Question

Find the volume \(V\) and surface area \(S\) of a sphere of radius 5 centimeters.

Short answer

Volume: \( \frac{500}{3} \pi \) cubic cm, Surface Area: \( 100 \pi \) square cm.

Step-by-step solution

- Identify the formula for volume

To find the volume of a sphere, use the formula \[ V = \frac{4}{3} \pi r^3 \]where \( r \) is the radius of the sphere.

- Substitute the radius into the volume formula

Substitute the given radius (5 cm) into the volume formula: \[ V = \frac{4}{3} \pi (5)^3 \]

- Calculate the volume

Calculate the volume by simplifying the equation: \[ V = \frac{4}{3} \pi (125) \] So, \[ V = \frac{500}{3} \pi \]

- Identify the formula for surface area

To find the surface area of a sphere, use the formula: \[ S = 4 \pi r^2 \]

- Substitute the radius into the surface area formula

Substitute the given radius (5 cm) into the surface area formula: \[ S = 4 \pi (5)^2 \]

- Calculate the surface area

Calculate the surface area by simplifying the equation: \[ S = 4 \pi (25) \] So, \[ S = 100 \pi \]

Key concepts

Sphere Volume Formula

The volume of a sphere is found using a specific formula. This formula is crucial for solving many geometry problems involving spheres. The formula is: \[ V = \frac{4}{3} \pi\ r^3 \]Here, \( V \) represents the volume, \( \pi \) is a constant approximately equal to 3.14159, and \( r \) is the radius of the sphere. To understand why this formula works, remember that it takes into account the sphere's three-dimensional space. In our specific exercise, the radius \( r \) is 5 cm. By substituting 5 cm into our formula, we get: \[ V = \frac{4}{3} \pi\ (5)^3 \]Next, you raise 5 to the third power, resulting in 125. Then, substitute back into the equation: \[ V = \frac{4}{3} \pi\ (125) \] Upon simplifying, the volume of our sphere becomes: \[ V = \frac{500}{3} \pi \] This formula is very efficient and always accurate, provided the radius value is correct!

Sphere Surface Area Formula

The surface area of a sphere can be found with another handy formula. This is important for determining the outer covering of a sphere, useful in many practical applications. The formula is: \[ S = 4 \pi\ r^2 \]In this equation, \( S \) stands for surface area, \( \pi \) is the same constant as above, and \( r \) is still the radius of the sphere. This formula considers all the points on the sphere's surface. For our exercise, we also use a radius \( r \) of 5 cm. Substituting 5 into the formula gives: \[ S = 4 \pi\ (5)^2 \] You square the 5, which equals 25, and substitute back into the equation: \[ S = 4 \pi\ (25) \] Simplifying results in: \[ S = 100 \pi \] This formula lets you find the sphere's surface area easily and accurately.

Geometry Calculations

Understanding geometry calculations is essential for solving problems involving shapes like spheres. Let’s break down some key points:

• The volume formula \( V = \frac{4}{3} \pi\ r^3 \) calculates the space inside the sphere.
• The surface area formula \( S = 4 \pi\ r^2 \) determines the area covering the sphere’s outer surface.
• Using accurate measurements for the radius is crucial for precise results in both formulas.

In the given exercise, with a radius of 5 cm:

• The volume calculation proceeds from \( V = \frac{4}{3} \pi\ (5)^3 \) to \( V = \frac{500}{3} \pi \).
• The surface area calculation moves from \( S = 4 \pi\ (5)^2 \) to \( S = 100 \pi \).

Always ensure your units (e.g., cm, m) remain consistent. These foundational geometry calculations are indispensable for various scientific and practical applications.