anonymous
  • anonymous
The lateral area of the right regular triangular prism is 18cm^2. Find the total surface area.Give answer in decimal form rounded of to 2 decimal places.
Geometry
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
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anonymous
  • anonymous
im looking for the area of the rectangle face and triangle base
terenzreignz
  • terenzreignz
This shouldn't be too hard :) Let's consider just one rectangular face... |dw:1377800484158:dw|

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terenzreignz
  • terenzreignz
You with me?
anonymous
  • anonymous
yes
terenzreignz
  • terenzreignz
Now, it'd be much nicer if we knew how long this is:|dw:1377800558650:dw|
terenzreignz
  • terenzreignz
Which we do, actually... we know that the area is 18...
terenzreignz
  • terenzreignz
And the area is equal to the product of the two side-lengths...3 and an unknown number... So... three times a number is 18, what is that number?
anonymous
  • anonymous
6
terenzreignz
  • terenzreignz
That's right :)|dw:1377800697260:dw|
anonymous
  • anonymous
how come looking at the picture 3cm looks alot longer then the 6cm side.
terenzreignz
  • terenzreignz
These things are rarely drawn to scale.
anonymous
  • anonymous
?
terenzreignz
  • terenzreignz
If it bothers you that much, I'll draw a slightly more accurate figure :P|dw:1377800780026:dw|
anonymous
  • anonymous
ohhh i see
terenzreignz
  • terenzreignz
Okay, that was the easy part, though... ready for the actual 'finding the surface area'?
anonymous
  • anonymous
yes
terenzreignz
  • terenzreignz
As you can see, there are three-rectangles around the sides, each with area 18.. so that makes how many total?
anonymous
  • anonymous
??? how many what?
terenzreignz
  • terenzreignz
total area around the three rectangles...
anonymous
  • anonymous
54 cm^2
terenzreignz
  • terenzreignz
That's right, and that's the total area around the three rectangles, now find the area of one triangular base.
anonymous
  • anonymous
okay one sec
anonymous
  • anonymous
gonna try it right now
terenzreignz
  • terenzreignz
Area of an equilateral triangle given a side-length s is \[\Large A_\Delta=\frac{s^2\sqrt3}{4}\]
anonymous
  • anonymous
idk i got about 7.8
anonymous
  • anonymous
oh ill use the formula u gave me
terenzreignz
  • terenzreignz
Yes, please do :P
anonymous
  • anonymous
wait can u draw that formula on the triangle so i can see it? i dont really get it
terenzreignz
  • terenzreignz
Well, when you have an equilateral triangle, you know all sides measure the same, right? So just square the side-length and mutltiply to \(\LARGE \frac{\sqrt3}{4}\)
anonymous
  • anonymous
Howd u get|dw:1377801669518:dw|
terenzreignz
  • terenzreignz
You want to derive formulas? :/ Okay.... but it can't be done without trigonometry... a fair warning XD Are you sure you want to derive it?
anonymous
  • anonymous
oh so thats the formula to find the area of an equilateral triangle?
anonymous
  • anonymous
I was trying to find the area of the right triangle first.
anonymous
  • anonymous
by splitting the triangle in half
terenzreignz
  • terenzreignz
yes... you know 30-60-90 triangles?
anonymous
  • anonymous
yea
terenzreignz
  • terenzreignz
Okay, let's have an arbitrary equilateral triangle...|dw:1377802040102:dw|
terenzreignz
  • terenzreignz
with side-length equal to s.
terenzreignz
  • terenzreignz
Now, the area of a triangle is always equal to \[\Large \frac{bh}2\]
terenzreignz
  • terenzreignz
Its base is here...|dw:1377802092050:dw| Which is equal to s, since this is just one side of the triangle... right?
anonymous
  • anonymous
yes
anonymous
  • anonymous
but isnt the base 6 since its an equilateral triangle?
terenzreignz
  • terenzreignz
We're just deriving the formula. This is the general case :)
terenzreignz
  • terenzreignz
So now, we need the height:|dw:1377802208651:dw|
terenzreignz
  • terenzreignz
We need the height h in terms of s. So, what we do, we remember that this angle measures 60 degrees (due to it being an equilateral triangle)|dw:1377802273303:dw|
anonymous
  • anonymous
ok
anonymous
  • anonymous
and the other angle is 30 degrees?
terenzreignz
  • terenzreignz
Of course :) So this side measures s/2, by the properties of the 30-60-90 triangle...|dw:1377802388850:dw|
terenzreignz
  • terenzreignz
And furthermore, by the properties of the 30-60-90 triangle, this side (h) measures|dw:1377802425279:dw|
terenzreignz
  • terenzreignz
So it turns out, \[\Large b= s\]\[\Large h = \frac{s\sqrt3}{2}\] Therefore, the area \[\Large A = \frac{bh}2= \frac{s^2\sqrt3}4\]
terenzreignz
  • terenzreignz
Understood?
anonymous
  • anonymous
but isnt the height just \[\sqrt{3}\]
terenzreignz
  • terenzreignz
That's if the side-length of the triangle is 2.
anonymous
  • anonymous
im sorry|dw:1377802649031:dw|
terenzreignz
  • terenzreignz
|dw:1377802675764:dw|
anonymous
  • anonymous
why over 2?
terenzreignz
  • terenzreignz
Because, in a 30-60-90 triangle, the longer leg is equal to the shorter leg times \(\large \sqrt3\) Now, in our case, the shorter leg measures \(\Large \frac{s}2\) (not s!)|dw:1377802763415:dw|
terenzreignz
  • terenzreignz
Understood?
anonymous
  • anonymous
|dw:1377802835960:dw| can u use this also.
terenzreignz
  • terenzreignz
No... you got it backwards...|dw:1377802980320:dw|
anonymous
  • anonymous
thats what i meant ^ is it the same as the other way of looking at it?
terenzreignz
  • terenzreignz
No, it's very different... |dw:1377803116229:dw| in fact, that's wrong^
anonymous
  • anonymous
not the way i wrote it im
anonymous
  • anonymous
|dw:1377803208015:dw|
anonymous
  • anonymous
is that drawing^ technically the same as this one|dw:1377803274662:dw|
terenzreignz
  • terenzreignz
Yes... just multiplied everything by 2.
anonymous
  • anonymous
so how do we use this formula in my problem?
terenzreignz
  • terenzreignz
To find the area of one of the triangle faces... one of the sides is 6, just apply it to the formula \[\Large A - \frac{s^2\sqrt3}4\]
terenzreignz
  • terenzreignz
\[\Large A \color{red}=\frac{s^2\sqrt3}4\] sorry
anonymous
  • anonymous
so about 15.588?
anonymous
  • anonymous
15.588 cm^2
terenzreignz
  • terenzreignz
Yeah, but there are two triangle faces, so double that.
anonymous
  • anonymous
about 31.176 cm^2 is the area od the two triangles.
anonymous
  • anonymous
*of
terenzreignz
  • terenzreignz
Okay, and add that to the area of the three rectangles, and that's the total surface area.
anonymous
  • anonymous
So the surface area is about 85.18 cm^2
terenzreignz
  • terenzreignz
Seems about right.
anonymous
  • anonymous
ALRIGHT! Thanks!

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