A greenhouse is a regular octagonal pyramid with a height of h=5 feet. The base has side lengths of 2 feet.
We are asked to find the volume of the greenhouse. To do this, first, we must find the area of its base. Then, we will use the formula for the volume of a pyramid.
Area of Base
Let's analyze the base of the greenhouse.
The base is a regular octagon. To find its area B we will use the formula for the area of a regular polygon.
B= 12aP
The variable a represents the apothem and P represents the perimeter of the polygon. Each base edge is 2 feet. This tells us that P=8* 2=16 feet. To find a, let's divide the base into eight congruent isosceles triangles.
Notice the following.
The angle ∠ ACB is a central angle, so m∠ ACB= 360^(∘)8=45^(∘).
Since CD is an apothem, it is an altitude of isosceles Δ ABC. This tells us that CD bisects ∠ ACB. Therefore, m∠ ACD= 12m∠ ACB=22.5^(∘).
The altitude CD bisects AB, too. Therefore AD= 12AB=1 foot.
To find a=CD, let's use the trigonometric ratios for right △ ADC.
The area of the base of the greenhouse is B=19.28 square feet, and its height is h=5 feet. Now, let's use the formula for the volume of a pyramid to find the volume of the greenhouse.
