Video Transcript
The given diagram shows a cylinder of radius 𝑟 and height ℎ. An expression for the total surface area of the cylinder is two 𝜋𝑟ℎ plus two 𝜋𝑟 squared. What does the term two 𝜋𝑟 squared represent? Factor the total surface area expression completely.
To better understand how the sum of these two expressions makes up the surface area formula, we will sketch a picture of the cylinder’s net. To do this, we think about unraveling the curved surface of the cylinder. The net is composed of a rectangular surface connected to two circular faces. The rectangle has a width of ℎ and a length that is equal to the circumference of one of the circles, two 𝜋𝑟. The top and bottom of the cylinder are both circles with radius 𝑟. The area of the rectangular piece is then length times width, or two 𝜋𝑟 times ℎ.
We recall that the area of a circle with radius 𝑟 is given by 𝜋𝑟 squared. By combining the expressions for the area of the rectangle and the areas of the two circles, we get two 𝜋𝑟ℎ plus 𝜋𝑟 squared plus 𝜋𝑟 squared. 𝜋𝑟 squared plus 𝜋𝑟 squared can be written as two 𝜋𝑟 squared. Therefore, the two 𝜋𝑟 squared term from the surface area formula represents the area of the two circular faces of the cylinder, whereas the two 𝜋𝑟ℎ term represents the area of the rectangular surface wrapped around the cylinder.
In the second part of the question, we are asked to factor the total surface area expression completely. The surface area expression is two 𝜋𝑟ℎ plus two 𝜋𝑟 squared. We want to factor this completely. First, we look for the highest common factor. In this case, the HCF is two 𝜋𝑟. Dividing each term in the formula by the HCF leaves us with ℎ plus 𝑟. Therefore, the complete factorization of the surface area expression is two 𝜋𝑟 times ℎ plus 𝑟. This means we can find the surface area of a cylinder by first adding its height and radius, then multiplying by two 𝜋 times the radius.
