Hemisphere is a three-dimensional shape generated by dividing a sphere into two halves. It is a half-spherical curve combined with a planar circular area. Generally, a hemisphere refers to half of the Earth, such as the northern or southern hemispheres.
In geometry, however, a hemisphere is a 3D shape formed by splitting a sphere into two halves, each with one flat side. In everyday life, we come across various items that have the form of a hemisphere. For example, if we cut a cherry in half, we obtain a hemisphere-shaped cherry.
In this article, we'll study more about hemispheres, their properties, formulas, and others in detail.
What is Hemisphere?
Hemisphere is a 3D shape which consist of a curved and a plane surface and resembles the shape of a bowl. Hemisphere means Half of Sphere. The word hemisphere is made up of two parts: hemi, which means half, and sphere, which is a mathematical 3D form in mathematics. A hemisphere is made by carefully cutting a sphere along its diameter, leaving two identical hemispheres behind. The flat side of the hemisphere is the base or face of the hemisphere. Hemispheres can be either hollow or solid.
Definition of Hemisphere
A hemisphere is a three-dimensional shape formed by cutting a sphere exactly in half along its diameter, resulting in two equal parts, with one side flat and the other forming a curved bowl. It represents half of a complete sphere.
Elements of a Hemisphere
Various Elements of Hemisphere are,
- A radius is a line that links a sphere's centre to a point on its perimeter.
- The diameter is twice the length of the line segment that runs through the centre from one side of the border to the other.
- π is the sign for pi, which is the circumference-to-diameter ratio of a circle. π is around 3.14.
Properties of Hemisphere
Some important properties of Hemisphere are,
- The radius of the hemisphere equals the radius of the sphere.
- A hemisphere has a flat circular base and a curved surface.
- A hemisphere, like a sphere, has no edges and no vertices.
- A hemisphere is not a polyhedron since it has a circular base and one curved surface.
- A line segment that connects two opposed locations on the base of a hemisphere and goes through its centre is said to have that object's diameter.
Volume of Hemisphere
The Volume of Hemisphere is determined by using the formula of volume of a sphere. The formula is used to compute the volume of a sphere, where r is the radius of the sphere.
Volume of a sphere = 4/3​πr3
To get the volume of a hemisphere, divide the volume of a sphere in half. The volume of a hemisphere formula is shown below for a hemisphere with radius, r.
Volume of a hemisphere = 4/3​πr3 /2
OR
Volume of a hemisphere = 2/3​πr3
where, r is the radius of the hemisphere and the value of ​π is 22/7 or 3.142.
Hemisphere Surface Area
A hemisphere's surface area is derived from that of a sphere and may be determined using particular methods. For a hemisphere with radius r, the total surface area comprises both the curved and base areas. Since, hemisphere has both plane and curved surfaces hence it has two types of surface areas mentioned below:
- Curved Surface Area of a Hemisphere
- Total Surface Area of a Hemisphere
Curved Surface Area of a Hemisphere
Curved surface area of a hemisphere is based on the surface area of a sphere. We know that surface area pf sphere is given as:
Surface Area of a Sphere = 4πr2
We know that hemisphere is half of sphere therefore, we divide the sphere's surface area in half to get the curved surface area of the hemisphere.
Curved Surface Area of a Sphere = 4πr2 /2 = 2πr2
where, r is the radius of the hemisphere and the value of ​π is 22/7 or 3.142
Total Surface Area of a Hemisphere
Total Surface Area of hemisphere consist of curved as well as planar surface area of hemisphere. If we want the total surface area, we must add the base area to the curved surface. The area of the circle base is provided by πr2 and curved surface area is 2πr2
Here is the formula for calculating the surface area of a hemisphere of radius r.
Total Surface Area of a Sphere = 2πr2 + πr2 = 3πr2
where, r is the radius of the hemisphere and the value of ​π is 22/7 or 3.142.
Hollow Hemisphere
A hollow hemisphere has two surfaces: an interior and exterior surface. As a result, a hollow hemisphere will have two separate radii, namely inner radii and outer radii. The thickness of a hollow hemisphere is defined as the difference between the radii of the inner and outer radii of a hollow hemisphere. A hollow hemisphere's area may be divided into two categories:
- Curved Surface Area of hollow hemisphere
- Total surface Area of hollow hemisphere
- Volume of hollow hemisphere
Curved Surface Area of Hollow Hemisphere
A hollow cylinder has two surfaces: an inner surface and an outer surface. It also has two radii, r for the inner hemisphere and R for the outer hemisphere.
Inner hemisphere curved surface area: 2Ï€r12
Outer hemisphere curved surface area: 2Ï€r22
Hollow Hemisphere Curved Surface Area: 2Ï€R2 + 2Ï€r2 = 2Ï€(R2+ r2)
Total Surface Area of Hollow Hemisphere
A hollow hemisphere's total surface area is the sum of the base of the circular ring and the curved surface areas of the inner and outer surfaces.
Total Surface Area of Hollow Hemisphere = Area of Circular Ring + Curved Surface Area of Hollow Hemisphere = 2π(R2+ r2) + π(R2 - r2)
Volume of Hollow Hemisphere
The volume of a hollow hemisphere is calculated by subtracting the volume of the smaller hemisphere (carved out) from the volume of the bigger hemisphere. To calculate the volume of a hollow hemisphere with outer radius R and inner radius r, use the following formula:
Volume of Hollow hemisphere = 2/3​π(R3 - r3)
Here are some essential formulae for the surface area and volume of hemispheres, presented in a tabular format:
Formula
| Expression
|
|---|
Total Surface Area of a Sphere
| 2πr2 + πr2 = 3πr2
|
|---|
Curved Surface Area of a Sphere
| 4Ï€r2 = 2Ï€r2
|
|---|
Volume of a sphere
| 2/3​πr3
|
|---|
Hollow Hemisphere Curved Surface Area
| 2Ï€r12 + 2Ï€r22 = 2Ï€(r12+ r22)
|
|---|
Total Surface Area of Hollow Hemisphere
| Area of Circular Ring + Curved Surface Area of Hollow Hemisphere
|
|---|
Volume of Hollow hemisphere
| 2/3​π(R3 - r3)
|
|---|
Hemisphere Shape Examples
Hemispheres can be seen in following examples in everyday life.
