Find the surface area of a sphere of diameter 21 cm. (Use π = \(\frac{{22}}{7}\) )
1386 cm 2
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:
\(441 \div 7\)
| Step | Calculation | Result |
|---|---|---|
| Divide 44 by 7 | \(44 \div 7\) | 6 with remainder 2 |
| Bring down 1, making 21 | \(21 \div 7\) | 3 |
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
| Property | Formula | Variables |
|---|---|---|
| Radius | \(r = \frac{d}{2}\) | \(r\) = radius, \(d\) = diameter |
| Diameter | \(d = 2r\) | \(d\) = diameter, \(r\) = radius |
| Surface Area | \(A = 4 \pi r^2\) or \(A = \pi d^2\) | \(A\) = surface area, \(r\) = radius, \(d\) = diameter, \(\pi \approx 3.14159\) or \(\frac{22}{7}\) |
| Volume | \(V = \frac{4}{3} \pi r^3\) or \(V = \frac{1}{6} \pi d^3\) | \(V\) = volume, \(r\) = radius, \(d\) = diameter, \(\pi \approx 3.14159\) or \(\frac{22}{7}\) |
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.
Important Questions from Solid Figures
If 3.96 cubic dm of lead is to be drawn in to a cylindrical wire of diameter 0.6 cm, then the length of the wire (in metres), is:
A cone and a hemisphere have equal bases and volumes. What is the ratio of the height of the cone to the radius of the hemisphere?
A metallic solid cuboid of dimensions 36 cm × 18 cm × 12 cm is melted and recast in the form of cubes of side 6 cm. Find the number of cubes so formed.
A solid cylinder has a radius of 9 cm and a height of 25 cm. What is the ratio of its total surface area to its curved surface area?
If the surface area of a sphere is 64 π cm 2, then the volume of the sphere is:
