A sphere is a perfectly symmetrical three-dimensional shape where every point on its surface is equidistant from its center. The distance from the center of the sphere to any point on its surface is called the radius (r).

The surface area of a sphere can be calculated using the following formula:

Surface Area (SA) = $4\pi r^2$

Where:

• $SA$ = Surface Area of the sphere
• $r$ = Radius of the sphere

Derivation of Surface Area Formula

To understand the derivation of the surface area formula, consider a sphere with radius $r$. Imagine wrapping the surface of the sphere with a thin layer of paint or wrapping it with a cloth. The total area covered would be the surface area of the sphere.

The formula can be derived by using the method of integration in calculus, but for Grade 9, we can approach it conceptually. Consider slicing the sphere into infinitely thin circular disks stacked on top of each other. The surface area can then be calculated by integrating the circumference of these circles.

The circumference of a circle at any point on the sphere is given by:

Circumference = $2\pi r$

As we stack these circles from the bottom to the top of the sphere, the height of each infinitesimal disk can be approximated. When integrating these circles, we would account for the radius at each point which changes depending on the height.

Properties of a Sphere

• The radius is constant from the center to any point on the surface.
• All diameters of a sphere are equal in length and measure $2r$.
• The surface area increases with the square of the radius, meaning if the radius doubles, the surface area increases by a factor of four.

Example Calculation

Let’s calculate the surface area of a sphere with a radius of $5$ cm.

Using the formula:

SA = $4\pi r^2$

Substituting $r = 5$ cm:

SA = $4\pi (5)^2$

SA = $4\pi \times 25$

SA = $100\pi$

Thus, the surface area of the sphere is $100\pi$ cm². For practical calculations, using $ \pi \approx 3.14$, we can further approximate the surface area:

SA $\approx 100 \times 3.14 = 314$ cm².

Applications of Surface Area of a Sphere

The concept of surface area is crucial in various real-world applications:

• In manufacturing spherical objects like balls and globes, ensuring adequate material use for covering.
• In science, calculating the surface area of celestial bodies such as planets.
• In biology, understanding the surface area of cells for nutrient absorption.

Understanding the surface area of a sphere is essential in solving real-life problems and applying mathematical concepts in various fields.