In geometry, a sphere is a solid, that is absolutely round in shape defined in three-dimensional space (XYZ space). Mathematically, a sphere is defined as the set of points that is at equal distances from a common point in three dimensional space. This constant distance is called radius of sphere and the common point is the center of sphere. An example of sphere in real life is a ball.
In the above figure, we can see, a sphere with radius ‘r’.
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Definition
As discussed in the introduction, sphere is a geometrical round shape in three dimensional space. It is three dimensional solid, that has surface area and volume. Just like a circle, each point of the sphere is at a equal distance from the center.
Unlike circle, which is a plane shape or flat shape, defined in XY plane, a sphere is defined in three dimensions, i.e. x-axis, y-axis and z-axis.
Important Facts:
- A sphere is a symmetrical object
- All the surface points of sphere are at equidistant from center
- A sphere has only curved surface, no flat surface, no edges and no vertices.
Shape of Sphere
The shape of a sphere is round. It is a geometrical three dimensional solid having curved surface. Like other solids, such as cube, cuboid, cone and cylinder, a sphere does not have any flat surface or a vertex or an edge.
The real life examples of sphere is:
- Basket balls
- World Globe
- Marbles
- Planets
- Moon
Equation of a Sphere
In analytical geometry, if “r” is the radius, (x, y, z) is the locus of all points and (x0, y0, z0) is the center of a sphere, then the equation of a sphere is given by:
(x -x_{0})^{2} + (y – y_{0})^{2} + (z-z_{0})^{2} = r^{2}
Properties of a sphere
The important properties of the sphere are:
- A sphere is perfectly symmetrical
- It is not a polyhedron
- All the points on the surface are equidistant from the centre.
- It does not have a surface of centres
- It has constant mean curvature
- It has a constant width and circumference.
Formulas
We know that, the diameter is twice the radius. Therefore, diameter of a sphere is given as:
D = 2r units
Since all the three-dimensional objects have the surface area and volume, the surface area and the volume of the sphere is explained here.
Surface Area of a Sphere
The surface area of a sphere is the total area covered by the surface of a sphere in a three dimensional space. The formula of surface are is given by:
The Surface Area of a Sphere(SA) = 4πr^{2} Square units
Where “r” is the radius of the sphere.
Volume of a Sphere
The amount of space occupied by the object three-dimensional object called a sphere is known as the volume of the sphere.
According to the Archimedes Principle, the volume of a sphere is given as,
The volume of Sphere(V) = 4/3 πr^{3} Cubic Units
Solved Examples
Example 1:
Find the volume of the sphere that has a diameter of 10 cm?
Solution:
Given, Diameter, d = 10 cm
We know that D = 2 r units
Therefore, the radius of a sphere, r = d / 2 = 10 / 2 = 5 cm
To find the volume:
The volume of sphere = 4/3 πr^{3} Cubic Units
V = (4/3)× (22/7) ×5^{3}
Therefore, the volume of sphere, V = 522 cubic units
Example 2:
Determine the surface area of a sphere having a radius of 7 cm
Solution:
Given radius = 7 cm
The Surface Area of a Sphere(SA) = 4πr^{2} Square units
SA = 4× (22/7)× 7^{2}
SA = 4 × 22 × 7
SA = 616 cm^{2}
Therefore, the surface area of a sphere = 616 square units.
What is Hemisphere?
A hemisphere is exactly half of a sphere. It has a curved surface and a flat surface.
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Frequently Asked Questions – FAQs
What is a sphere?
How many sides a sphere have?
Is sphere a circle?
What are the characteristics of a sphere?
What are the examples of sphere?
What is surface area and volume of sphere?
SA = 4πr^2 Square units
Volume of sphere is the space occupied by sphere in three dimension space. The formula is:
V = 4/3πr^3