- A bowl from which we eat is a hollow hemisphere, similar to a half-cut coconut shell.
- When round fruits are cut, they form solid hemispheres, such as oranges, watermelons, and tamarinds.
- The Earth is divided into the Northern Hemisphere and the Southern Hemisphere, separated by the equator.
- The human brain is divided into two hemispheres, the left and the right, each governing distinct cognitive functions.
- Objects like basketballs or soccer balls are three-dimensional with two hemispheres, contributing to their spherical shape.
These forms are all around us, from dishes to fruit slices. They seem like half-balls or domes that we see all the time, whether in the kitchen or out and about.
Difference Between Hemisphere and Sphere
Various important differences between Hemisphere and Sphere are,
Hemisphere Vs Sphere
|
|---|
Hemisphere
| Sphere
|
|---|
It's a 3D figure made by splitting a spherical in two.
| This is a 3D spherical shape with no edges or vertices that is used in geometry.
|
The Earth is split into two hemispheres: Northern and Southern.
| Represents a whole object with no divisions.
|
It features a curving surface and a circular base that is flat.
| The sphere has no flat surface and is totally curved.
|
Examples: Bowls, Planets cut in half
| Examples: Balls, Planets, completely round objects
|
Hemisphere Volume: 2/3 πr3
| Sphere Volume: 4/3 πr3
|
Total Surface Area of Hemisphere: 3 πr2
| Total Surface Area Sphere: 4 πr2
|
Also Read,
Example 1: Determine the total surface area for a hemisphere having a radius of 25 cm.
Solution:
Total surface area of hemisphere = 3Ï€r2
Given Radius = 25 cm
Total surface area of hemisphere = 3Ï€r2
= 3 × 22/7 × 25 × 25
= 41250/7
= 5892.85 cm2
Thus, the total surface area of hemisphere = 5892.85 cm2
Example 2: Determine the volume of a hemisphere having a radius of 22.6 inch.
Solution:
Volume of hemisphere = 2/3Ï€r3
Given Radius = 22.6 inch
Volume of hemisphere = 2/3Ï€r3
= 2/3 × 3.14 × 22.6 × 22.6 × 22.6
= 72,491.14/3
= 24,163.71 inch3
Thus, the volume of hemisphere is 24,163.71 inch3
Example 3: The radius of a hemisphere is 15.4 cm. Determine its curved surface area.
Solution:
Curved surface area of hemisphere = 2Ï€r2
Given Radius = 15.4 cm
Curved surface area of hemisphere = 2Ï€r2
= 2 × 22/7 × 15.4 ×15.4
= 10,435.04/7
= 1,490.72 cm2
Thus, the Curved surface area of hemisphere is 1,490.72 cm2
Example 4: What is the volume of water that a hemispherical bowl with a radius of 18 cm can hold?
Solution:
Volume of hemisphere = 2/3Ï€r3
Given Radius = 18 cm
Volume of hemisphere = 2/3Ï€r3
Volume of hemisphere = 2/3 × 3.14 × 18 × 18 × 18
Volume of hemisphere = 36,624.96/3
Volume of hemisphere = 12,208.32 cm3
Thus, the volume of hemisphere is 12,208.32 cm3
Example 5: If the area of the hemispherical bowl's base is 201.14 unit square, find its radius.
Solution:
Area of the hemispherical bowl's base = 201.14 unit square
We know that Area of base of hemisphere = πr2
201.14 = 22/7 × r2
r2 = 201.14 × 7/22
r2 = 1407.98/22
r2 = 64
r = √64
r = 8
The hemispherical bowl has a radius of 8 units.
Practice Questions on Hemisphere
Q1. The radius of a hemisphere is 12 meters. Determine its total surface area.
Q2. Determine the volume of a 5.8-centimeter-radius hemisphere.
Q3. Determine the curved surface area of a 9.5-inch-radius hemisphere.
Q4. The radius of a hemispherical tank is 30 feet. How much liquid can it hold?
Q5. Determine the total surface area of a 7.2-meter-radius hemisphere.
Q6. A fountain features a 14-inch-radius hemispherical bowl. Determine the amount of water it can hold.
Hemisphere Frequently Asked Questions
Define Hemisphere.
A hemisphere is a half of a spherical or roughly spherical object, such as a sphere or a planet, divided by a plane that passes through its center.
What are the 4 hemispheres of the Earth?
The Earth is divided into four hemispheres: Northern Hemisphere, Southern Hemisphere, Eastern Hemisphere, and Western Hemisphere, each representing half of the planet based on different orientations.
What shape is a Hemisphere?
A hemisphere has a semicircular or half-sphere shape, resembling half of a full sphere.
What are Sphere and Hemisphere?
A sphere is a three-dimensional geometric entity with all points equidistant from its centre that resembles a ball. A hemisphere is a half-sphere split by a plane that passes through its centre.
What is Hemisphere Formula?
The Hemisphere Formula determines the volume or surface area of a hemisphere, which is half of a sphere. The formulas for volume and surface area are (2/3)Ï€r3 and 2Ï€r2, respectively, with "r" being the radius.