Norbert W. answered • 07/25/16
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The regular heptagon inscribed in a circle can be represented by 7 isosceles triangles whose equal sides are radii of the circle r and the third side s.
The angle at the center of the between the equal sides of each triangle is 2π/7.
Using the law of cosines, s2 = r2 + r2 - 2r2cos(2π/7)
= 2r2 - 2r2cos(2π/7)
= 2r2(1 - cos(2π/7))
Also cos(2π/7) = 1 - 2sin2(π/7), from the double angle for cosines
Substituting this into the equation above:
s2 = 2r2(1 - cos(2π/7))
= 2r2(1 - 1 + 2sin2(π/7))
= 4r2sin2(π/7)
- (2rsin(π/7))2
Then s = 2r * sin(π/7)
This is the size of the side of the heptagon and its
perimeter is P = 7s = 14r * sin(π/7)
π/7 = (π/7) * (180/π) = 180/7 ≈ 25.714°
With this angle and the radius being 7, plug these numbers into the
formula for the perimeter and you have your answer.
Nicolas M.
But, I will try to guide you as best as possible for me.
Because a rectangular heptagon is inscribed in a circle, we have 7 inner triangles like shown here:
/\
/ | \
r / | \ r
/ | h \
/ | \
--------------
a
Where "r" is the radius of the circle, and "h" is the triangle's height
The base of this triangle "a" is the length of one side of the heptagon (in total, we have seven sides of length "a")
The upper angle of this triangle is: 51.4° (= 360°/7)
Because the triangle's height divide this triangle in two rectangular triangles, we use the concept of sin(θ)
sin(θ) = (a/2)/r But θ = 51.4°/2 = 25.7° Then, a = 2*r*sin(25.7°)
a = 2*7*sin(25.7°)
a = 6.07
Because you have a total of 7 sides of length "a" (heptagon), the perimeter of this regular heptagon is given by:
Rectangular Heptagon Perimeter = 7*a = 7*6.07 ∼ 42.49
07/25/16