Please help me find the height of this pyramid?

Question

I FOUND THE FOLLOWING QUESTION IN YAHOO!ANSWER THAT POSTED BY "NHI D" RECEIVED 3 DIFFERENT ANSWER. PLS HELP. I QUITE CURIOUS IF THERE IS ANY CORRECT ANSWER FROM THE 3.

Please help me find the height of this pyramid?
The base of the pyramid is an equilateral triangle. Each side of the base is 6.70 cm long.
The slant height of the pyramid is 3.35 cm (half of the base)
I need to find the volume of this pyramid. Please help. Thanks so much. (IF YOU CAN, HELP TO STAR THIS QUESTION, SO THAT I GET SOME POINTS BACK. TQ)

ok, now got 3 same answers. but see what 'friends' said, this measurement cannot form a pyramid. ok, i need the original asker "NHI D" back to answer the question from 'friends'. Anyway, thanks ! I will choose one for best answer after 4 hours the question been posted.

tricky tricky, the more answer i get, the more confused i am ! Anyway, just give the 10 points for the long winded answer. Sorry, the rest, i really don't know what is the correct answer. If next time got chance, will give the rest 10points in other answer. Cheers !!!

Answer 1

Hi,

If the base of the pyramid is an equilateral triangle with a side of 6.70 cm and you drop an altitude down from one vertex, it forms 2 30-60-90 degree triangles. Since the side across from 30° is half of a side, its length is 3.35 cm. The third side or altitude is the side across from the 60° angle and is 3.35√3 or 5.80237 cm. So the 30-60-90° sides are 3.35, 5.80237, and 6.7 respectively.

If we drew all 3 altitudes in our 6.7 cm equilateral triangle, they would split the equilateral triangle into 6 tiny congruent 30-60-90° triangles. Since each 60° angle in the original triangle was bisected, those little angles are now 30°. The altitudes make right angles with the sides of the triangle and the 6 adjacent angles where the altitudes all intersect are all 360°/6 or 60°. In this little 30-60-90° triangle we know the side across from the 60° angle is 3.35 cm, half the length of the original triangle's length. Since this is in a 30-60-90° triangle, we could say the 30-60-90° sides of this little triangle are x, 3.35, and y respectively. Since this triangle is similar to the triangle we formed by dropping one altitude in the original triangle, we can set up a proportion to solve for x:

3.35..............x
------------ = -------
5.80237.....3.35

This is 30° side over 60° side equals 30° side over 60°.
Solving this for x gives x = 1.93412. This is the perpendicular distance from any side of the triangle to the center of the base. This is needed to find the height of the pyramid on our way to the pyramid's volume.

If the slant height of the pyramid is 3.35 cm and this distance to the center of the base is 1.93412, then the pyramid's height can be found by a² + b² = c², where a = 1.93412 and c = 3.35. Solving for the height, we get

1.93412² + b² = 3.35²
3.7408 + + b² =
2.73527 = b This is the height of the pyramid.

The formula for the volume of a pyramid is:

V = ⅓Bh, where B is the area of the base and h is the height of the pyramid. Since the base is an equilateral triangle and the formula for the area of an equilateral triangle is A=s²√(3)/4, then this triangle's area is A=6.70²√(3)/4 = 19.4379 cm². This is B in our volume equation. Since our pyramid's height is 2.73527 cm, then the volume is

V = ⅓Bh = ⅓(19.4379)(2.73527) = 17.7227 cm³

I hope that is clear and that it helps!! :-)

Answer 2

Volume of a Pyramid is equal to:

Volume = 1/3 * B * H

Where:
B = The area of the base of the Pyramid
H = The height of the Pyramid

Now you have an equilateral triangle at the base with each of the sides equal to 6.70 CM. We have to now calculate B which is the area of our triangle that is equal to:

Area of a triangle = 1/2 * Base * Height

But we do not have the height of the triangle. The height of a triangle can be calculated in this way:

Height^2 + (Side/2)^2 = Side^2

That is the formula I just found myself using the Pythagorean theorem so the height will be:

6.70^2 = 3.35^2 + Height^2
44.89 = 11.2225 + Height^2
Height^2 = 44.89 - 112225
Height^2 = 33.6675
Height = √33.6675
Height = 5.80237

Now we have the height of our base triangle so let's calculate its area:

Area of the base = 1/2 * 6.70 * 5.80237
Area of the base = 19.43794 CM
B = 19.43794 CM


Now that we have the area of the base triangle, we have to calculate the Volume of the Pyramid using the formula I just gave you:

Volume = 1/3 * B * H
Volume = 1/3 * 19.43794 * 3.35 = 21.70569 CM

We found the volume! Tada! To be exact, the volume of the Pyramid will be:

21.705699876534926726383139333764 CM

Good luck.

