a hemisphere is:
Curved Surface Area = 2πr²
Where:
π = 3.14
r = radius of the hemisphere

Example: If the hemisphere has a radius of 4 cm, the curved surface area would be?

Solution: Curved surface area = 2πr² (half of the surface area of a sphere)
= 2 x 3.14 x 4²
= 100.48 cm²

Example: A hemisphere has a radius of 5 cm. What is the curved surface area of the hemisphere?

Solution: First, substitute the given values into the formula: 2πr²
Given radius(r) = 5 cm
Curved surface area = 2π(5)²
= 2π (25)
= 50π cm²
Since π= 3.14, we can approximate the surface area to be approximately 157 cm².

Total Surface Area of Hemisphere

The total surface area of a hemisphere is calculated by the sum of the surface area of the curved part (half of a sphere) and the area of the circular base.

Formula of Total Surface Area of a Hemisphere

The formula for the total surface area of a hemisphere is:
Total Surface area = 2πr² + πr²
= 3πr²
Where r is the radius of the hemisphere.

Example: Let's say we have a hemisphere with a radius of 5 cm, if we need to calculate the total surface area of the hemisphere then:

Solution: We know that
The surface area of the curved part of the hemisphere would be 2πr².
= 2π (5)²
= 50π cm²
The surface area of the circular base would be πr².
= π (5)² = 25π cm²

So the total surface area of the hemisphere would be 2πr² + πr²
= 50π + 25π
= 75π cm²

Note: In this case, the surface area of the curved part is twice the surface area of the circular base. This is because the curved part covers the entire top half of the sphere, while the circular base only covers the bottom half.

Hollow Hemisphere

A hollow hemisphere is a three-dimensional geometric shape that is created by taking a sphere and cutting off one-half of it along a plane that passes through the center of the sphere. A hollow sphere is one that has been flattened out, is not solid, has interior space, and has a cavity.

Surface Area of Hollow Hemisphere

A hollow hemisphere is a half-spherical shape that is open on one side, similar to a bowl or a cup. The surface area of a hollow hemisphere refers to the total area of the outer surface of a sphere that has been cut in half along its equator. The surface area of a hollow hemisphere is calculated by subtracting the surface area of the inner hemisphere of radius r₁ from the surface area of an outer hemisphere of radius r₂.

Formula of Surface Area of a Hollow Hemisphere

The surface area of a hollow hemisphere can be calculated using the formula:

Surface area of Hollow Hemisphere = Area of External hemisphere - Area of Inner Hemisphere
Area of External Hemisphere = 2πr₂²
Area of Inner Hemisphere = 2πr₁²
Surface area of Hollow Hemisphere = 2πr₂² - 2πr₁²
= 2π( r₂² - r₁²)
Area of the ring = π( r₂² - r₁²).

Example: A hollow hemisphere has an outer diameter of 20 cm and an inner diameter of 15 cm. Find the surface area of the hollow hemisphere.

Solution: First, we need to find the radius of the outer hemisphere and the inner hemisphere.
The outer hemisphere has a diameter of 20 cm, so the radius is 20 / 2 = 10 cm.
The inner hemisphere has a diameter of 15 cm, so the radius is 15 / 2 = 7.5 cm.

Next, we need to find the surface area of the outer hemisphere and the inner hemisphere.
The surface area of a sphere can be found using the formula 2πr², where r is the radius of the sphere.

The surface area of the outer hemisphere is 2π x 10² = 200π cm²
The surface area of the inner hemisphere is 2π x 7.5² = 112.5π cm²

Finally, we subtract the surface area of the inner hemisphere from the surface area of the outer hemisphere to find the surface area of the hollow hemisphere.

The surface area of the hollow hemisphere is 200π - 112.5π = 87.5π cm²

Total Surface Area of Hollow Hemisphere

The total surface area of a hollow hemisphere refers to the sum of the outer surface area and the inner surface area of a spherical object that has been cut in half and hollowed out. It is calculated by multiplying the circumference of the base (outer surface area) and the curved surface area of the sphere (inner surface area).

Formula of Total Surface Area of Hollow Hemisphere

The formula for the total surface area of a hollow hemisphere is given by:
2π( r₂² + r₁²) + π( r₂² - r₁²)
Where r₂ is the radius of the outer surface and r₁ is the radius of the inner surface of the hollow space.

Example: If the outer radius of the hollow hemisphere is 5 cm and the inner radius of the hollow hemisphere is 3 cm, then the total surface area would be:

Solution: The total surface area of hollow sphere = 2π (r₂² + r₁²) + π ( r₂² - r₁²)
= 2π (52 + 32) + π (52 - 32)
= 84π cm²