What is the surface area of this complex figure?

Question

a) 7,701.8 cm2 b) 5,492.8 cm2 c) 6,597.3 cm2 d) 8,806.2 cm2

Answer 1

Hello, in order to find the surface are of this complex figure you need to find the Total surface and the subtract off areas that do not touch the air ( The bottom of the cylinder - a circle)

For this problem you will need a few formulas:

Area of a circle = π•r2

circumference = 2πr

area of a rectangle = l•w

Area of a parallelogram - all 6 sides = bxh (where the length and width meet at a 90 degree angel)of the parallelogram

Surface area of a Cylinder is 2 circles (top and bottom) & the side that goes around (circumference • height

area base (circle) (note diameter is 92.49)

π • (46.245)² (radius is half = 46.245)

area of 1 circle = π • 2138.600025 yd² = 2138.6 π yd²

area of the side = 2πr • h

rearrange terms = 2•r•h•π•

2•46.245•112 π = 10358.9 π yd²

add areas together for 12497.5π yd^{2 (I did not add in a second circle because it does not touch the air)}

Now the parallelogram.

Base area = 8• 92.49 = 739.92 yd² ( this side is facing or looking at you)

opposite side same = 739.92 yd²

4 sides going around : one unique side has area of 72 • 92.49 = 6659.28 yd²

bottom

you must subtract off circle area from the top 6659.28-2138.6 = 4520.68 yd²

I'm assuming the missing side length of the parallelogram facing us is the same as the bottom (this may not be the case) so the other 2 sides would also be 6659.28 yd²

The parallelogram SA = 6659.28*5 + 4520.68 = 37817.08 yd²

Total SA = cylinder + parallelogram = 12497.5π yd² + 37817.08 yd² = 50314.58 yd²

Answer 2

Hello Isbella,

I will get you started on this one. How many surfaces are there and what shapes are they?

The prism on the bottom has 6 surfaces that are all parallelograms (three sets of two congruent parallelograms). The formula is A = bh for a parallelogram.

The cylinder covers a circle in the surface of the prism, but then the same surface is on top (so it is subtracted and then added so it can be ignored). The lateral (side) surface of the cylinder is a rolled-up rectangle with area A = 2(pi)rh.

So there are 7 surfaces to compute (6 parallelograms and a rolled up rectangle) and add together.

The height for each of the 4 lateral parallelograms is 8 yd. The diameter of the circle in the cylinder is

92.49 yd so you should then find the radius. See if you can work it out from there.

Tom S.

Answer 3

2(92.49)(8) +

2(72)(8) +

2(92.49)(72) +

46.245π +

(92.49)(112)π

Answer 4

Hi Isabella:

Let's do this one! You can see the desired area comprises the surface area of the cylinder minus its circular base plus the surface area of the parallelepiped (3 dimensional parallelogram) minus the circular base of the cylinder, again.

The diameter of the cylinder is 92.49 yards (same as the width of the parallelepiped.)

Its radius, R, is half of this = 46.245 yds.

The surface area of the cylinder = 2 Π R H = 2 Π x 46.245 x 112

= 92.49 x 112 Π

Surface area of the parallelepiped = 2 x (front + side + bottom)

= 2 x (92.49 x 8 + 72 x 8 + 72 x 92.49)

= 2 x (92.49 x 80 + 8 x 72)

= 2 x 8 x (92.49 x 10 + 72)

= 16 x 996.9 sq yds

So, total SA = (92.49 x 112 Π + 16 x 996.9 - 2 Π x 46.245² ) sq yds. [subtracting the cylinder base twice

I leave the arithmetic to you!

Answer 5

The approach would be to compute the surface area of the lower figure + surface area of cylinder - 2 * the surface area of the circle that is hidden.

The surface area of the lower figure = base perimeter * height + 2 * area of base

The base perimeter is 2 * 92.49 yd + 2 * 72 yd. The height is 8 yd and the area of the base is 92.49 yd * 8 yd (area of a parallelogram)

The surface area of a cylinder 2 * PI * R * H + 2 PI* R^2, where height is 112 yd and radius is 46.235 yd.

The surface area of the hidden circle is PI* R^2.