Answer 3

The answer I get is 21.704cm^3

I got this by taking what we have and drawing pictures immediately.

Draw an aerial view of the pyramid, then you need to find the height of the base of the pyramid. If you draw a vertical line from one of the points, you can make a right triangle and then use Pythagorean theorem.

The hypotenuse is 6.7cm, the bottom base is half this, 3.35cm, and by solving you get......

(3.35)^2+(b^2)=(6.7)^2
b^2=33.6675
b=5.802
b=height of the base of the pyramids triangle

The Volume of a pyramid equation is V=(1/3)Ah; where A is the area of the base, so......

A (base of pyramid)= (1/2)bh
(1/2)(6.7)(5.802)=19.437

Now take this and use the Volume formula.....
V=(1/3)Ah
V=(1/3)(19.437)(3.35)
>>>>>3.35 is from the given information
V=21.704cm^3

Answer 4

If I remember, V(sq. pyramid)= b^2 (h/3), where b is the length of a square. This is a lot of dog-work with English system units, but here goes: Convert density from pounds per cubic foot to pounds per cubic inch by dividing 655/1728, which is appx 0.4 lb/in^3. Now, find the max. no of cubic inches, divide 2.5 lb by 0.4 lb/in^3, to get 10 in^3 appx. Now you can find the height of the pyramid: 10 = 4 (h/3), h is about 7 inches. Note, that's actually a very skinny pyramid.

Answer 5

THE SLANT HEIGHT OF THE PYRAMID IS NOT ACTUALLY THE HEIGHT OF THE PYRAMID!!! SO 21.705 IS NOT THE CORRECT ANSWER!!!


(1/3)(Area of base * Height)
Area of base= bh/2
6.7^2 = 3.35^2 + a^2 (pythagorean theorem)
a=height of the base= 5.802 cm
Area = 6.70(5.802)/2 = 19.4367 cm

slant height = sqrt (h^2+ (1/12)a^2) formula
3.35^2 - (1/12)(height of the base/3)^2 = h^2
11.2225 - 0.3117= h^2
h = 3.3032 = height of the pyramid

Volume = (1/3)(19.4367)(3.3032)
= 21.401 cm^3

Answer 6

this seems fairly simple, so i will try and see i i mess up! :)

Firstly Area of a pyramid = Area Base * Height *1/3

we know the height, given, so we need to work out the area of base.


base is an equilateral triangle, so we know that each angle is 60% we also know that the area = 1/2 base * perp. height.

so we have a rt angle triangle with an angle of 60 and a hypotenuse of 6.7cm.

lenght of perp height = sin 60 * 6.7 = 0.866 *6.7 = 5.8022

Therefore area of base = 1/2(6.7)(5.8022) = 19.43737

Therefore area of pyramid = 19.43737 * 3.35 *1/3 = 21.705 or 21.71?!

did i miss something?

Answer 7

Hey the question is wrong. U cannot form a pyramid with those lengths.

Imagine one base 6.7 cm in length is AB. and one slant edge is 3.35 called AC. So C is the topmost point on the pyramid.
if u have a traingle ACD,where D is the midpoint of AB then length of AD and AC would be 3.35. So its and isocelles triangle and hence angle ADC and angle ACD would be same.But ADC is 90 degree . hence you cannot have a triangle with 2 90degrees. Hence the measurementsgiven are wrong.


Hope u got it.. else message me ...

Answer 8

THe formula for the volume of a pyramide is

area of the base x height/3

the formula for the base area is:

side^2 X sqrt3 / 4 so for this one it is

6.7^2 x sqrt3 /4 x 3.35 / 3 = 21.705 cubic cm