Equation Of Sphere

A sphere is defined as a completely round geometrical object in a three-dimensional space just like a round ball.

To be geometrical, a sphere is a set of points that are equidistant from a point in space. The distance between the outer point and center of the sphere is called the radius, denoted by r and the maximum straight distance between any two sides of the sphere through the center is known as the diameter, denoted by d.

Sphere

A hemisphere is exactly half of a sphere which can only be obtained when a sphere is split from the middle. The biggest circle of a sphere is a circle that has the same center and radius of a sphere.

A great circle of the sphere is a circle that has the same radius and center as the sphere itself.

Volume of sphere:

The formula to calculate the volume of a sphere is given by:

Volume = \(\frac{4}{3} \pi r^{3}\)

Where r is the radius of the sphere.

Surface area of sphere:

The formula to calculate the area of sphere is given by:

Surface area = \( 4 \pi r^{2}\)

Equation of a sphere:

The equation of a circle of radius r is given by:

x2 + y2 = r2

You can relate it to algebraic method of starting the Pythagoras theorem.

Equation of a sphere

The point (x,y) lies on the circle only when the right triangle has sides of length |x| and |y| and hypotenuse of length r, which can be written as:

x2 + y2 = r2

Pythagoras theorem can be used twice for the equation of a sphere. In the below figure, O is the origin and P(x,y,z) is a point in three space. P is on the sphere with radius r only when the distance from O to P is r.

Equation of a sphere

Since OAB is a right angle triangle, x2 + y2 = s2. The triangle OBP is another right triangle and therefore, s2 + z2 = r2. Hence, the distance between O and P can be expressed by:

x2 + y2 + z2 = |OP|2

Hence, we can conclude that (x,y,z) lies on the sphere with radius r only if,

x2 + y2 + z2 = r2

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Practise This Question

Which of the following is the sample space for the following experiment?
We toss a coin until we see two consecutive tails. We record the total number of coin tosses.