Explore the formula for calculating the surface area of a sphere, with examples and detailed solutions.

A sphere is a set of points equidistant from a fixed point O (center of the sphere) at a constant distance R (radius). The formula for calculating the surface area of a sphere is also a straightforward and easy-to-remember concept. In this article below, let's explore the detailed calculation formula of this geometric shape.

How is the surface area of a sphere calculated?

Calculating the surface area of a sphere

- General formula for calculating the surface area of a sphere:

S_sphere = 4 π.R² (1)

or:

S_sphere = π. d² (2)

- Explanation of symbols for the variables:

S_sphere : Symbol for the surface area of the sphere
R is the radius of the sphere
d is the diameter of the sphere
π : Pi (π = 3.14)

- Unit of area: square meter (m²

Proving the formula for calculating the surface area of a sphere

The formula for the surface area of a sphere is proven as follows:

Illustrative example of calculating the formula for the surface area of a sphere :

Exercise 1. Calculate the surface area of a sphere with a radius extending from the center O as follows:

a) 8 m
b) 1.3 dm
c) 2 cm
d) 15 cm

Solution

Applying formula (1)

a) The surface area of the sphere is:
4x 3.14 x 8² = 6430.72 (m²)

b) Surface area of the sphere is:
4 x 3.14 x 1.3² = 27.59432 (dm²)

c) Surface area of the sphere is:
4 x 3.14 x 2² = 100.48 (cm²)

d) Surface area of the sphere is:
4 x 3.14 x 15² = 42390 (cm²)

Exercise 2. Calculate the surface area of the sphere given the diameter's length:

a) 2.1 cm
b) 9 cm
c) 1⁄2 cm
d) 4.5 cm

Solution: 

Applying formula (2)

a) The surface area of the sphere is:
3.14 x 2.1² = 13.8474 (cm²)

b) Surface area of the sphere is:
3.14 x 9² = 254.34 (cm²)

c) Surface area of the sphere is:
3.14 x (1/2)² = 0.785 (cm²)

d) Surface area of the sphere is:
3.14 x (4.5)² = 63.585 (cm²)

* Hint 

For these exercises, simply substitute the values into the formula for calculating the surface area of a sphere (1) or (2) and perform the calculations (you can do mental math for simple numbers or use a handheld calculator for complex ones).

How is a spherical surface different from a spherical shape?

- A spherical surface is the outer shell of a sphere; in other words, it is the hollow sphere
+ The spherical surface exists in 3D
+ Its characteristic is surface area

- A spherical shape encompasses both the spherical surface, including the outer part, and the inner part limited by that surface.
+ The spherical shape is in 2D and is a solid sphere
+ Its characteristic is volume.

Before tackling exercises related to calculating the surface area of a sphere, it's crucial to differentiate between the concepts of a spherical surface and a spherical shape. Additionally, the most important step in solving problems is to be familiar with the formulas. Reviewing how to calculate the area of a circle is essential preparation for solving problems related to circles. We hope that our comprehensive article here will help you in your learning and problem-solving journey. We wish you continued enjoyment in studying mathematics in general and geometry in particular.

In addition to understanding the formulas for calculating the surface area of a sphere, it's essential to master the formula for finding the volume of a sphere in the series of exercises related to spheres and spherical surfaces. These are fundamental and crucial concepts that you need to study thoroughly.