12.4 Surface Areas of Cylinders By: Kristy Truong and Laura Blair

Objectives  Find lateral areas of cylinders  Find surface areas of cylinders

Lateral Areas of Cylinders  The axis of the cylinder is the segment with endpoints that are centers of the circular bases.  If the axis is also the altitude, then the cylinder is a right cylinder.  Otherwise, the cylinder is an oblique cylinder. Axis Base Altitude Base

Net of a Cylinder  The net of a cylinder is composed of two congruent circles and a rectangle. The area of this rectangle is the lateral area. The length of the rectangle is the same as the circumference of the base, 2πr. So, the lateral area of a right cylinder is 2πrh. r 2πr2πr h

Lateral Area of a Cylinder  If a right cylinder has a lateral area of L square units, a height of h units, and the bases have radii of r units, then…  L=2πrh h 2πr2πr

Example 1  A zoo has recycling barrels for storing animal feed. The barrels are cylindrical with cardboard sides and plastic lids and bases. Each barrel is 4 feet tall, and the diameter is 60 inches. How many square feet of cardboard are used to make each barrel?

 The cardboard section of the barrel represents the lateral area of the cylinder. If the diameter is 60, then the radius is 30 inches. The height is 4 feet, or 48 inches. L=2πrh =2π(30)(48) = Each barrel uses about 9048 square inches of cardboard. 144 sq. inches = 1 sq. ft. there are about 9048/144 or 62.8 sq. ft. per barrel.

Surface Areas of Cylinders  To find the surface area of a cylinder, first find the lateral area and then add the areas of the bases. This leads to the formula for the surface area of a right cylinder.

Surface Area of a Cylinder  If a right cylinder has a surface area of T sq. units, a height of h units, and the bases have radii of r units, then….  T= 2πrh + 2πr 2 r h

Surface Area of a Cylinder  Find the surface area of the cylinder. 5.3 ft 7.5ft

Cont…  T= 2πrh + 2πr  = 2π(5.3)(7.5) + 2π(5.3)  = sq ft 5.3 ft 7.5ft 2 2

Find Missing Dimensions  Find the radius of the base of a cylindrical cup if the surface area is 57π sq inches and the height is 6 inches.

Cont…  T=2πrh + 2πr  57π = 2π(6)r + 2πr  57π =12πr +2πr  28.5 = 6r + r (Divide each side by 2π)  0 = r + 6r – 28.5  r = -6 +√6 -4(1)(-28.5) / 2(1)  r =3.1  Radius of base = 3.1inches

Assignment  Pg 657 #9-20 all, 22, 24