Question

Find the surface area of the composite figure.
SA = [ ? ] cm^2

Answer 1

SA = 436
`cm^(2)`

Explanation:

Hi Allenwarren! I noticed that you were having a lot of trouble with surface area through a friend who answered one of your questions, so I wanted to see if I could try to help you out with this topic! I'll try to explain this as best as I can, and hopefully, your confusion with the topic of surface area will be cleared. However, if you still have questions after this, please comment, and I'll be more than happy to help :D

Okay, so first of all, we need to know what exactly surface area means:

Surface area is the measurement of all of the outside area of a shape/composite figure (which is a figure made up of more than one simple shape). It's like making a custom wrap for it. Don't forget to count the bottom too!

There will be six faces, like a die. Attached below, I have drawn all 6 faces and labeled them with their measurements. Please refer to these as we walk through the steps.

Before we start, I want to quickly address the entire reason why the measurements for the top and bottom are different. We see that the width of the top of the green wedge and the width of the bottom of it changes from 9 cm to 6 cm. This is because the 9 cm is actually the length of a diagonal cut of the wedge if it was a rectangular prism, so it's longer than the actual width.

Okay! So, since I've already written down the measurements, let's calculate the area of the shapes shown on each face and add them:

--------------------------------------------------------------------------------------------------------------

TOP Face Area:

Yellow:

8 × 6 = 48
`cm^(2)`

Green:

9 × 6 = 54
`cm^(2)`

Top Total Surface Area:

48 + 54 = 102
`cm^(2)`

--------------------------------------------------------

BOTTOM Face Area:

Yellow:

8 × 6 = 48
`cm^(2)`

Green:

6 × 6 = 36
`cm^(2)`

Bottom Total Surface Area:

48 + 36 = 84
`cm^(2)`

--------------------------------------------------------

FRONT Face Area:

Yellow:

8 × 7 = 56
`cm^(2)`

Green:

(area of a triangle is
`(bh)/(2)` b=6 and h=7. Remember that the 9 is not the height.)

7 × 6 = 42
`cm^(2)`

↓

42 ÷ 2 = 21
`cm^(2)`

Front Total Surface Area:

56 + 21 = 77
`cm^(2)`

--------------------------------------------------------

As you can see from the pictures, the front and back faces have the same area, so we'll just double this calculation once we add to get the surface area.

--------------------------------------------------------

LEFT FACE Face Area:

Yellow:

6 × 7 = 42
`cm^(2)`

Left Face Total Surface Area:

Only the yellow portion is shown on the left face, so the total for the left face will be 42
`cm^(2)`

--------------------------------------------------------

RIGHT FACE Face Area:

Green:

6 × 9 = 54
`cm^(2)`

Right Face Total Surface Area:

Only the green portion is shown on the right face, so the total for the right face will be 54
`cm^(2)`

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Time for the grand total of all 6 faces to find the surface area of the entire figure:

Top - 102

Bottom - 84

Front - 77

Back - 77

Left Face - 42

Right Face - 54

↓

102 + 84 + 77 + 77 + 42 + 54 = 436
`cm^(2)`

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That's all! If there are any inconsistencies in my answer or I've made a mistake, please let me know. Again, feel free to comment if you need any clarification! I hope this helps :)