Question

Earth is approximately a sphere of radius \(6.37 \times 10^{6} \mathrm{~m} .(a)\) What is its circumference in kilometers? (b) What is its surface area in square kilometers? ( \(c\) ) What is its volume in cubic kilometers?

Short answer

The circumference of the Earth is approximately \(40.03 \times 10^3\) km, the surface area is approximately \(510.2 \times 10^6\) km², and the volume is approximately \(10.86 \times 10^{11}\) km³.

Step-by-step solution

Calculate the Circumference

Use the formula for the circumference of a sphere, which is \(C = 2\pi r\). The radius is given in meters, so we need to convert this to kilometers by dividing by \(10^3\) before applying the formula. The radius in kilometers is \(6.37 \times 10^3\) km, so the circumference is \(C = 2\pi\times 6.37 \times 10 ^3\) km.

Calculate the Surface Area

Apply the formula for the surface area of a sphere, which is \(A = 4\pi r^2\). We already have the radius in kilometers from the previous step, so we can directly substitute into the formula to get the surface area: \(A = 4\pi\times (6.37 \times 10^3)^2\) km².

Calculate the Volume

Finally, the volume of a sphere can be calculated using the formula \(V = \frac{4}{3}\pi r^3\). Substitute the converted radius value from step 1 into the formula: \(V = \frac{4}{3}\pi\times (6.37 \times 10^3)^3\) km³.

Key concepts

Circumference of a Sphere

Understanding the circumference of a sphere can be a key aspect in diverse fields, such as geography when estimating the distance around our planet. The circumference is essentially the distance around the sphere at its widest part. To calculate it, we use the formula \( C = 2\pi r \), where \( \pi \) (Pi) is a constant approximately equal to 3.14159, and \( r \) is the radius of the sphere. For Earth, with a radius of about \( 6.37 \times 10^6 \mathrm{m} \), we must convert meters to kilometers by dividing by \( 10^3 \) to ensure the units are consistent with how we’ll express the result.

Keep in mind that precise calculations are critical here, especially when dealing with such large numbers, as small errors can lead to large deviations in your results. Once converted, the radius and formula lead us to the circumference of Earth, which showcases a fundamental application of spherical geometry in real-world measurements.

Surface Area of a Sphere

The surface area of a sphere represents the total region covered by the surface of the sphere. This is important in fields such as astronomy and physics, as it can affect heat absorption and radiation, among other phenomena. To determine the surface area, we apply the formula \( A = 4\pi r^2 \). It requires the square of the radius, which significantly increases the effect of the radius on the surface area. A critical observation should be made here: the square of the radius means that if the radius doubles, the surface area increases by four times.

Practical Application:

For our home planet, we've already converted the radius to kilometers earlier. Square this value and multiply by 4\pi to get Earth's surface area in square kilometers. This calculation reflects how much of space our planet occupies, calculating the area over which Earth's atmosphere, oceans, and land spreads.

Volume of a Sphere

When we talk about the volume of a sphere, we're referring to the three-dimensional space it contains. This information is paramount for applications like modeling the liquid content in a spherical tank or estimating the quantity of material required to create a spherical object. The formula for volume of a sphere is given as \( V = \frac{4}{3}\pi r^3 \), where you take the cube of the radius. This third power illustrates an exponential growth: when the radius is multiplied, the volume increases by the cube of that factor.

Why is this important?

Knowing the volume of a sphere such as Earth gives us an understanding of its size in a more tangible way than simply knowing its radius. For example, scientists can derive the mass of the Earth by using its volume alongside the average density of the materials that compose it. Truly, comprehending how to calculate the volume of a sphere enriches our grasp of the spatial properties that shape our world.