Surface Area & Volume of Sphere

Surface Area of a Sphere

The surface area of a sphere is the measure of the region covered by its outer surface in three-dimensional space. A sphere is a three-dimensional solid whose surface consists of all points at a fixed distance from a fixed point (the centre). The fixed distance is called the radius (r) of the sphere.

A circle is a two-dimensional figure while a sphere is three-dimensional; therefore the formula for the area of a circle is different from the surface area of a sphere.

The area of a circle is πr².

The total (outer) surface area of a sphere is given by the formula

A = 4 π r²

Interpretation: this formula gives the area of the entire curved surface of the sphere.

For three-dimensional solids we commonly refer to three related area measures:

• Curved Surface Area: the area of all curved regions of the solid.
• Lateral Surface Area: the area of the surface excluding bases (top and bottom) where applicable.
• Total Surface Area: the area of all external faces and curved surfaces of the solid.

Since a sphere has no flat faces or bases, its total surface area equals its curved surface area.

Solved Examples - Surface Area

Q.1. Calculate the cost required to paint a football which is in the shape of a sphere having a radius of 7 cm. If the painting cost of football is INR 2.5/square cm. (Take π = 22 / 7)

Solution:

Given: Radius, r = 7 cm.

Total surface area of a sphere = 4 π r².

4 × (22/7) × 7 × 7.

4 × (22/7) × 7 × 7 = 616 cm².

Cost of painting = rate × area.

Cost = 2.5 × 616 = Rs. 1540.

Q.2. Calculate the curved surface area of a sphere having radius equals to 3.5 cm. (Take π = 22 / 7)

Solution:

Given: Radius, r = 3.5 cm.

Curved surface area of a sphere = Total surface area = 4 π r².

4 × (22/7) × 3.5 × 3.5.

4 × (22/7) × 3.5 × 3.5 = 154 cm².

Volume of a Sphere

The volume of a sphere is the amount of three-dimensional space enclosed by it; it measures the sphere's capacity. The volume depends on the radius of the sphere because every cross-section through the centre is a circle whose area depends on the radius.

The formula for the volume of a sphere of radius r is

V = (4/3) π r³

Remarks on origin: the formula can be derived by methods of integral calculus (summing areas of circular slices) or by classical geometric arguments (for example, Archimedes used comparisons with cylinders and cones to find the relation). For school purposes it is sufficient to remember and apply this formula.

Solved Examples - Volume

Q.1. Find the volume of a sphere whose radius is 3 cm?

Solution:

Given: Radius, r = 3 cm.

Volume of a sphere = (4/3) π r³.

V = (4/3) × π × 3³.

3³ = 27.

V = (4/3) × π × 27 = 36 π.

If π = 3.14 then V = 36 × 3.14 = 113.04 cm³.

Q.2. Find the volume of sphere whose diameter is 10 cm.

Solution:

Given: Diameter = 10 cm.

Radius = diameter / 2 = 10 / 2 = 5 cm.

Volume = (4/3) π r³.

V = (4/3) × (22/7) × 5 × 5 × 5.

V = (4/3) × (22/7) × 125.

Evaluating gives V ≈ 523.81 cm³ (approximately).

Notes and Applications

• Units: Surface area is expressed in square units (cm², m², etc.). Volume is expressed in cubic units (cm³, m³, etc.).
• Use of π: Use the value of π as given in the question (commonly 3.14 or 22/7). Keep consistency when calculating.
• Practical examples: Problems on paint, coatings, material required for hollow spheres, capacity of spherical containers and sports balls use these formulas directly.
• Related quantities: If diameter d is given, use r = d/2. If surface area is given, radius can be found by rearranging A = 4 π r². If volume is given, radius can be found by rearranging V = (4/3) π r³.

Summary

• Surface area of a sphere: A = 4 π r².
• Volume of a sphere: V = (4/3) π r³.
• Convert diameter to radius when necessary: r = d / 2.