Chapter 12.4Surface Area of Cylinders

ObjectivesObjectives

Find lateral areas of cylinders Find lateral areas of cylinders

Find surface areas of cylindersFind surface areas of cylinders

The The axisaxis of the cylinder is the of the cylinder is the segment with endpoints that are segment with endpoints that are

centers of the circular bases. If the centers of the circular bases. If the axis is also the altitude, then the axis is also the altitude, then the cylinder is called a cylinder is called a right cylinderright cylinder. .

Otherwise, the cylinder is an Otherwise, the cylinder is an oblique cylinderoblique cylinder..

Base

Altitude Axis

Base Base

Base

Altitude

Axis

Right Cylinder Oblique Cylinder

2 2 rr

•A cylinder is composed of two A cylinder is composed of two congruent circles and a rectangle. congruent circles and a rectangle.

The area of this rectangle is the The area of this rectangle is the lateral area. The length of the lateral area. The length of the rectangle is the same as the rectangle is the same as the

circumference of the base, 2circumference of the base, 2ππrr. . So, the lateral area of a right So, the lateral area of a right

cylinder is 2cylinder is 2ππrhrh

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.

22ππrr

hh

Lateral Area of a Lateral Area of a CylinderCylinder

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

6 in6 in

14 ft14 ft(168 in)(168 in)

The lateral area can’t be more than The lateral area can’t be more than 267(3204 in) square ft. Does his pole 267(3204 in) square ft. Does his pole

qualify for the track meet?qualify for the track meet?

L = 2ππrh Lateral Area of a Cylinder

Substitution

Simplify

L = 2ππ(3)(168)

L = 3166.7 in

L = 263.9 Ft Divide

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 1Example 1

3 Ft

30 In

L = 2L = 2ππrhrh Lateral Area of Cylinder

r = 15, h = 36

Use a

Calculator

L = 2L = 2ππ(15)(36)(15)(36)

L = 3392.9L = 3392.9(36) In

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

• If a right cylinder has a surface area of T If a right cylinder has a surface area of T square units, a height of square units, a height of hh units, and the units, and the bases have radii of bases have radii of rr units, then T = 2 units, then T = 2ππrhrh + + 22ππr r ²²

rrhh

Surface Area of CylindersSurface Area of Cylinders

Surface Area of a Surface Area of a CylinderCylinder

6.6

8.3

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 = 2T = 2ππrhrh + 2 + 2ππrr²²

8.3

6.6

T = 2T = 2ππrhrh + 2 + 2ππr r ²²

T = 2T = 2ππ(8.3)(6.6) + (8.3)(6.6) + 22ππ(8.3(8.3²²))

T = 777.0 ftT = 777.0 ft

The surface area is approximately 777.0 square feetThe surface area is approximately 777.0 square feet

Surface Area of a Cylinder

r = 8.3, h = 6.6 Substitution

Use a calculator

Find Missing Parts of the Find Missing Parts of the EquationsEquations

Find the radius of the base of a right Find the radius of the base of a right cylinder if the surface area is 128cylinder if the surface area is 128ππ

square centimeters and the height is 12 square centimeters and the height is 12 centimeters centimeters

12 cm12 cm

128128ππ square centimeters square centimeters128128ππ = 2 = 2ππ(12)(12)rr + 2 + 2ππrr ²²

128128ππ = 2 = 2ππ(12)(12)rr + 2 + 2ππrr ² ²

T = 2T = 2ππrhrh + 2 + 2ππr r ²²

128128ππ = 24 = 24ππrr + 2 + 2ππrr ²²

64 = 1264 = 12rr + + rr ²²

0 = 0 = rr ² + 12² + 12rr - 64 - 64

0 = (0 = (rr - 4)( - 4)(rr + 16) + 16)

rr = 4 or -16 = 4 or -16

Surface Area of a Cylinder

Substitution

Simplify

Divide each side by 2ππ

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

AA

Page six hundred fifty seven through page six hundred fifty eight.

657-658 9-20, 22-25, Omit 23

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