Earth is approximately a sphere of radius 6.37 ‘ 106 m. What are (a) its circumference in kilometers, (b) its surface area in square kilometers, and (c) its volume in cubic kilometers?
Step 1
Knowing that the perimeter (or circumference) of a circle with radius r is given by the formula:
p = 2\pi rSo, the circumference in kilometers, knowing that the radius converted from meters is:
r = \dfrac{6.37 \times 10^6\ \text{m}}{1000\ \left[\dfrac{\text{m}}{\text{km}}\right]} = 6.37 \times 10^3\ \text{km}Or simply by using the unit conversion:
1\ \text{km} = 1000\ \text{m} r = 6.37 \times 10^6\ \text{m}Since 1 km has 1000 meters, then we have 1000 meters per kilometer. Therefore:
p = 2\pi \times 6.37 \times 10^3\ \text{km} = 4 \times 10^4\ \text{km}Step 2
The surface area of a sphere is given by:
S = 4\pi r^2Using r = 6.37 \times 10^3\ \text{km} from the previous step, we have:
S = 4\pi \left(6.37 \times 10^3\right)^2 = 509.9 \times 10^6\ \text{km}^2Putting it in scientific notation:
S = 5.1 \times 10^8\ \text{km}^2Note that we square the radius because we are squaring a unit of length.
Step 3
The volume of a sphere is given by:
V = \dfrac{4\pi}{3} r^3Using r = 6.37 \times 10^3\ \text{km} , we get:
V = \dfrac{4\pi}{3} \left(6.37 \times 10^3\right)^3 = 1.08 \times 10^{12}\ \text{km}^3Since we’re cubing a length, the result will be in cubic kilometers.
Final Answer
a) 4 \times 10^4\ \text{km}
b) 5.1 \times 10^8\ \text{km}^2
c) 1.08 \times 10^{12}\ \text{km}^3
