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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
(A) 12√31
(B) 12√109
(C) 24√31
(D) 24√109
(E) 48√31
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?
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
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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pyramid.gif [ 3.24 KiB | Viewed 9040 times ]
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.
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.
