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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 futaiScholar
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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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