Chapter 12.4 Surface Area of Cylinders

Objectives Find lateral areas of cylinders Find lateral areas of cylinders Find surface areas of cylinders Find surface 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 called a right cylinder. Otherwise, the cylinder is an oblique cylinder. Base Altitude Axis Base Altitude Axis Right Cylinder Oblique Cylinder

2 r 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 2πrhA 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

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 = 2 π rh. 2πr2πr2πr2πr h Lateral Area of a Cylinder

Sawyer needs to find out the lateral area for his pole vault pole to know if the pole qualifies for the state track meet. He knows that the diameter of the bases are 6 inches and that the height is 14 ft. The lateral area can’t be more than 267(3204 in) square ft. Does his pole qualify for the track meet?

6 in 14 ft (168 in) The lateral area can’t be more than 267(3204 in) square ft. Does his pole qualify for the track meet? π L = 2πrh Lateral Area of a Cylinder Substitution Simplify π L = 2π(3)(168) L = in L = FtDivide

An office has recycling barrels for cans and paper. The barrels are cylindrical with cardboard sides and plastic lids and bases. Each barrel is 3 feet tall, and the diameter is 30 inches. How many square feet of cardboard are used to make each barrel? Example 1

3 Ft 30 In L = 2πrh Lateral Area of Cylinder r = 15, h = 36 Use a Calculator L = 2π(15)(36) L = (36) In

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

Surface Area of a Cylinder Find the surface area of the cylinder The radius of the base and the height of the cylinder are given. Substitute these values in the formula to find the surface area. T = 2πrh + 2πr²

T = 2πrh + 2πr ² T = 2π(8.3)(6.6) + 2π(8.3²) T = ft The surface area is approximately square feet Surface Area of a Cylinder r = 8.3, h = 6.6 Substitution Use a calculator

Find Missing Parts of the Equations Find the radius of the base of a right cylinder if the surface area is 128 π square centimeters and the height is 12 centimeters Find the radius of the base of a right cylinder if the surface area is 128 π square centimeters and the height is 12 centimeters 12 cm 128π square centimeters 128π = 2π(12)r + 2πr ²

128π = 2π(12)r + 2πr ² T = 2πrh + 2πr ² 128π = 24πr + 2πr ² 64 = 12r + r ² 0 = r ² + 12r = (r - 4)(r + 16) r = 4 or -16 Surface Area of a Cylinder Substitution Simplify π Divide each side by 2π Subtract 64 from each side Factor Since the radius of a circle cannot have a negative value, -16 is eliminated. So the radius of the base is 4 centimeters

A Page six hundred fifty seven through page six hundred fifty eight , 22-25, Omit 23 s s ign m e n t Assignment