Area of Hemisphere

Introduction

The area of a hemisphere is of two types: The curved surface area of a hemisphere = 2𝜋r² square units. The total surface area of a hemisphere = 3𝜋r²square units. Two-dimensional geometry is concerned with the x-y plane in mathematics. Three-dimensional geometry is an extension of two-dimensional geometry that deals with three axes in the Cartesian plane, namely x, y, and z.

3D shapes have three dimensions: length, breadth, and height.

In most cases, three-dimensional objects are created by rotating two-dimensional objects. Our planet's spherical shape is one of the best examples of 3D shapes. The sphere is formed by rotating a 2D shape known as the circle.

In this tutorial, we will discuss the area of the hemisphere.

Hemisphere

• The word "hemisphere contains two words, i.e., “Hemi” (means half) and "sphere" (a three-dimensional geometry analogous to a two-dimensional circle).

• In other words, a hemisphere is defined as the half of a sphere.

• If a sphere is cut into two equal parts, each part is called a hemisphere (as shown in the figure).

• The hemisphere contains one curved surface and a flat surface. The flat surface is known as the base or face of the hemisphere.

A hemisphere has several properties that are summarized below.

• It does not contain any edges or vertices.

• It has a curved surface area.

• The dimension of the hemisphere is defined as the diameter or radius of the hemisphere.

• The diameter of a hemisphere is defined as the line segment connecting any two points located on the circumference and passing through the center of the hemisphere.

• It is not considered a polyhedron as no polygon is enclosed in a hemisphere.

• The radius of the hemisphere is half of the diameter of the hemisphere. It is defined as the line segment joining any point on the circumference with the center.

CSA of a Hemisphere

A hemisphere's curved surface area is the area covered by its curved surface. It is exactly half of a sphere's surface area. The following formula can be used to calculate the curved surface area of a hemisphere with radius 'r'.

$$\mathrm{Curved\: surface\: area\: of\: hemisphere= \frac{1}{2}(a\: sphere's\: curved\: surface\: area)}$$

Where r is the hemisphere's radius.

TSA of a Hemisphere

The total surface area of the hemisphere is defined as the space occupied by the curved surface and the hemisphere's base surface. A hemisphere's total surface area can be calculated by adding the areas of its curved and base surfaces. It should be noted that a hemisphere's base is a circle. If the radius of a hemisphere is known, the total surface area can be calculated using the formula −

$$\mathrm{Total\: surface\: area\: of\: hemisphere = Curved\: Surface\: Area\: + Base\: Area }$$

$$\mathrm{= 2πr^2+πr^2}$$

$$\mathrm{= 3πr^2}$$

Where r is the hemisphere's radius.

Solved Examples

1) Find the curved surface area of a hemisphere whose radius is 2 cm.

Answer: It is given that the radius of the hemisphere is 2 cm and we know that the following formula can be used to calculate the curved surface area

The curved surface area of the hemisphere= 2πr²

$$\mathrm{ = 2π×2^2}$$

$$\mathrm{= 8 π}$$

2)Find the curved surface area of a hemisphere whose radius is 7 cm.

Answer: It is given that the radius of the hemisphere is 7 cm and we know that the following formula can be used to calculate the curved surface area of a hemisphere with radius 'r'.

The curved surface area of the hemisphere= 2πr²

$$\mathrm{ = 2π×7^2}$$

$$\mathrm{ \Rightarrow 308}$$

3) Find the volume of a sphere whose radius is √3.

Answer: It is given that the radius of the sphere is √3 and we know that the formula for the volume of a sphere is $$\mathrm{\frac{4}{3} π r^3 }$$

$$\mathrm{The\: volume\: of\: a\: sphere =\frac{4}{3} π r^3 }$$

$$\mathrm{=\frac{4}{3} π (√3)^3}$$

$$\mathrm{=4√3 π }$$

4) Find the volume of a hemisphere whose radius is 1.

Answer: It is given that the radius of the hemisphere is 1 and we know that the formula for the volume of a hemisphere is $\mathrm{\frac{2}{3} π r^3 }$

$$\mathrm{The\: volume\: of\: a\: sphere =\frac{2}{3} π r^3 }$$

$$\mathrm{ =\frac{2}{3} π (1)^3 }$$

$$\mathrm{ =\frac{2}{3} π }$$

5) Find the total surface area of a hemisphere whose radius is 5 cm.

Answer: It is given that the radius of the hemisphere is 5 cm.

We know that the following formula can be used to calculate the total surface area of a hemisphere with radius 'r'

$$\mathrm{Total\: surface\: area\: of\: hemisphere = 3πr^2}$$

$$\mathrm{=3π×5^2}$$

$$\mathrm{ =75π}$$

6) Find the total surface area of a hemisphere whose radius is 11 cm.

Answer: It is given that the radius of the hemisphere is 11 cm.

We know that the following formula can be used to calculate the total surface area Total surface area of hemisphere = 3πr²

$$\mathrm{=3π×11^2}$$

$$\mathrm{=363π}$$

Conclusion

A hemisphere's surface area is defined as the region covered by the hemisphere's faces. A hemisphere's total surface area includes the curved surface area as well as the hemisphere's base area.

FAQs

1. Define sphere.

A sphere is a circular, three-dimensional solid form known as a sphere in geometry. It consists of a combination of a group of points connected in three dimensions by a common point at equal distances.

2. Define hemisphere.

The words hemisphere and sphere can be separated into hemi, which denotes half, and sphere, which refers to the mathematical 3D shape. Consequently, a hemisphere is a 3D geometric object that is made up of half of a sphere, with one side being flat and the other being a bowl-like shape.

3. What is a Hemisphere's Surface Area?

A hemisphere's surface area is the sum of all its faces. Because a hemisphere is made up of a curved surface and a flat circular surface, we must add these areas together to get the total surface area of a hemisphere.

4. What is the Formula for a Hemisphere's Surface Area?

The formula for calculating the total surface area of a hemisphere is the total surface area of a hemisphere = 3πr^2 where 'r' is the radius of the hemisphere.

5. What is a Hemisphere's Lateral Surface Area?

A hemisphere's lateral surface area is also known as the curved surface of the hemisphere. If the radius (r) of a hemisphere is known, the lateral surface area can be calculated as follows: lateral surface area = 2πr²

6. What is the Difference Between Curved Surface Area and Hemisphere Surface Area?

The Curved Surface Area (CSA) is the curved portion of the hemisphere. The formula for calculating a hemisphere's curved surface area is CSA of hemisphere = 2πr².

The total surface area of the hemisphere, on the other hand, includes the curved area as well as the rest of the circular face of the hemisphere’s area. TSA of hemisphere 3πr² is the formula used to calculate the total surface area of the hemisphere.