Deriving the Formula for Surface Area

Oct 13, 2025 by thesweetdesigns.com 81 views

The Surface Area of a Sphere: Deriving the Formula and Solving for Radius

The surface area of a sphere is a fundamental concept in mathematics and physics, with numerous applications in fields such as engineering, architecture, and astronomy. The surface area of a sphere is given by the formula $s = 4\pi r^2$ , where $s$ is the surface area and $r$ is the radius of the sphere. In this article, we will derive the formula for the surface area of a sphere and then solve for the radius of a sphere with a given surface area.

Deriving the Formula for Surface Area

The surface area of a sphere can be derived by considering the sphere as a collection of infinitesimally small circular rings. Each ring has a radius $r$ and a width $dr$ . The surface area of each ring is given by the circumference of the ring multiplied by the width of the ring, which is $2\pi r dr$ . To find the total surface area of the sphere, we need to integrate the surface area of each ring over the entire sphere.

The surface area of the sphere can be expressed as:

s = \int_{0}^{2\pi} 2\pi r dr

Evaluating the integral, we get:

s = 2\pi r \int_{0}^{2\pi} dr

s = 2\pi r [r]_{0}^{2\pi}

s = 2\pi r (2\pi - 0)

s = 4\pi r^2

Solving for Radius

Now that we have derived the formula for the surface area of a sphere, we can solve for the radius of a sphere with a given surface area. We are given that the surface area of the sphere is $154$ .

We can set up the equation:

154 = 4\pi r^2

To solve for $r$ , we can divide both sides of the equation by $4\pi$ :

\frac{154}{4\pi} = r^2

r^2 = \frac{154}{4\pi}

Taking the square root of both sides, we get:

r = \sqrt{\frac{154}{4\pi}}

Simplifying the expression, we get:

r = \sqrt{\frac{77}{2\pi}}

Numerical Solution

To find the numerical value of $r$ , we can use a calculator or a computer program to evaluate the expression:

r = \sqrt{\frac{77}{2\pi}}

Using a calculator, we get:

r \approx 3.98

Therefore, the radius of a sphere with a surface area of $154$ is approximately $3.98$ .

In this article, we derived the formula for the surface area of a sphere and solved for the radius of a sphere with a given surface area. We showed that the surface area of a sphere is given by the formula $s = 4\pi r^2$ , where $s$ is the surface area and $r$ is the radius of the sphere. We then solved for the radius of a sphere with a surface area of $154$ and found that the radius is approximately $3.98$ . This result has numerous applications in fields such as engineering, architecture, and astronomy.
The Surface Area of a Sphere: Q&A

In our previous article, we derived the formula for the surface area of a sphere and solved for the radius of a sphere with a given surface area. In this article, we will answer some frequently asked questions about the surface area of a sphere.

Q: What is the surface area of a sphere?

A: The surface area of a sphere is the total area of the surface of the sphere. It is given by the formula $s = 4\pi r^2$ , where $s$ is the surface area and $r$ is the radius of the sphere.

Q: How do I calculate the surface area of a sphere?

A: To calculate the surface area of a sphere, you need to know the radius of the sphere. You can then use the formula $s = 4\pi r^2$ to find the surface area.

Q: What is the relationship between the surface area and the radius of a sphere?

A: The surface area of a sphere is directly proportional to the square of the radius of the sphere. This means that if the radius of the sphere is doubled, the surface area will increase by a factor of four.

Q: Can I use the surface area formula to find the radius of a sphere?

A: Yes, you can use the surface area formula to find the radius of a sphere. If you know the surface area of the sphere, you can rearrange the formula to solve for the radius.

Q: What is the surface area of a sphere with a radius of 5?

A: To find the surface area of a sphere with a radius of 5, you can use the formula $s = 4\pi r^2$ . Plugging in $r = 5$ , you get:

s = 4\pi (5)^2

s = 4\pi (25)

s = 100\pi

Therefore, the surface area of a sphere with a radius of 5 is $100\pi$ .

Q: What is the radius of a sphere with a surface area of 200?

A: To find the radius of a sphere with a surface area of 200, you can use the formula $s = 4\pi r^2$ . Rearranging the formula to solve for $r$ , you get:

r = \sqrt{\frac{s}{4\pi}}

Plugging in $s = 200$ , you get:

r = \sqrt{\frac{200}{4\pi}}

r = \sqrt{\frac{50}{\pi}}

Therefore, the radius of a sphere with a surface area of 200 is $\sqrt{\frac{50}{\pi}}$ .

Q: Can I use the surface area formula to find the volume of a sphere?

A: No, the surface area formula cannot be used to find the volume of a sphere. The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$ , where $V$ is the volume and $r$ is the radius of the sphere.

In this article, we answered some frequently asked questions about the surface area of a sphere. We showed how to calculate the surface area of a sphere, how to find the radius of a sphere with a given surface area, and how to use the surface area formula to solve for the radius of a sphere. We also discussed the relationship between the surface area and the radius of a sphere, and how to use the surface area formula to find the volume of a sphere.