Area of Hemisphere - Surface and Volume of Area of Hemisphere
A sphere is a three dimensional version of a circle. The definition of a sphere is the collection of all points in space that are equidistant from a given fixed point known as the centre. A ring and a ball are similar in a sense. Both are round, but can we say they are the same shape? Clearly not. One is a sphere the other is a circle.
Circles and spheres are both studied using their radii. Then what is the difference between them? A circle can be drawn on paper, but a sphere cannot.
Diagram of a Sphere
A circle is the set of all the points in a plane that are equidistant from a given fixed point, known as its centre. The distance between any point on the circle and its centre is known as its radius. Any line segment passing through the centre of the circle and having its two endpoints on the circle is known as its diameter. If we spin the circle keeping its diameter fixed, the object formed in 3D space is a sphere.
Surface Area
For finding the surface area of any round thing we can do this:
A ball of any game is the best case of a sphere. Take a ball and insert a nail into it. Tie a string from the nail and wind it over the ball. Wind the string in such a way that the two layers of string don't cover one another. We have to cover the ball with only one layer of string. On achieving this, use pins to keep the strings flawless. Spread the entire ball along these lines. Then mark the beginning and end points of the string. Estimate the length of the string used. The width of the ball gives us the diameter of the ball. With the same diameter, draw four circles on plain paper. With the assistance of the string that was utilized to cover the ball, fill the circles on paper successively. You will see that the string used to cover the ball, covers the four circles on paper. This conveys to us that the surface area of the ball is equal to the sum of the areas of the four circles. Consequently, the surface area of a sphere = 4 × area of a circle = 4 × πr2. So,
SA=4πr2
SA=4πr2
Solved Examples
Example 1: What is the total surface area of a sphere whose radius is 5.5 meters?
Solution:
Given that,
r=5.5m
Surface area of the sphere
SA=4×π×r2
SA=4×π×r2
SA=4×π×(5.5)2
SA=4×π×(5.5)2
SA=4×3.14×30.25
SA=4×3.14×30.25
SA=379.94m2
SA=379.94m2
The surface area of the sphere is 379.94 m2.
Example 2: Find the surface area of a sphere whose radius is given as r = 11 cm.
Solution:
SA=4πr2
SA=4πr2
SA=4×3.14×112
SA=4×3.14×112
SA=1519.76
SA=1519.76
The surface area of sphere is 1519.76 cm2.
Example 3: A hemisphere has a radius of 8.3 cm. Find its curved surface area.
Solution:
r=8.3cm
The surface area of half of the sphere without the base is calculated as follows.
SA=2πr2
SA=2π×8.32
SA=432.62 cm2
The surface area of the half of the sphere is 432.62 cm2.
Example 4: Find the surface area of a sphere whose radius is 6 cm.
Solution:
SA=4πr2
SA=4π×62
SA=4π×36
SA=452cm2
Volume
For finding the volume of a circular body we utilize the Archimedes rule. What is Archimedes' guideline? As indicated by the Archimedes rule, when a solid figure is drenched in a compartment loaded up with water, at that point the volume of water that floods from the holder is equivalent to the volume of that solid figure. Archimedes' rule essentially gives you the volume, however following this guideline is painstaking. For finding the equation for the volume of the sphere, we embed the round body in a barrel-shaped holder.
The figure above demonstrates a circular body inside a cylindrical compartment. We see that the radius of the round bases of the cylinder is equal to the radius of the spherical body. What's more, since the spherical body contacts the top and base of the compartment, its diameter is equal to the height of the holder.
Hemisphere
A hemisphere is the half piece of a sphere. If we cut the sphere into two equal parts, each part will have a curved surface as well as a flat surface. So, there are two types of areas associated with a hemisphere, which are the curved surface area and the total surface area, which is the sum of the curved and flat surface areas. Hemispheres can either be of a solid kind or can be hollow.
Diagram of a Hemisphere
Elements of a Hemisphere
The radius is a line that connects the centre of a sphere with a point on the circumference.
The diameter is double the line segment from one side of the boundary to the other through the centre.
π is the symbol we use for pi, which is the ratio of the circumference of a circle to its diameter. π is approximately 3.14.
The Surface Area of a Hemisphere
The surface area of a sphere is determined by the formula 4πr2.
Now, since a hemisphere is half of a sphere, the surface area of the hemisphere ought to be 2πr2. However, as we are aware that a hemisphere additionally has a flat surface, we divide the total surface area of the hemisphere into two parts, which are the curved and flat surface areas.
The curved surface area of a hemisphere is half of the surface area of a sphere.
= ½ of (surface area of the circle)
= ½ of 4πr2
= 2πr2
Now, to find the total surface area of a hemisphere we additionally need to determine the surface area of the circular surface of the flat base of the given hemisphere, which is equal to πr2.
Subsequently, total surface area of a hemisphere = flat surface area + curved surface area. Hence, it will be as follows.
Total Surface Area=πr2+2πr2
=3πr2
Total Surface Area=πr2+2πr2=3πr2
Practical Examples of calculating the Area of a Hemisphere
Example: Find the surface area of a hemisphere with radius 10 cm.
Solution:
Since it isn't mentioned whether we have to find the total surface area of curved surface area, let us compute both.
