Question Video: Calculating the Surface Area of a Cylinder given Its Net Mathematics • Second Year of Preparatory School

Video Transcript

The diagram below shows the net of a cylinder where 𝐴𝐵𝐶𝐷 is a rectangle with 𝐴𝐵 equals 20 centimeters and 𝐴𝐷 equals 44 centimeters. The net is formed into a cylinder by joining line segment 𝐴𝐵 with line segment 𝐷𝐶, then folding over the two circles of radii seven centimeters to make the top and the base. What is the total surface area of the cylinder? Use 𝜋 as 22 over seven.

We’ve been given the net of a cylinder and asked to calculate its total surface area. The total surface area of any solid is simply the sum of the areas of all its faces, which can easily be found from its net as this is a two-dimensional representation of the solid. From the given net, we can identify that a cylinder has a rectangular face, which forms the curved surface of the cylinder when the net is folded, and two circular faces, which form its ends. We’ve been given the dimensions of each of these shapes. So we can calculate their areas by applying standard formulae.

The area of a rectangle is found by multiplying its length by its width. So the area of this rectangle is 44 multiplied by 20. The area of a circle is 𝜋 multiplied by its radius squared. The two circles at either end of the cylinder are congruent, and both have a radius of seven centimeters. So their combined area is two multiplied by 𝜋 multiplied by seven squared.

We’ve been told to use the value 22 over seven as an approximation for 𝜋. So before we evaluate, we’ll make this substitution. 44 multiplied by two is 88, so 44 multiplied by 20 is 880. For the second part of the expression, we can cancel a factor of seven from the numerator and denominator, leaving two multiplied by 22 multiplied by seven. That’s 14 multiplied by 22. And using any method for multiplying two two-digit numbers without a calculator, we find that this is equal to 308. Finally, we sum these two values and include the units for this area, which are square centimeters.

We’ve found that the total surface area of the given cylinder using 𝜋 as 22 over seven is 1,188 square centimeters.