Copyright

Derivation of Formula for Total Surface Area of the Sphere by Integration Video

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Derivation of Formula for Volume of the Sphere by Integration

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:04 Surface Area of a Sphere
  • 0:37 Radius, Angle, & Arclength
  • 2:38 Using Integration
  • 4:55 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

The total surface area of a sphere is found using an equation. In this lesson, we derive this equation using the arclength definition, radius relationships, and integral calculus.

The Surface Area of a Sphere

The idea of a surface area is easy to grasp when the object is flat like a square or a triangle. Even a three-dimensional object made up of flat surfaces has an easy to understand equation for the surface area. But what about a curved surface? What about a sphere?

A sphere is a special kind of three-dimensional object. If we know the radius, R, we can find other features of the sphere like its volume and surface area.

For example, the total surface area, A, is 4π times the square of R. In this lesson, we show how to derive this equation.

Radius, Angle, & Arclength

The radius, R, goes from the center of the sphere to it's outer surface. We could also draw another radius, r. This radius also touches the surface, but it starts perpendicular to a vertical line passing through the center of the sphere:

An angle, θ, provides a relationship between R and r:

\

We can displace r:

Use the definition of cosine (adjacent side over the hypotenuse):


cos_theta=r/R


Solving for r:


r=Rcos_theta


Great! We will use this r-R relationship later on. Now, lets delve into the idea of arclength.

We know once around a circle is 360o which is the same as 2π radians. We also know, the perimeter of a circle as being 2πR.

If θ is some arbitrary angle less than 2π, then θ / 2π is a fraction. The length along the arc (better known as the arclength), s, subtended by θ is a fraction of the total length 2πR. This fraction is given by s/2πR. These fractions are equal:


theta/2pi=s/2piR


The 2π factors cancel from each side of the equation, leaving:


theta=s/R


Solving for s:


s=R_theta


We can take the derivative of both sides:


ds=d(R_theta)


On the second line, we used the product rule which gives two terms. But dR is 0 because R is a constant. Meaning, θ dR is 0. Thus, we are left with only one term on the third line, giving us another very useful result:


ds=Rd_theta


Using Integration

With these results, we'll integrate to derive the total surface area equation. But first, lets revisit the circle traced out by r.

If we draw another circle immediately below it with the same radius we get the circle appearing here:


Two_circles_separated_by_ds


The separation between the two circles is an infinitely small arclength, ds:


Extracting_this_cylindrical_band_from)the_sphere


We can open this band to get a rectangle:


A_rectangle_with_known_dimensions


The rectangle area, da, is a differential area equal to the length times the width. This rectangle has a length, 2πr, and a width, ds. Thus, the area, da, of this infinitely thin cylinder is:


da=2pi_r_ds


To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support