Question Video: Calculating the Base Diameter of a Cylinder given Its Lateral Surface Area and Its Height Mathematics • Second Year of Preparatory School

Video Transcript

Find the diameter of the base of a cylinder if the lateral surface area is 434𝜋 square centimeters and the height is 31 centimeters.

We’ve been given the lateral surface area of a cylinder, along with its height, and asked to determine the diameter of the cylinder’s circular base. Let’s begin by recalling how the lateral surface area is calculated.

The lateral surface area of a cylinder is the area of the curved surface that wraps around the cylinder. This surface is a rectangle, with a length equal to the circumference of the circular base and a width equal to the height of the cylinder. The area of this surface is therefore equal to two 𝜋𝑟ℎ. We can substitute the lateral surface area of 434𝜋 and the height of 31 to give the equation 434𝜋 equals two 𝜋𝑟 multiplied by 31.

Now, as it is the diameter of the circular base that we’re asked to calculate, and we know that the diameter is always twice the radius, we can replace two 𝑟 with 𝑑 to give an equation in terms of 𝑑. Alternatively, we could solve our previous equation for 𝑟 and double its value at the end.

To solve our equation for 𝑑, we can first cancel a factor of 𝜋 from both sides to give 434 is equal to 31𝑑. Finally, we divide both sides of the equation by 31. And we have that 𝑑 is equal to 14. Including the length units, we’ve found that the diameter of the base of the given cylinder is 14 centimeters.