Surface Area of a Sphere

The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space. A sphere is a three-dimensional solid having a round shape, just like a circle. The difference between a sphere and a circle is that a circle is a two-dimensional figure or a flat shape, whereas a sphere is a three-dimensional shape. A circle is also the two-dimensional cross-section of a sphere.

From a visual perspective, a sphere has a three-dimensional structure formed by rotating a circular disc with one of the diagonals that cross its center.

Consider a situation where a spherical ball is painted. To paint the whole surface without wasting paint, the paint quantity required has to be known beforehand. Therefore, the area of the surface has to be known to calculate the paint quantity for painting it. We define this term as the total surface area.

The Greek mathematician Archimedes discovered that the surface area of a sphere is the same as the lateral surface of a cylinder having the same radius as the sphere and a height the length of the diameter of the sphere.

The lateral surface area of the cylinder is $\mathrm{LSA}=2\pi rh$ where $h=2r$ .

$2\pi r\left(2r\right)=4\pi r^2$

The surface area of a sphere with radius r equals $4\pi r^2$ .

For any three-dimensional shape, the area of the object can be categorized into three types. They are:

• Curved surface area: The curved surface area is the area of all the curved regions of the solid
• Lateral surface area: The lateral surface area is the area of all the regions except bases (top and bottom, for example)
• Total surface area: The total surface area is the area of all the sides, top, and bottom of the solid object.

In the case of the sphere, there are no flat sides.

Therefore, the total surface area of a sphere is the curved surface of a sphere.

Example:

Find the surface area of a sphere with a radius of 5 inches.

$\mathrm{Surface\; area}=4\pi {\left(5\right)}^2$

$\mathrm{SA}=100\pi {\mathrm{inches}}^2$

Approximately $314.16{\mathrm{inches}}^2$

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