Basic Math and Pre-Algebra

PART 3. The Shape of the World

 

CHAPTER 16. Surface Area and Volume

 

Cylinders

Prisms and pyramids are solids formed from polygons, but there are also solids whose bases are circles. The good news is that they have a lot in common with prisms and pyramids.

A solid with two parallel bases that are circles (instead of polygons) is called a cylinder and has a lot in common with prisms.

A cylinder is a solid with two circles as parallel bases and a rectangle wrapped around to join them.

Surface Area

The surface area of a cylinder is the total of the areas of the two circles at the ends, plus the area of the rectangle that forms the sides. (Think about a label on a can. If you remove it from the can and unroll it, it’s a rectangle.) The area of each circle is πr2. The rectangle has a height equal to the height of the cylinder and a base equal to the circumference of the circle, so its area is circumference x h or 2πrh. The total surface area is SA = πr2 + 2πrh.

Suppose that a can is going to be made in the shape of a cylinder with a radius of 4 inches and a height of 10 inches. How much metal will be needed to make the can? The metal needed to make the can is the surface area of the cylinder. Each of the two circular bases has an area of 16π. The curved surface of the can unrolls to a rectangle whose base is the circumference of the circle, 8π, and whose height is 10. The total surface area is 2 x 16π + 8π x 10 = 32π + 80π = 112π square inches. That’s approximately 351.8 square inches.

Volume

The similarities between prisms and cylinders hold up when volume is concerned. The volume of a prism is the area of the base times the height, and the volume of a cylinder is the area of its base, a circle, times its height. That makes the volume formula V = πr2h.

To find the volume of the cylinder whose diameter and height are both 4 inches, first find the radius. If the diameter = 4 inches, the radius = 2 inches, so the area of the base = 4π square inches. The volume is the area of the base times the height, or 4π∙4 = 16π cubic inches. That’s approximately 50.24 cubic inches.

CHECK POINT

Find the surface area and volume of each cylinder. SA = πr2 + 2πrh and V = πr2h

21. A cylinder 14 cm high with a radius of 5 cm.

22. A cylinder 8 inches high with a diameter of 6 inches.

23. A cylinder 2 m high with a circumference of 2π m.

24. A cylinder 82 cm high with a diameter of 90 cm.

25. A cylinder 20 inches high with a circumference of 20π inches.