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find the surface area that a best octagonal pyramid v 0 votes
height 2.5 yards, and its base has apothem length 1.5 yards
pyramid-surfaceareaaskedJun 14, 2013in ALGEBRA 2by futaiScholarplease log in or it is registered to include a comment. you re welcome log in or register to answer this question. 1 price 0 votes
The pyramid consists of an octagonal base and also eight isosceles triangles.
You are watching: Surface area of a octagonal pyramid
Let the center of the basic be A. Any type of apothem will meet a side of the octagon in ~ the midpoint of the side.
Let that midpoint it is in B. Permit one end of the side be C.
Then alphabet is a right-angled triangle. It is just one of 16 comparable triangles making up the octagon.
The edge CAB = 2pi/16 = pi/8
octagonal pyramid apothem size is ab = 1.5 yards
So CB = 1.5 tan (pi/8)
Apply half-angle formula tan (a/2) = (1-cos a)/sin a
tan (pi/8) = <1-cos(pi/4)>/sin (pi/4)
= <1- (1/√2)/(1/√2)
=√2 -1
Substitute tan (pi/8) = √2 -1
CB = 1.5 (√2 -1)
So base area = 16 x 1.5 x 1.5(√2 - 1) x (1/2)
= 18(√2 - 1)
= 7.456 sq. Yds. (approx.)
Now, permit the optimal of the pyramid be D.
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octagonal pyramid with height 2.5 yards
BD = √<(2.5)^2+(1.5)^2>
= √(8.5)
The full area that the sloping deals with of the pyramid = 8 x √(8.5) x (1.5) x (√2 - 1)
= 6 √34 (√2 - 1)
= 14.492 sq. Yds. (approx.)
If you are counting the base, total area the pyramid = 18(√2 - 1)+ 6 √34 (√2 - 1)
= (18 + 6 √34)(√2 - 1)
= 6(3 +√34 )(√2 - 1)
= 21.948 sq. Yds. (approx.).
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