As we know, curved surface area of hemisphere=2πr2
=2×3.14×10×10=628cm2
Curved Surface Area of hemisphere=2πr2=2×3.14×10×10=628cm2
Subsequently, total surface area of a hemisphere=3πr2
=3×3.14×10×10=942cm2
Subsequently, Total Surface Area of a hemisphere=3πr2=3×3.14×10×10=942cm2
Example: Find the radius of a hemisphere whose the total surface area is 1000 cm2?
Solution:
Total surface area of a hemisphere=3πr2
Total surface area of a hemisphere=3πr2
1000=3×3.14×r2
1000=3×3.14×r2
r2=1000/9.42
r2=106.16
r=10.30cm
Example: If the radius of a hemisphere is tripled, what will be its impact on its total surface area?
Assume that the radius is 1 unit
Now, Total Surface Area=3πr2=3π
In the case where the radius is tripled, the total surface area will be =3×π×32=27π
3×π×32=27π
In this way, the surface area of half of the sphere will increase by a factor of 9.
Example: If the radius of a hemispherical balloon increases from 5 cm to 10 cm as air is being pumped into it, find the ratio of the surface areas of the new balloon to its original.
Solution:
Beginning hemisphere, r=5cm
Total surface area=3πr2
=3×π×52
=75πcm2
Total surface area=3πr2=3×π×52=75πcm2
In the case where the radius is 10 cm,
Total surface area=3πr2
=3×π×102
=300πcm2
Total surface area=3πr2=3×π×102=300πcm2
Therefore the ratio is as follows.
⇒ S1 : S2 = 75π : 300 π = 1 : 4
Applications of Mensuration:
Mensuration is a subpart of geometry which deals with the measurement of things, especially geometric figures. Measuring the TSA, LSA and the volume of a hemisphere is also a part of mensuration. Following are the applications of mensuration:
When you have to paint a certain surface area, one needs mensuration to calculate the total surface area or the lateral surface area. This will help in planning and mobilising resources.
Generally, mensuration is used by architects to measure the interiors and plan the resources accordingly.
Volume measurement in mensuration is used to measure the capacity of an object like an oil tanker, container etc,
It is used in agriculture to measure the amount of soil needed to fill an empty area.
Mensuration also helps in travelling, navigation and reading a road map to find out the distance.
FAQs on Area of Hemisphere
1. Is surface area refers to the total surface area of curved surface area?
A hemisphere is where there are a set of points and every point is equidistant from the centre and a hemisphere is a two-dimensional figure. It can also be formed when a plane cuts off the sphere in the middle. For example, the equator is an imaginary line on the earth that cuts the entire earth into two equal hemispheres i.e, the northern and southern hemispheres. Lateral surface area is the measurement of the external surface area of any geometrical figure. In the case of the hemisphere, Surface area refers to the total surface area of a solid. If it refers to curved surface area it will be mentioned as “CSA” if it is not mentioned then it is understood as a total surface area “TSA”.
2. What is the formula to calculate the total surface area of the hemisphere?
A hemisphere is where there are a set of points and every point is in equidistant from the centre and a hemisphere is a two dimensional figure. It can also be formed when a plane cuts off the sphere in the middle. For example, the equator is an imaginary line on the earth that cuts the entire earth into two equal hemispheres i.e, the northern and southern hemisphere. Total surface area is the measurement of the entire area of any geometrical figure. In the case of the hemisphere, Total surface area is a combination of the curved surface area including the area of the circle formed on the top surface. Formula to calculate the total surface area of the hemisphere is 3πr2
3. What is the difference between curved surface area and total surface area of the hemisphere?
A hemisphere is where there are a set of points and every point is equidistant from the centre and a hemisphere is a two-dimensional figure. It can also be formed when a plane cuts off the sphere in the middle. For example, the equator is an imaginary line on the earth that cuts the entire earth into two equal hemispheres i.e, the northern and southern hemisphere. Curved Surface Area (CSA) only includes the area of all the curved parts of the hemisphere whereas Total Surface Area (TSA) includes the curved surface area as well as surface area formed on the top of the hemisphere. To know the formula to measure the TSA and LSA of a hemisphere, refer to the above article.
4. What is the volume of a hemisphere?
A hemisphere is where there are a set of points and every point is equidistant from the centre and a hemisphere is a two-dimensional figure. It can also be formed when a plane cuts off the sphere in the middle. For example, the equator is an imaginary line on the earth that cuts the entire earth into two equal hemispheres i.e, the northern and southern hemispheres. Archimedes derived the formula to measure the volume of the hemisphere. The volume of an object is the entire space it occupies when it is on a surface. The other two important measurements of any geometrical figure are TSA (total surface area) and LSA (lateral surface area).
The volume of a hemisphere = (2/3)πr3 cubic units, here, r is the radius of the hemisphere.
5. Is it important to study Mensuration?
Mensuration is a subpart of geometry that deals with the measurements. In modern society, measuring things is a very important and undeniable skill. We need measurement to measure distances, time, seed, area of a figure to build a house, apartment or any construction. When we sleep, we set an alarm and count the ideal and minimum hours of sleep. Even the refrigerator we use measures ideal temperature to store the food we keep in it and any miscalculation would only spoil the entire food we store. Even at work, computers and all electronic devices use measurement to function. Long-distance travelling, navigation is possible only through measurement. Mensuration plays a very important role in our day-to-day life.
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