Calculating Surface Area of a Sphere

The question asks us to find the surface area of a sphere given its diameter. We are also given the value of \(\pi\) to use in our calculation.

Understanding the Sphere and its Properties

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball. Key properties include:

• Diameter: The distance across the sphere passing through its center.
• Radius: The distance from the center of the sphere to any point on its surface. The radius is half of the diameter.
• Surface Area: The total area of the outer surface of the sphere.

Given Information

• Diameter of the sphere = 21 cm
• Value of \(\pi = \frac{22}{7}\)

Finding the Radius

The radius (\(r\)) of a sphere is half of its diameter (\(d\)).

Radius \(r = \frac{\text{Diameter}}{2}\)

Substituting the given diameter:

\(r = \frac{21}{2}\) cm

Formula for Surface Area of a Sphere

The formula to calculate the surface area (\(A\)) of a sphere is:

\(A = 4 \pi r^2\)

Where:

• \(A\) is the surface area
• \(\pi\) is the mathematical constant (given as \(\frac{22}{7}\))
• \(r\) is the radius of the sphere

Step-by-Step Calculation

Now we substitute the value of \(\pi\) and the calculated radius (\(r = \frac{21}{2}\) cm) into the surface area formula:

\(A = 4 \times \pi \times r^2\)

\(A = 4 \times \frac{22}{7} \times \left(\frac{21}{2}\right)^2\)

First, calculate the square of the radius:

\(\left(\frac{21}{2}\right)^2 = \frac{21^2}{2^2} = \frac{21 \times 21}{2 \times 2} = \frac{441}{4}\)

Now substitute this back into the formula:

\(A = 4 \times \frac{22}{7} \times \frac{441}{4}\)

We can cancel out the '4' in the numerator and the denominator:

\(A = \frac{22}{7} \times 441\)

Next, divide 441 by 7:

\(441 \div 7\)

So, \(441 \div 7 = 63\).

Now multiply 22 by 63:

\(A = 22 \times 63\)

Let's perform the multiplication:

\(22 \times 63 = (20 + 2) \times 63 = (20 \times 63) + (2 \times 63)\)

\(20 \times 63 = 1260\)

\(2 \times 63 = 126\)

\(1260 + 126 = 1386\)

So, the surface area \(A = 1386\) cm².

Final Answer

The surface area of the sphere with a diameter of 21 cm is 1386 cm².

Revision Table: Sphere Calculations

Additional Information on Sphere Geometry

Understanding the properties of a sphere is crucial in geometry and physics. Here are some key points:

• The surface area formula \(A = 4 \pi r^2\) is four times the area of a circle with the same radius (\(\pi r^2\)). This is a fascinating geometric relationship.
• The surface area can also be calculated directly using the diameter with the formula \(A = \pi d^2\). Let's check this for our example: \(A = \pi \times (21)^2 = \frac{22}{7} \times 441 = 22 \times 63 = 1386\) cm². This matches our previous result.
• Spheres have the smallest surface area among all surfaces enclosing a given volume, and the largest volume among all closed surfaces with a given surface area. This makes them efficient shapes in nature (like bubbles or planets).
• The concept of surface area is important in various applications, such as calculating the amount of paint needed to cover a spherical tank or determining heat transfer from a spherical object.