Answer:
Volume of the given pyramid = 1122.37 cubic feet
Step-by-step explanation:
Volume of the regular hexagonal pyramid = [tex]\frac{1}{3}(\text{Area of the base})(\text{Base})[/tex]
Measure of internal angle of a polygon = [tex]\frac{(n-2)\times 180}{n}[/tex]
Here, n = number of sides of the polygon
For a hexagon, n = 6
Measure of interior ∠C = [tex]\frac{(6-2)\times 180}{6}[/tex]
= 120°
Measure of ∠BCD = [tex]\frac{120}{2}[/tex]
= 60°
By applying tangent rule in ΔCED,
tan(∠ECD) = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
tan(60°) = [tex]\frac{DE}{CE}[/tex]
[tex]\sqrt{3}=\frac{DE}{6}[/tex]
DE = [tex]6\sqrt{3}[/tex] feet
And cos(60°) = [tex]\frac{EC}{CD}[/tex]
[tex]\frac{1}{2}=\frac{6}{CD}[/tex]
CD = 12 feet
Area of ΔBCD = [tex]\frac{1}{2}(\text{Base})(\text{Height})[/tex]
= [tex]\frac{1}{2}(6\sqrt{3})(12)[/tex]
= [tex]36\sqrt{3}[/tex] feet
Area of hexagonal Base of the pyramid = [tex]6(36\sqrt{3})[/tex]
= 216√3 square feet
Since, lateral height of the pyramid (AC) = 15 feet
By applying Pythagoras theorem in ΔADC,
AC² = AD² + CD²
(15)² = AD² + (12)²
AD = [tex]\sqrt{225-144}[/tex]
AD = 9 feet
Volume of the given pyramid = [tex]\frac{1}{3}(216\sqrt{3})(9)[/tex]
= 648√3 cubic feet
= 1122.37 cubic feet
