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In this unit, you calculated the surface area of solid figures composed of polygons, such as rectangles and triangles. But imagine finding the surface area of a cylinder—a solid figure that has several curved surfaces. A cylinder is made up of two circular bases and a curved (lateral) surface in between. Develop a formula for finding the surface area of the outside of a cylinder based on what you know about the circumference of a circle and the areas of circles and rectangles. (Hint: Think about rolling a rectangular piece of paper into the shape of a cylinder.) Discuss your approach and why it works.
Answers
The surface area of a cylinder is the surface area of rectangular piece of paper as you roll it into a cylinder and the surface area of the base and top areas:
Assume that you had a rectangular piece of paper with the long side h, and short side w, let h be the height of the cylinder as you roll it, the surface area generated is the circumference of the circle with radius r multiplied by the height = 2 ¶ r h . The area of the top and bottom circular bases is 2 ¶ r².
The total surface area = 2 ¶ r h + 2 ¶ r2 = 2 ¶ r ( h+ r ) to express r in terms of w , w = 2 ¶ r the width of the rectangle equal to the circumference of the circle , then the total surface area = w ( h+ r )
but r = w / 2 ¶ , the surface area = w(h+w/ 2¶) = (wh+ w2/2¶)
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