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Question

A regular octagon is inscribed in a circle of radius 15.8 cm. Find the perimeter of the octagon?

Answer 1

16*15.8*cos(67.5)
= (approx) 96.74237

Divide the octagon into 8 isosceles triangle with 67.5 deg for each base angle.

Each side of the octagon is the "base" of the triangle, and half the base can be found using cosine:

cos (67.5 deg) = (1/2*b)/15.8

Find 8*b because there are 8 sides to the octagon.

Answer 2

I come up with a perimiter of 96.064 cm approximately.

You have to consider each side of the octagon as a side of a triangle to which the other two sides are formed by the radius line of the circle. This means that one side of the octagon, which we will call 'd', is the base of an equilateral triangle which the other two sides, which we will call 'e' and 'f' (and e=f), measure 15.8 as they are two radius of the circle which converge at the point O, of the center of the circle.

Then to calculate the angles of this triangle you first have to calculate the angle that is formed by the two radius lines converging at O, let's call this angle A, and the other two at the base can be B and C. A is formed by 1/8 of the complete angle of the circle 360degrees, so A=360/8=45. (Now you know that all the angles of a triangle always add up to 180 degrees. So A+B+C=180, so B+C=180-A, and A=45, so B+C= 135, but seeing as we know this is an equilateral triangle we also know that the 2 angles at hte base are equal, so B=C, which means that B=C=135/2=67.5. )

Then imagine a line perpendicular to the line d (the octagon side base) which crosses through the center of the circle at O, dividing your equilateral triangle into two right angle triangles, dividing 'd' in two parts, 'g' and 'h', where g=h=d/2. This means that the angle at the peak of the triangle is halved within the right angle triangle. This new angles which will be called A1=A/2 and A=45, so A1=45/2=22.5.

Once you know this angle it is a simple case of applying trigonometry. You are looking for the measurement of the opposite in this equilateral triangle, which is g or h. And you know the measurement of the hypothenuse e or f, to be 15.8.
And you know that in trigonometry
opposite
sin= -------------------
hypothenuse

so here

g
sinA1=---------
e

and g is your unknown so

g=sinA1xe

and A1=22.5 and e=15.8

so

g=sin22.5x15.8

From here its a simple case of using your calculator to calculate the the sine of 22.5, then solving for g.

So here sin22.5=0.38 (rounded off)

so g=0.38x15.8

g=6.004

Then since g is half of an octagon side because g=d/2
solve:

d=gx2

d=6.004x2

d=12.008

and then dx8 to obtain the perimiter of the octagon p:

p=dx8

p=12.008x8

p=96.064cm.

Answer 3

Join vertices of octagon to centre to form 8 triangles.
Angles at centre are then 45°
Sides of each triangle are 15.8 ,15.8 and x cm, say.
By cosine rule:-
x² = 15.8² + 15.8² - (2 x 15.8 x 15.8 x cos 45°)
x² = 146.24
x = 12 . 09 cm
Perimeter = 8 x 12.09 cm
Perimeter = 96.7 cm

Answer 4

Make it a hexagon and I will tell you.