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Question

Find the area given a regular polygon with 7 sides and your given the line that goes from the center of the polygon to a vertex on the polygon that is equal to 6.8

Answer 1

use the formula A=(1/2)(r^2)(n)(sin[360/n]) whenever the radius is given

A=(1/2)(6.8^2)(7)(sin51.42857º)  (360/7=51.42857º)

A=(1/2)46.24)(7)(0.78183146692)

A=(161.84)(0.78183146692)

A=126.5316 square units

Answer 2

Hi Brooklyn, this problem is fairly difficult to explain with the aid of a diagram but I will try. 

 

Draw a line from the center to any vertex (corner) of the polygon. Draw another line from the center to an adjacent vertex. 

These two lines should form an isosceles triangle with one of the edges of the polygon. 

The angle between these two lines (interior angles) can be found by the following equation:

angle = (n-2)*180°/n

where n is the number of sides. 

 

In this case, we have n = 7

interior angle = (7-2)180/7  = 128.6°

 

We can split the triangle we formed into two right angle triangles by drawing a line from the center to the edge of the polygon. 

 

From this we can determine the center angle in the right triangle which is 128.6/2 = 64.3°

We can now determine the base of  the triangle: 

6.8 cos(64.3) = base = 2.95

 

we now know the base of the triangle and the height :

base = 2.95

height = 6.8sin(64.3)

 

area of one triangle  = (base x height )/2  = (2.95*6.8sin(64.3))/2 = 9.037 

 

Now, we have 7x2  =  14 triangles so we multiple the area of one triangle by 14

 Total area of polygon = 14*9.037 =126.5 

 

I hope this helps you understand how to solve this kind of problem Brooklyn!

 

 

 

Answer 3

roughly 126

 

drawing in all seven segments from center to the vertex, each side is the base of an isosceles triangle.

Dropping the height from the vertex to the side of the triangle, two congruent right triangles are formed

within this isosceles right triangle, with each hypotnuese measuring 6.8 as given in the problem. Label

the intersection point of the height and the side of the 7-gon point D, which is fact is the midpoint.

All 7 isosceles triangles are congruent.

 

Inside the regular 7-gon resides 180*(7-2) = 900 degrees. Dividing by 7 rations 128 and 4/7 degrees

at each vertex. This is split in 2 equal pieces within each of the congruent isosceles right triangles at 64 and 2/7

 in the corner pocket.  

 

 So from the vertex to D, it is 6.8* cos(64 and 2/7) which is approximately 2.95

 the height is 6.8 * sin(64 and 2/7) which is approximately 6.126652

 The area of this right triangle is just over 9, which makes the area of the isosceles triangle just over 18.

  Multiplying by 7 gives a result just over 126