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Surface Areas and Volumes

Sphere

A sphere is a three-dimensional circle or we can also say that it is a circle in space. A ring and a ball have something in common. Both are round but are they the same? Obviously not. One is a circle the other is a sphere. Space below shall help you delve into the measurements of surface area and volume of the sphere.

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Sphere

Circle and sphere, both are round, both are measured using radius. Then what is the difference between the two? A circle can be drawn on a plane, but can a sphere be drawn on a paper?  The answer is no! This is because a sphere is a 3-dimensional circle.

Sphere

(source: pixabay)

A circle is a closed figure which can be drawn using a constant length from a fixed point center. This fixed point is called the center of the circle and the distance with which a circle is drawn is called the radius. The line passing through the center from one end to the other is called the diameter.

If a circular ring is tied to a string and rotated then we see a change in the shape. The changed shape is called the Sphere. When a circle changes into a sphere, nothing in it changes, neither the radius nor the diameter. It just becomes a three-dimensional version of a planar circle. So a spherical shape is a three-dimensional counterpart of a circle, with all its points lying in space at a constant distance from the fixed point or the center, called the radius of that sphere.

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Surface Area

For finding the surface area of any spherical item we need to perform the following activity: A ball of any game is the best example of a sphere. Take a ball and drive a nail into it. Tie a string from the nail and wind it across the ball. Wind the string in such a manner that the two layers of string do not overlap each other.

We need to cover the ball with just one layer of string. On reaching the center use pins to keep the string intact. Cover the whole ball in a similar fashion. Now mark the starting and end points of the string wrapped around the ball. Try measuring the diameter of the ball with the help of a scale.

The diameter gives us the radius of the ball. With the same radius, draw four circles on plain paper. With the help of the string that was used to cover the ball, fill the circles on paper sequentially.

You will notice that the string used to cover the ball, covers the four circles on paper. This brings us to the conclusion that the surface area of the ball is equal to the area of four circles. Therefore, the Surface area of a ball = 4 × Area of a circle = 4 × πr2. Therefore,

The Surface Area of a Sphere = 4πr2

Hemisphere

A hemisphere is the half of a sphere. When a sphere is cut into two halves, then the shape we get is called the hemisphere. A hemisphere has a curved surface and a flat base. The curved surface area of the hemisphere is half of the surface area of the sphere. Therefore,

The Curved Surface Area of Hemisphere =1/2 × 4 × πr2

Curved surface area of a hemisphere = 2πr2. Since a sphere is a combination of a curved surface and a flat base, to find the total surface area we need to sum up both the areas. The flat base being a plane circle has an area πr2. Total surface area of a hemisphere is 2πr2+πr2. So,

The Total surface area of Hemisphere = 3πr2

Volume

For finding the volume of a spherical body we use the Archimedes principle. What is Archimedes principle? According to the Archimedes principle, when a solid figure is immersed in a container filled with water, then the volume of water that overflows from the container is equal to the volume of that solid figure.

Similarly when we place a spherical figure into a container filled with water, then the amount of water that overflows is equal to the volume of that figure. Archimedes principle practically gives you the volume, but following this principle is not possible every time. For finding the formula for the volume of the sphere, we insert the spherical body in a cylindrical container.

The figure above shows a spherical body inside a cylindrical container. We notice that the radius of the circular bases of the cylinder is equal to the radius of the spherical body. And since the spherical body touches the top and bottom of the container, its diameter is equal to the height of the container.

The volume of a spherical body is assumed to be 2/3 of the cylindrical container, this gives us: Vsphere = 2/3 Vcylinder or 2 / 3 π r2h. Thus, we have: h = 2r = 2 / 3 πr(2r) = 4 / 3 πr3. Thus,

The Volume of the Sphere = 4 / 3 πr3

The Volume of Hemisphere

As already said a hemisphere is half the sphere, hence its volume will also be half the volume of the sphere. The volume of the hemisphere = 1 / 2 × 4 / 3 πr3.

The Volume of the Hemisphere = 2 / 3 π r3

Solved Examples for You

Question: The diameter of the moon is approximately 1/4th of the diameter of the earth. What fraction of the volume of the earth is equal to the volume of the earth?

Solution: Let the diameter of Earth be x, and the radius of the earth be = x/2

Volume of Earth = 4 / 3π (x / 2)3
The diameter of moon = 1 / 4x = x / 4
Radius of moon = 1 / 2 × 1 / 4x =  x / 8

Volume of Moon = 4 / 3 π (x / 8)3
Volume of moon/Volume of Earth =   { 4 / 3 π  (x / 8)} / { 4 / 3 π  (x / 2)}
= (x / 8)3 / (x / 2)= x/ 8× 23 /x= 23 / 8= 8 / 8 × 8 × 8 = 1/64.
Therefore, the volume of the moon is 1/64th of Earth.

Ques. Is sphere a circle?

Ans. Any circle of a sphere is basically a circle that lies on a sphere. Circle-like this can be formed as the meeting of a sphere and a plane, or of 2 spheres. On a sphere a circle’s plane passes through the middle of the sphere is known as a great circle.

Ques. How many circles make a sphere?

Ans. A diagram can show us this; where the intersection of any sphere and a cylinder contains 2 circles. Would the radius of the cylinder be equal to the radius of the sphere? The intersection would be a circle, where both the surfaces will be tangent.

Ques. What is the equation for the sphere?

Ans. Equation of a Sphere is given in mathematics as follows:

The general equation of a sphere is:

(x – a)² + (y – b)² + (z – c)² = r²

Here, a, b, and c symbolizes the middle of the sphere.

Here, r represents the radius and x, y, and z are respectively the coordinates of the points on the surface of the sphere.

Ques. How many sides does a sphere have?

Ans. A sphere is having 2 sides, an inside and an outside respectively. However, it is not very easy to prove this.

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