Problem 1
L = 10 = length
W = 5 = width
H = 4 = height
SA = surface area of a rectangular prism, aka block or box
SA = 2*(LW+LH+WH)
SA = 2*(10*5+10*4+5*4)
SA = 220
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LA = lateral area, aka wall area
LA = (perimeter of the base)*(height)
LA = (2L+2W)*H
LA = (2*10+2*5)*4
LA = 120
or
LA = SA - (floor area) - (ceiling area)
LA = SA - 2*(area of the floor)
LA = 220 - 2*(LW)
LA = 220 - 2*(10*5)
LA = 220 - 100
LA = 120
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Answers:
Surface Area = 220 square feet
Lateral Area = 120 square feet
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Problem 2
Imagine rotating the object so that the triangular face is resting on the flat ground. This means the triangular faces are the floor and ceiling. The walls of any prism are always rectangles.
Focus on the right triangle with leg b = 4 and hypotenuse c = 5.
Use the pythagorean theorem to find that a = 3 is the missing leg length.
The triangle has base = 4 and height = 3, so its area is base*height/2 = 4*3/2 = 6 square inches. This is the area of the floor and ceiling each.
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This is the summary so far
- base or floor area = 6 square inches
- depth of the prism = 8 inches
- (a,b,c) = (3,4,5) are the sides of the right triangle
Those values are then plugged into the formulas below.
SA = surface area of a triangular prism
SA = 2*(base area) + depth*(a+b+c)
SA = 2*(6) + 8*(3+4+5)
SA = 108
LA = lateral area
LA = depth*(a+b+c)
LA = 8*(3+4+5)
LA = 96
or
LA = SA - 2*(floor area)
LA = 108 - 2*(6)
LA = 96
Note that a+b+c = perimeter of the triangular base.
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Answers:
Surface Area = 108 square inches
Lateral Area = 96 square inches
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Problem 3
Imagine rotating the figure so that the octagon is resting on the flat ground.
Refer to the screenshot below to see the steps on computing the area of the regular octagon, aka the floor area.
The floor area is approximately 120.710678 square cm. The same goes with the ceiling because the floor and ceiling are congruent regular octagons that are parallel.
Each rectangular wall has an area of 5*22 = 110 square cm. Eight identical walls give a total lateral area of 8*110 = 880 square cm.
Add the floor and ceiling area to the wall area to get the total surface area
floor + ceiling + walls = 120.710678 + 120.710678 + 880 = 1121.421356
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Answers:
Surface Area = 1121.421356 square cm (approximate)
Lateral Area = 880 square cm