Question Video: Interpreting and Factoring the Expression for the Surface Area of a Cylinder | Nagwa Question Video: Interpreting and Factoring the Expression for the Surface Area of a Cylinder | Nagwa

Question Video: Interpreting and Factoring the Expression for the Surface Area of a Cylinder Mathematics • Second Year of Preparatory School

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The given diagram shows a cylinder of radius π‘Ÿ and height β„Ž. An expression for the total surface area of the cylinder is 2πœ‹π‘Ÿβ„Ž + 2πœ‹π‘ŸΒ². What does the term 2πœ‹π‘ŸΒ² represent? Factor the total surface area expression completely.

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Video Transcript

The given diagram shows a cylinder of radius π‘Ÿ and height β„Ž. An expression for the total surface area of the cylinder is two πœ‹π‘Ÿβ„Ž plus two πœ‹π‘Ÿ squared. What does the term two πœ‹π‘Ÿ 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 β„Ž and a length that is equal to the circumference of one of the circles, two πœ‹π‘Ÿ. The top and bottom of the cylinder are both circles with radius π‘Ÿ. The area of the rectangular piece is then length times width, or two πœ‹π‘Ÿ times β„Ž.

We recall that the area of a circle with radius π‘Ÿ is given by πœ‹π‘Ÿ squared. By combining the expressions for the area of the rectangle and the areas of the two circles, we get two πœ‹π‘Ÿβ„Ž plus πœ‹π‘Ÿ squared plus πœ‹π‘Ÿ squared. πœ‹π‘Ÿ squared plus πœ‹π‘Ÿ squared can be written as two πœ‹π‘Ÿ squared. Therefore, the two πœ‹π‘Ÿ squared term from the surface area formula represents the area of the two circular faces of the cylinder, whereas the two πœ‹π‘Ÿβ„Ž 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 πœ‹π‘Ÿβ„Ž plus two πœ‹π‘Ÿ squared. We want to factor this completely. First, we look for the highest common factor. In this case, the HCF is two πœ‹π‘Ÿ. Dividing each term in the formula by the HCF leaves us with β„Ž plus π‘Ÿ. Therefore, the complete factorization of the surface area expression is two πœ‹π‘Ÿ times β„Ž plus π‘Ÿ. This means we can find the surface area of a cylinder by first adding its height and radius, then multiplying by two πœ‹ times the radius.

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