Area of Polygon?

Question

Please solve this problem: Find the area of a regular hexagon with sides of 4 cm. I know that I need to use the Pathagorean Theorem to find the length of the radius or the apothem. However, I do not know if I can assume that the length of the radius is equal to one side of the hexagon, which would be four. Help!!!!

Answer 1

I have a new equation for you... take this back to your math teacher and teach him a thing or two. :-P

The area of any and all REGULAR POLYGONS has the same equation... and only needs the numerical values that you already know (that being the number of sides and the length of a side)

A = (l^2 • n) / ( 4 • tan [ 180°/n ] )

That is, the area, A, is equal to: the length of a side, l, squared... multiplied by the number of sides, n... divided by four... and divided again by the tangent of the quotient of a full 180 degrees divided by the number of sides, n.

It works for any real regular polygon. And, like I said, all you need to know is the number of sides and the length of a side.

In the shape you described, a hexagon, we already know it has n=6 sides... and you have measured the length of a side to be l=4

A = [ (4)^2 • 6 ] / [ 4 • tan (180° / 6) ]
A = [ 96 ] / [ 4 • tan (30°) ]
A = [ 96 ] / [ 4 • √(3)/3 ]
A = 24•√(3)
A approx = 41.569

Answer 2

In a hexagon the length of a radius is the same as a side because when you draw two radii to consecutive vertices, you will form an equilateral triangle. (You know the central angle is 60 and the radii are equal so the base angles are 60) Now you can draw the apothem and you get a 30 - 60 - 90 triangle. Since the leg opposite the 30 degree angle is 2 (an altitude of an isosceles/equilateral triangle bisects the side it's drawn to), the apothem is 2 sqreroot 3.

Finally, the Area = (1/2) apothem * Perimeter
= (1/2) 2sqreroot3 * 24
=24 squareroot 3

Answer 3

Pathagorean Theorem? That wins the prize. Well anyway, I don't know why you need a radius.
Simply draw a rectangle through the middle of the hexagon, leaving 2 equilateral triangles, one above and one below the rectangle. If you have done the 30/60/90 triangle routine, you can figure that the width of the rectangle is 4 (the rectangle is actually a square). Also you can figure that the height of either triangle is 2*sqrt(3). So the area of either triangle (1/2*base*ht) is 4*sqrt(3), and since you have two triangles, the sum of areas is 8*sqrt(3) for the triangles, AND the area of the hexagon is
8(2+sqrt(3))

Answer 4

Polygon is something it is shaped while some strains intersect and close around, making it impossible which you could flee in case you have been interior it. There could properly be polygons of many factors. next we come to section. this is in basic terms like length, yet with one extra measurement. while we degree length (permit's say in inches), what we are doing is actual making a reference length and then measuring what share situations the scale is larger than the reference length. eg. 4 inches long pencil actual skill that somebody defined that one inch be this plenty.. and we are saying that in case you're taking for this muches and place them one after the different, you get the scale of the pencil. section is the comparable situation, in basic terms we define it to the scale expanded by using length and we are saying that the popular section is of that of a sq. which this is section x section. next, we degree what share such squares the polygon can accommodate, you could take a million/2 a sq. too adn even a quater etc..... that's what we call section.. you could generalize the comparable situation to quantity too... desire it helps!!!

Answer 5

area is 41.569 sq. cm......We can find the area of a regular hexagon by splitting it into 6 equilateral triangles. Yes, you are right that the length of the radius is equal to one side of the hexagon (for regular hex) but I want you to do it by yourself :-) .......go here for clear illustration....http://www.drking.plus.com/hexagons/misc/area.html