Video: Finding the Surface Area of a Circular Cylinder Given the Volume and the Height

Video Transcript

Find the surface area of a circular cylinder whose volume is 1256 centimeters cubed and whose height is 16 centimeters. Use 𝜋 is equal to 3.14.

With these type of questions, what I like to do first is to write down the information I’ve been given. So first of all, I’ve got the volume, 𝑣, is equal to 1256 centimeters cubed. Then, I know the ℎ, the height, is 16 centimeters. And we’re told to use 𝜋 is equal to 3.14. So, the next thing I do is draw a sketch to help me visualize what’s going on. So here, I’ve shown our cylinder. And I’ve got the height and also got the radius on there cause we don’t know the radius. And then next, what we need to do is consider what formulae we might use to help us solve the problem.

The first one is the volume of a cylinder, and this is equal to 𝜋𝑟 squared ℎ. And then next, what we need to consider is the formula for surface area. And that is two 𝜋𝑟ℎ plus two 𝜋𝑟 squared. And we get that cause the two 𝜋𝑟ℎ is the surface area of our curved surface. And then, the two 𝜋𝑟 squared is the surface area of our two end points, which are both circles. And the area of a circle is 𝜋𝑟 squared, and we have two of them.

So now, what we’re gonna do is find 𝑟. And we can do that using our volume formula. If we substitute in our values to that formula, we get 1256, cause that’s our volume, is equal to 3.14, the value that we’re using for 𝜋, multiplied by 𝑟 squared multiplied by 16. So next, if we multiply the 3.14 and 16, we get 50.24. So therefore, we’ve got 1256 equals 50.24𝑟 squared. Then, if we divide through both sides by 50.24, we get 25 is equal to 𝑟 squared. And then, if we take the square root of both sides, we get five is equal to 𝑟. And we’re only interested in the positive result because we’re looking at a length, because we’re looking at radius. So, we now know that the radius is equal to five.

So now, what we can do, because we’ve got the radius, is move on and find the surface area cause that’s what we need to find, because we’ve got all the information we need. So now, what we do is to find the surface area, substitute in our values into two 𝜋𝑟ℎ plus two 𝜋𝑟 squared.

When we do that, we get two multiplied by 3.14, our value for 𝜋, multiplied by five, which is our radius, multiplied by 16 and then plus two multiplied by 3.14 multiplied by five squared. And when we calculate this, what we get is 502.4 plus 157, which gives us our final answer of 659.4. And this is centimeters squared because we’re looking at an area. So therefore, we can say that the surface area of a circular cylinder whose volume is 1256 centimeters cubed and whose height is 16 centimeters is 659.4 centimeters squared.