Activity 2.7.7.
Fun Activity Idea
🔹Use an orange or a ball and cover it with small square sticky notes.
🔹 Estimate how many squares fit over the sphere’s surface.
🔹Then compare with other group members’ results to the actual formula!
Subsection 2.7.4 Surface Area of a Sphere
The surface area of a sphere is the total area covering its curved outer surface.
Formula for Surface Area of a Sphere
Surface Area \(= 4 \pi r^2\) where r is the radius of the sphere.
The \(= 4 \pi r^2\) comes from integrating small patches over the sphere’s curved surface. We can compare the sphere to how a sphere fits inside a cylinder of the same radius and height.
Example 2.7.12.
Find the surface area of the following sphere (correct to 1 decimal place
Example 2.7.13.
Hint.
Solution.
a) Total Surface area.
\begin{align*}
= \amp 4 \times \frac {22}{\cancel {7}}\times \cancel{14} \, \text{cm} \times 14\, \text{cm} \\
= \amp 4 \times 22 \times 2\, \text{cm} \times 14\, \text{cm} \\
= \amp 4 \times 22 \times 28\\
= \amp 4 \times616\, \text{cm}^2\\
= \amp 2464 \,\text{cm}^2
\end{align*}
The surface area of the sphere is \(2464\, \text{cm}^2\) .
b) The \(\textbf{area of the sphere painted}\) would be the entire surface area.
so the Painted area =\(2464 \,\text{cm}^2\text{.}\)
c) If the radius is doubled i.e,
( \(r = 14 \, \text{cm}+ 14\, \text {cm} = 28\, \text{cm}\)),
the new surface area would be:
\begin{align*}
\text{ New surface area} = \amp 4 \frac {22}{\cancel {7}}\times \cancel{28} \, \text{cm} \times 28 \text{cm} \\
= \amp 4 \times 22 \times 4 \, \text{cm} \times 28 \,\text{cm} \\
= \amp 4 \times 22 \times 112 \text{cm}^2\\
= \amp 4 \times616 \, \text{cm}^2\\
= \amp 9856 \, \text{cm}^2
\end{align*}
The surface area of the sphere is \(9856 \, \text{cm}^2\) if the radius is doubled.
\(\textbf{Exercise}\)
Find the surface areas of the figure below.
1. A football used in a tournament has a radius of 11 cm. Find the total surface area of the football, assuming it is a perfect sphere.
2. The surface area of a spherical ornament is measured to be 452.16 cm². Using the formula for the surface area of a sphere, determine the radius of the ornament.
3. A planetarium is constructing a dome in the shape of a hemisphere with a radius of 20 m. Since the dome covers only half of a full sphere, determine its total surface area, including the flat circular base.
4. A spherical metal ball with a radius of 7 cm is to be coated with a layer of paint. Determine the total area that needs to be covered with paint.
5. A company is designing a spherical water tank with a diameter of 24 cm. Compute the total surface area of the tank, which represents the external surface that will be painted.
