Approximate the volume of the regular triangular pyramid if the height is approximately 11.5. (Remember that the base of a regular triangular pyramid must be an equilateral triangle, not necessarily congruent to the sides of the pyramid.)
828
239
717

- BeccaB003

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- BeccaB003

V = 1/3 (base area)(height)
V = 1/3 (base area)(12)
I need help finding the base area

##### 1 Attachment

- taramgrant0543664

Since the base is an equilateral triangle you can solve for the base area by using the formula A=(bxh)/2

- BeccaB003

So, A = (12*12)/2 = 72

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## More answers

- BeccaB003

V = 1/3 (72)(12) = 288 <------------ But that isn't the correct answer. I got that answer last time I did this problem and it wasn't correct. I was told that I needed to use Hero's formula but I don't know how or why?!

- NathalyN

SA: 1/2bh(number of sides)
SA: 1/2(12)(12)(4)
SA: 1/2(144)(4)
SA: 72*4
SA: 288

- BeccaB003

That solves for the Surface area not the volume. @NathalyN

- BeccaB003

Hero's formula is needed to solve this problem but I don't know how.

- BeccaB003

@jim_thompson5910 Can you please help solve this problem?

- taramgrant0543664

Here is a link to how to use that formula:
https://www.mathsisfun.com/geometry/herons-formula.html
I've never heard of this before but I hope it helps

- jim_thompson5910

`Remember that the base of a regular triangular pyramid must be an equilateral triangle`
the length of the bottom side is 12, so the other 2 sides must also be 12 too
plug s = 12 into the area of an equilateral triangle formula
\[\Large A = \frac{\sqrt{3}}{4}*s^2\]

- BeccaB003

Okay so,
\[\frac{ \sqrt{3} }{ 4 }*12^2\]
= 62.4

- jim_thompson5910

62.3538290724796 which turns into 62.4 if you round to 1 decimal place
looks good

- jim_thompson5910

that's the area of the base

- jim_thompson5910

Volume of pyramid = (1/3)*(area of base)*(height of pyramid)

- BeccaB003

So, (1/3) * 62.4 * 12 = 249.6 <--------- that isn't one of the answer options. It is closest to 239. Also, how did you find that formula for an equalateral triangle. Also, can I do this problem using Hero's formula because that is what I was told to use to get the correct answer. @jim_thompson5910

- jim_thompson5910

You can use hero's formula, but you'd get the same result if you used the formula I posted above
this page shows how the formula is derived
http://mathcentral.uregina.ca/QQ/database/QQ.09.03/jared1.html

- jim_thompson5910

also why did you use 12 for the height, when the height is 11.5?

- BeccaB003

Okay, thank you. Sorry, I didn't mean to use 12. I recalculated it and it and got 239. Thank you so much for your help! :)

- BeccaB003

Do you mind helping me with one more problem using the same image?
Calculate the surface area of a regular triangular pyramid with the base edges of length 12 and a slant height of 12. (Remember that the base of a regular triangular pyramid must be an equilateral triangle, not necessarily congruent to the sides of the pyramid.)

- BeccaB003

SA = Base area + 1/2 bln

- BeccaB003

SA = 62.4 + 1/2 (12)(12)(4)

- jim_thompson5910

you have one base face
and 3 lateral faces

- jim_thompson5910

we already have the area of the base face from last time
the area of each face is (1/2)*b*h
b = base of lateral face triangle = 12
h = height of lateral face triangle = 12

- jim_thompson5910

the area of each lateral face is (1/2)*b*h
b = base of lateral face triangle = 12
h = height of lateral face triangle = 12

- BeccaB003

Wait, I thought it had 4 sides not including the base. It is really hard to tell from the picture... So, it is SA = 62.4 + (1/2)(12)(12)(3)

- BeccaB003

So the answer is 278.4?

- jim_thompson5910

Rule for a pyramid (any pyramid)
there is always one base face (usually on the ground)
the lateral faces are triangles and there are n of them where the base is an n-gon (example: n = 4, square pyramid has 4 lateral faces)

- jim_thompson5910

yeah approximately

- BeccaB003

Okay, so if it doesn't say that it is a square pyramid then it has 3 lateral faces?

- jim_thompson5910

the key word "triangular" in "triangular pyramid" means the base is a triangle

- jim_thompson5910

if they said "rectangular pyramid", then the base is a rectangle (4 lateral sides)

- jim_thompson5910

if they said "pentagonal pyramid" then the base is a pentagon with 5 lateral sides

- BeccaB003

Oh! I see. I was really confused because by the picture it looks like there are four lateral sides. Thank you so much for your help! You clear everything up brilliantly! :D

- jim_thompson5910

yeah it's hard to convey 3D shapes on a flat 2D page

- jim_thompson5910

if you had these (maybe from a kit or something)
http://bit-player.org/wp-content/uploads/2012/10/tetrahedron-geomags-06601.jpg
then it might be easier to visualize

- BeccaB003

I do have something similar to those. Thanks!

- jim_thompson5910

no problem

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