A regular decagon(12 sides) in inscribed in a circle with the radius r. The decagon has an area of108 in^2. What is the radius of the cirlce?
First of all a decagon has 10 sides, not 12.
Let's split it up into 10 isosceles triangles
Each triangle will have an area of 10.8 in^2 and a central angle of 36 degrees.
Area of triangle = (1/2)(r)(r)sin36
21.6 = r^2(sin36)
r^2 = 21.6/sin36
r = 3.56
To find the radius of the circle, we can use the formula for the area of a regular decagon:
Area = (5/4) * s^2 * cot(180/10),
where 's' is the length of each side of the decagon.
Since we know that the area is 108 in^2, we can set up the equation:
108 = (5/4) * s^2 * cot(180/10).
To simplify things, let's find the value of cot(18), which is cot(180/10):
cot(18) = 1 / tan(18).
Using a calculator, we find that the tangent of 18 degrees is approximately 0.3249.
cot(18) ≈ 1 / 0.3249 ≈ 3.0777.
Now, let's substitute this value back into the equation:
108 = (5/4) * s^2 * 3.0777.
Next, we can simplify the equation:
108 = (15.3877/4) * s^2.
Multiply both sides by 4/15.3877:
(4/15.3877) * 108 = s^2.
The left side simplifies to:
27.7152 = s^2.
Finally, take the square root of both sides to solve for 's':
s ≈ √27.7152 ≈ 5.2625.
Now, we know the length of each side of the decagon. To find the radius 'r' of the circle, we can use the formula:
r = s / (2 * sin(180/10)).
Using a calculator, we find that sin(18) ≈ 0.3090.
Now, let's substitute the values into the formula:
r = 5.2625 / (2 * 0.3090).
Simplifying the equation:
r ≈ 5.2625 / 0.6180,
r ≈ 8.5092.
Therefore, the radius of the circle is approximately 8.5092 inches.
To find the radius of the circle, we need to use the formula for the area of a regular decagon:
Area = (n × s²) / (4 × tan(π/n))
Where:
- Area is the given area of the decagon (108 in²).
- n is the number of sides of the decagon (12).
- s is the length of each side of the decagon.
- π is the mathematical constant pi (approximately 3.14159).
In this case, since we want to find the radius of the circle, we need to find the length of each side of the decagon first.
A regular decagon is split into 12 congruent isosceles triangles, with each triangle having two sides of equal length. The angle between these sides in each triangle is 360°/12 = 30°. For convenience, we can bisect this angle to form a right-angled triangle.
In the right-angled triangle, the hypotenuse is the length of each side of the decagon (s), and one of the other sides is equal to the radius of the circle (r). The angle opposite the radius (r) is half of the angle between the sides of the decagon, which is 30°/2 = 15°.
Using trigonometry, we can determine the length of each side of the decagon (s) in terms of the radius (r):
sin(15°) = opposite / hypotenuse
sin(15°) = r / s
Since the length of each side of the decagon (s) is equal to 2r (twice the radius), we can substitute 2r for s in the equation:
sin(15°) = r / (2r)
sin(15°) = 1 / 2
Using a calculator or reference table, we can find that sin(15°) = 0.2588 (approximately).
So, we have:
0.2588 = 1 / 2
0.2588 × 2 = 1
0.5176 = 1
This equation is not true, which means there must be a mistake in the calculations.
However, I have noticed an error in the question. A regular decagon has 10 sides, not 12. Therefore, I will provide the correct solution based on a regular decagon with 10 sides.
We can use the same formula for the area of a regular decagon:
Area = (n × s²) / (4 × tan(π/n))
Given Area = 108 in² and n = 10:
108 = (10 × s²) / (4 × tan(π/10))
Evaluating tan(π/10) using a calculator or reference table, we find that tan(π/10) = 0.1763 (approximately).
Now we can solve for s²:
108 = (10 × s²) / (4 × 0.1763)
108 = (10 × s²) / 0.7052
Multiplying both sides of the equation by 0.7052:
108 × 0.7052 = 10 × s²
76.0456 = 10 × s²
7.60456 = s²
Taking the square root of both sides:
√(7.60456) = s
s ≈ 2.759
Since s is the length of each side of the decagon, we know that s = 2r (twice the radius of the circle). Therefore:
2.759 = 2r
Dividing both sides by 2:
r ≈ 1.3795
So, the radius of the circle is approximately 1.3795 inches.