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Chapter 1: Problem 1

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

Short Answer

Expert verified
Circumference: \( 4.00 \times 10^{4} \mathrm{~km} \), Surface Area: \( 5.10 \times 10^{8} \mathrm{~km^2} \), Volume: \( 1.08 \times 10^{12} \mathrm{~km^3} \).

Step by step solution

01

Formula for Circumference

The circumference of a sphere is given by the formula \( C = 2\pi r \). Here, \( r = 6.37 \times 10^{6} \mathrm{~m} \) is the radius of the Earth.
02

Calculate Circumference

Substitute the radius into the formula: \( C = 2 \times \pi \times 6.37 \times 10^{6} \mathrm{~m} \). When calculated, this yields approximately \( 4.00 \times 10^{4} \mathrm{~km} \) as the circumference of the Earth.
03

Formula for Surface Area

The surface area of a sphere is given by the formula \( A = 4\pi r^2 \). Again, \( r = 6.37 \times 10^{6} \mathrm{~m} \).
04

Calculate Surface Area

Substitute the radius into the formula: \( A = 4 \times \pi \times (6.37 \times 10^{6} \mathrm{~m})^2 \). This results in approximately \( 5.10 \times 10^{8} \mathrm{~km^2} \).
05

Formula for Volume

The volume of a sphere is determined by the formula \( V = \frac{4}{3}\pi r^3 \). Here we again use \( r = 6.37 \times 10^{6} \mathrm{~m} \).
06

Calculate Volume

Substitute the radius into the formula: \( V = \frac{4}{3} \times \pi \times (6.37 \times 10^{6} \mathrm{~m})^3 \). This results in approximately \( 1.08 \times 10^{12} \mathrm{~km^3} \) for the Earth's volume.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circumference Calculation
Understanding how to calculate the circumference of Earth is crucial when studying its geometry. The circumference represents the distance around the widest part of the sphere. For any sphere, it can be calculated using the formula \( C = 2\pi r \). Here, \( r \) symbolizes the radius of the sphere.

When dealing with the Earth, we're given that its radius is \(6.37 \times 10^{6} \mathrm{~m}\). To find the circumference, we plug this value into our formula, giving us \( C = 2 \times \pi \times 6.37 \times 10^{6} \mathrm{~m} \). By performing the calculation step-by-step, you multiply the components together:
  • First, multiply 2 by \(\pi\), which is approximately 3.14159.
  • Next, multiply the result by the Earth's radius in meters.
This simplifies to give a circumference of around \( 4.00 \times 10^{4} \mathrm{~km} \), after converting the result from meters to kilometers. This means if you were to travel around the widest point of Earth without deviating, you would travel approximately 40,000 kilometers.
Surface Area of a Sphere
The surface area of a sphere is important in understanding the total area that makes up its outer layer. For the Earth, this is equivalent to the entire surface that includes land, water, and ice. The formula for surface area is \( A = 4\pi r^2 \). Here, \( r \) is again the radius.

To calculate the Earth's surface area, use the given radius of \(6.37 \times 10^{6} \mathrm{~m}\):
  • Square the radius to get \( (6.37 \times 10^{6} \mathrm{~m})^2 \).
  • Multiply the squared radius by 4 and then by \(\pi\).
This results in a surface area of approximately \( 5.10 \times 10^{8} \mathrm{~km^2} \) once converted into square kilometers. Therefore, the Earth's surface area is roughly 510 million square kilometers, encompassing all continents and oceans.
Volume of a Sphere
The volume of a sphere gives us an idea of how much space it occupies. For Earth, this volume includes the entire mass of its inner layers. To find the volume, use the formula \( V = \frac{4}{3}\pi r^3 \). The radius \( r \) remains as \(6.37 \times 10^{6} \mathrm{~m}\) for Earth.

Here's how you calculate its volume:
  • Cube the radius to get \( (6.37 \times 10^{6} \mathrm{~m})^3 \).
  • Multiply the result by \(\pi\).
  • Finally, multiply by \(\frac{4}{3} \).
The result is the Earth's volume, approximately \( 1.08 \times 10^{12} \mathrm{~km^3} \) when converted into cubic kilometers. This immense volume underscores the vastness of Earth's internal mass.

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Most popular questions from this chapter

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