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A pyramid with four equal-sized flat surfaces and a base of 36 feet^2

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exploringm
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A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   16 Feb 2013, 13:30
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A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 has a height of 10 feet. What is the total surface area of the pyramid, excluding the base?

(A) 12√31
(B) 12√109
(C) 24√31
(D) 24√109
(E) 48√31

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the book's answer description sort of says it's a right triangle without really explaining how we know that..can anyone help out with understanding that as well?
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   23 Jun 2013, 20:07
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From Figure,
Apex to base be height 'h' =10
we know base b*b =36 => b=6

S = (h^2+ (b/2)^2)^1/2 = (10^2+3^2)^1/2 =109 ^0.5
SO area of the flat surface with height(S) = 109^0.5 * 3
Area of four flat surfaces = 4*3*109^0.5 =12*109^0.5
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   16 Feb 2013, 16:52
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You are very welcome! I looked at this too quickly though. Ignore the above, see below.

So the base is 36. Therefore the area of the square is 36. So if we call each side of the square "S" then S*S = 36 and S = 6. Now we need to find the area of one of the triangles and then multiply that area by 4 and we will have the surface area of the pyramid (without counting the base). So drop the height from the the top of the pyramid straight down to the base and draw a line there connecting to the side of the square (this will be right at the midpoint of the side of the square). That is a right triangle and you can use Pythagorean theorem to find the third side which is the height of our triangle faces. So 10^2 + 3^2 = C^2 109 = C^2 C=109^1/2.

Now find the area: BH/2 6*109^1/2/2 = 3*109^1/2. And multiply that area by 4 because there are four equal triangular faces. 12*109^1/2

Let me know if you have any questions. I know that it can be difficult to visualize without a diagram.

HG.
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   16 Feb 2013, 15:32
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Hi there,

If it is a pyramid with four equal faces then the bases of all the faces are equal and therefore the base of the pyramid is a square. Because the base is a square it has four 90 degree angles. Because the triangles are equal, each one the the squares right angles is being cut in half (bisected) so that the triangles are each 45-45-90 or right isosceles. You can also look at it this way: the four angles at the top the pyramid added together mus equal 360 degrees. The 4 triangles are equal therefore the angles are equal and 360/4 is 90.

Let me know if you need more advice on this. Happy studies!

HG.
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   16 Feb 2013, 16:07
HerrGrau wrote:
Hi there,

If it is a pyramid with four equal faces then the bases of all the faces are equal and therefore the base of the pyramid is a square. Because the base is a square it has four 90 degree angles. Because the triangles are equal, each one the the squares right angles is being cut in half (bisected) so that the triangles are each 45-45-90 or right isosceles. You can also look at it this way: the four angles at the top the pyramid added together mus equal 360 degrees. The 4 triangles are equal therefore the angles are equal and 360/4 is 90.

Let me know if you need more advice on this. Happy studies!

HG.


Thanks as always. Could you also show the steps to solve for the answer?? Geometry really isn't my thing. :/
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   18 May 2013, 19:49
HerrGrau wrote:
You are very welcome! I looked at this too quickly though. Ignore the above, see below.

So the base is 36. Therefore the area of the square is 36. So if we call each side of the square "S" then S*S = 36 and S = 6. Now we need to find the area of one of the triangles and then multiply that area by 4 and we will have the surface area of the pyramid (without counting the base). So drop the height from the the top of the pyramid straight down to the base and draw a line there connecting to the side of the square (this will be right at the midpoint of the side of the square). That is a right triangle and you can use Pythagorean theorem to find the third side which is the height of our triangle faces. So 10^2 + 3^2 = C^2 109 = C^2 C=109^1/2.

Now find the area: BH/2 6*109^1/2/2 = 3*109^1/2. And multiply that area by 4 because there are four equal triangular faces. 12*109^1/2

Let me know if you have any questions. I know that it can be difficult to visualize without a diagram.

HG.


Okay let's see if I understand this. The reason for applying the Pythagorean Theorem is because the sides are at an angle. Thus the 10 ft. height of the pyramid is not the height of the sides?
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   23 Jun 2013, 08:15
can someone please explain the solution? I am unable to visualize this problem.
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   23 Jun 2013, 15:38
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So this is what I get from this question.

1 Square with 36 sq feet or 6' x 6' (doesn't count because question stem excludes it)
4 Right Triangles with a height of 10 when they are connected by all sides and base.

The question tries to trick you because when the 4 triangles are tilted and connected the height is 10, but the real height is the hypotenuse. Think about when you see a pyramid from the side view.

The height from the apex (top of the pyramid) straight down would be 10. We're trying to find the hypotenuse.
So use the pyth theorem. a^2 + b^2 = c^2 or 10^2 + 3^2 (half of the base because we're looking for hyp not area)=sq 109

Now we have real height of the triangle. This is the real height of the face of the triangle because it's laid down on a slant (I'm repetitive but it's important)

From here we can just do find the area. 1/2(b)(h)= .5(6)(sq109) = 3 sq(109)
Since there are 4 triangles, 4 x 3 sq(109) = 12 sq (109)
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   24 Jun 2013, 04:47
thank you james & butwhy
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   30 Mar 2015, 12:30
I am trying to understand why during Pythagoras theorem we used 3 and again in area we used base as 6, please help.
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   09 Jun 2018, 07:19
exploringm wrote:
A pyramid with four equal-sized flat surfaces and a base of 36 ft2 has a height of 10 feet. What is the total surface area of the pyramid, excluding the base?

12 31^1/2
12 109^1/2
24 31^1/2
24 109^1/2
48 31^1/2

the book's answer description sort of says it's a right triangle without really explaining how we know that..can anyone help out with understanding that as well?


Surface area of symmetrical triangular pyramid (SA)=0.5*circumference*l (where l is slant height)
l=(10^2+3^2)^1/2 = (109)^1/2
so SA=0.5*(4*6)*(109)^1/2
SA=12(109)^1/2
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   12 Sep 2019, 22:02
Couldn't solve it the first time, so I walked away and came back and solved it.

Here it is.
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Re: A pyramid with four equal-sized flat surfaces and a base of 36 feet^2 [#permalink] New post   02 Feb 2020, 15:20
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exploringm wrote:
A pyramid with four equal-sized flat surfaces and a base of 36 ft2 has a height of 10 feet. What is the total surface area of the pyramid, excluding the base?

12 31^1/2
12 109^1/2
24 31^1/2
24 109^1/2
48 31^1/2

the book's answer description sort of says it's a right triangle without really explaining how we know that..can anyone help out with understanding that as well?


The total surface area of the pyramid, excluding the base, is also known as the lateral surface area of the pyramid. It is the total area of the 4 triangular faces of the pyramid.

Since the base of the pyramid is 36 ft^2, a side of the base is 6 ft. Recall that the height of the pyramid, half of a side of the base and the slant height of the pyramid form the three sides of a right triangle where the slant height is the hypotenuse. Since the height of the pyramid is 10 ft, half of a side of the base is 3 ft, so the slant height, s, can be determined by the equation:

10^2 + 3^2 = s^2

109 = s^2

s = 109^½

Recall that the slant height of the pyramid is also the height of the triangular faces of the pyramid, and a side of the base of the pyramid is the base of the triangular faces. So a triangular face has height = 109^½ and base = 6 and hence an area of 1/2 x 6 x 109^½ = 3 x 109^½. Since there are 4 triangular faces, the total area is 4 x 3 x 109^½ = 12 x 109^½.

Answer: B
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