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Find the volume of a regular hexagonal pyramid with side length $15$ meters and a lateral edge of $17$ meters.

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Volume of cubes and rectangular prisms: word problems

Full solution

Q. Find the volume of a regular hexagonal pyramid with side length

15

meters and a lateral edge of

17

meters.

Calculate Hexagonal Base Area: Step $1$ : Calculate the area of the hexagonal base. $\newline$ The formula for the area of a regular hexagon is $\text{Area} = \frac{3\sqrt{3}}{2} s^2$ , where $s$ is the side length. $\newline$ Area = $\frac{3\sqrt{3}}{2} \times 15^2$ $\newline$ Area = $\frac{3\sqrt{3}}{2} \times 225$ $\newline$ Area = $337.5\sqrt{3}$ square meters.
Calculate Pyramid Height: Step $2$ : Calculate the height of the pyramid. $\newline$ Using the Pythagorean theorem in the triangle formed by the height of the pyramid, half the side length of the base, and the lateral edge. $\newline$ Let $h$ be the height of the pyramid, $a = \frac{15\sqrt{3}}{2}$ (half the side length of the base times $\sqrt{3}$ ). $\newline$ $17^2 = h^2 + \left(\frac{15\sqrt{3}}{2}\right)^2$ $\newline$ $289 = h^2 + \left(\frac{15\sqrt{3}}{2}\right)^2$ $\newline$ $289 = h^2 + 97.5$ $\newline$ $h^2 = 191.5$ $\newline$ $h = \sqrt{191.5}$ $\newline$ $h \approx 13.84$ meters.
Calculate Pyramid Volume: Step $3$ : Calculate the volume of the pyramid. $\newline$ The volume $V$ of a pyramid is given by $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$ . $\newline$ Volume = $\frac{1}{3} \times 337.5\sqrt{3} \times 13.84$ $\newline$ Volume = $1554.6\sqrt{3}$ cubic meters.

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