A tabletop has the shape of a regular decagon with a radius of 9.5 inches. What is the area of the tabletop to the nearest square inch?
To find the area of the tabletop, we need to know the formula for the area of a regular decagon. A regular decagon is a polygon with ten equal sides and ten equal angles.
The formula for the area of a regular decagon is given by:
Area = (5/4) * s^2 * cot(π/10),
where s is the length of each side of the decagon. In this case, since we know the radius of the tabletop, we need to find the length of each side.
The radius of the tabletop is the distance from the center to any corner of the decagon. We can find the length of each side by using trigonometry.
Since a regular decagon has ten equal angles, each angle at the center of the decagon is 360/10 = 36 degrees.
By drawing two radii connecting the center of the decagon to two consecutive corners, we form an isosceles triangle where the angle at the center is 36 degrees and the base angles are equal.
To find the length of each side of the decagon, we can use the Law of Cosines. Considering one of the side lengths as c, with the other two side lengths as a and b, and the angle opposite side c as C, we have:
c^2 = a^2 + b^2 - 2ab*cos(C).
In our case, the side length c is the radius of the tabletop (given as 9.5 inches), and the angle C is half of the base angle (18 degrees).
Let's calculate the length of each side:
c^2 = a^2 + b^2 - 2ab*cos(C),
(9.5)^2 = a^2 + b^2 - 2ab*cos(18).
We have two unknowns, a and b, but we know that they are equal since the decagon is regular. So, we can simplify the equation to:
(9.5)^2 = 2a^2 - 2a^2*cos(18),
90.25 = 2a^2(1 - cos(18)),
a^2 = 45.125 / (1 - cos(18)),
a ≈ √(45.125 / (1 - cos(18))).
Now, we can substitute the value of a into the formula for the area of a regular decagon:
Area = (5/4) * s^2 * cot(π/10),
Area ≈ (5/4) * (√45.125 / (1 - cos(18)))^2 * cot(π/10).
Using a calculator, we can now find the approximate value of the area of the tabletop to the nearest square inch.