Download - Chapter 12.4 Surface Area of Cylinders
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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