## Volume & Surface Area: 2 cong. reg. hex. pyramids

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jordan
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### Volume & Surface Area: 2 cong. reg. hex. pyramids

A decorative ornament is made of solid wood. It is comprised of two congruent regular hexagonal pyramids that share the same base. The perimeter of the hexagon is 72 centimeters. The slant height of each of the faces if 16 centimeters. The apothem is 10.4, and the height of one of the pyramids is 12.2.

a) What is the volume of the ornament to the nearest 10th?

b) What is the surface area of the ornament?

stapel_eliz
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### Re: Volume & Surface Area: 2 cong. reg. hex. pyramids

jordan wrote:A decorative ornament is made of solid wood. It is comprised of two congruent regular hexagonal pyramids that share the same base. The perimeter of the hexagon is 72 centimeters. The slant height of each of the faces if 16 centimeters. The apothem is 10.4, and the height of one of the pyramids is 12.2.

a) What is the volume of the ornament to the nearest 10th?

The volume V of a pyramid with base area B and height h is given by:

. . . . $V\, =\, \frac{1}{3}Bh$

The the area A of an n-sided regular polygon with apothem a is given by:

. . . . $A\, =\, na^2 \tan\left(\frac{\pi}{n}\right)$

Plug the numbers they gave you (namely, a = 10.4 and n = 6) into the second formula above to find the area of the base of the pyramids. Then plug this value in for B" in the first formula above, along with the other value they gave you (namely, h = 12.2) to find the volume of one of the pyramids.

To complete the solution, multiply by "2".

jordan wrote:b) What is the surface area of the ornament?

The surface area SA of a pyramid whose base is an n-sided polygon with perimeter P and having a slant-height s (and ignoring the shared "base" area) is given by:

. . . . $SA\, =\, \frac{1}{2}Ps$

Plug the numbers they gave you (namely, P = 72 and s = 16) into the formula to find the surface area of one of the pyramids (and thus half the total surface area). To complete the solution, multiply by "2".

If you get stuck, please reply showing how far you have gotten. Thank you!