Find the area of a six sided regular polygon inscribed inside a circle. The circles diameter is 5.?

Question

Find the area of a six sided regular polygon inscribed inside a circle. The circles diameter is 5. Show all steps.

Answer 1

The polygon is built up of 12 right-angled triangles of equal area. Each triangle has hypotenuse length 2.5 (from the centre to a vertex) and one side length 1.25 (half the length of a polygon side), so by Pythagoras the other side is 1.25 times sqrt(3). Multiplying the sides and halving, each triangle has area 25 times sqrt(3) over 32, and multiply by 12 for the polygon area 75 times sqrt(3) over 8 or approximately 16.238 square units of whatever the diameter is measured in.

Answer 2

A six-sided regular polygon can be divided up into 6 isosceles triangles (in the case of 6, of course, they are also equilateral) with their apices at the centre of the circle. Since we know that the base and sides are equal to the radius of the circumscribed circle (that can be proved, of course, by the construction of the polygon) and that the angle at the apex of each triangle is 360/6 = 60, one can find the area of the triangle from the formula Area = (base x height)/2

The height can be found by constructing a perpendicular from the base to the apex. This gives a right angled triangle with sides (in this case) of 2.5, 1.25 and h (= height of isosceles triangle) where 2.5 is the length of the hypotenuse.

So:

2.5^2 = h^2 + 1.25^2 and h^2 = 2.5^2 - 1.25^2

Therefore h = square root (4.6875) = 2.1650635094610966169093079268823

So area of each of the 6 equilateral triangles = 2.5 x 2.1650635094610966169093079268823 /2

= 2.7063293868263707711366349086029

and the total area of the polygon is 6 times this:

= 16.237976320958224626819809451618 square units

(With apologies for the significant figures - I was copying and pasting from Windows Calculator!)

The more general formula for a regular polygon of n sides inscribed in a circle of radius r, which can be derived by dividing the polygon into 2n equal right-angled triangles, is:

n * (r^2) * cos(360/(2*n)) * sin(360/(2*n))

Answer 3

The general formula to find the area of a regular polygon of n number of sides in this case is,
An = {n tan(theta(n)/2)}{R cos(theta(n)/2}^2
Here n is the number of sides of the regular polygon. accordingly An denotes the area of the corresponding regular polygon. Theta(n) indicates the angle subtended by a side of the regular polygon. R represents the radius of the circumscribing circle.
By substituting the relevant values we can get the corresponding area. For example in this case , R=2.5, Theta (n)= 60 , n=6
Then, An = {6 *tan(60/2)}*{2.5*cos(60/2}^2
=16.23797632 square units

Answer 4

The regular hexagon can be divided into 6 "congruent" equilateral triangles. Since the diameter is 5, then the radius is 2.5, and all sides (of the hexagon and the 6 triangles) measure 2.5 .

Taking one triangle, the base measures 2.5 and by the Pythagorean theorem, the altitude measures 1.25 sqrt 3. The area of 1 triangle is
A = bh/2
A = 2.5 · 1.25 sqrt 3 /2
A = 1.5625 sqrt 3

Since there are 6 "congruent" triangles, then the area of the hexagon (using the Area Addition Postulate) is:
A(hex) = 6A
A(hex) = 6(1.5625 sqrt3)
A(hex) = 9.375 sqrt3

Therefore, the area of the hexagon is 9.375 sqrt 3 square units or approximately 16.2380 square units.

^_^

Answer 5

A six sided regular polygon is known as regular hexagon.
Its side is equal to the radius of the circle, in which it is inscribed.

So, the hexagon in question has each side 5/2

And, the using formula (6*sqrt(3)/4)*s^2, we can calculate the area as

A = (6*sqrt(3)/4)*(5/2)^2
= (6*1.732/4)*(25/4)
= 150 * 1.732 / 16
= 150 * 0.10825
= 16.23750

Answer 6

Six Sided Circle

Answer 7

this tells you that from the centre to each corner of the polygon is 5. So split the polygon into 6 triangles. If you then cut one of these into two you obtain a right angled triangle of which the hyponotuse is 5. so its area is 5*cos(pi/6), in radians. now multiply this by 12 to get the area of the polygon.61.1

Answer 8

d = 5, r = 2.5

area = 6 equilateral triangles with sides 2.5

area = 6 s² root(3)/4

plug in the numbers with s = 2.5

area = 16.2375

Answer 9

R = 5/2 = 2.5
Area K = 3R^2 * sqrt(3)/2
K = 3 * (2.5)^2 * sqrt(3)/2
K = 3 * 6.25 * sqrt(3)/2
K = (18.75sqrt(3))/2
K = (1875sqrt(3))/(200)
K = (75sqrt(3))/8

K = about 16.24

Answer 10

try google or ask.com