finding the apothem, area, and perimeter of regular polygon

472flj

New member
Joined
Jun 2, 2007
Messages
2
okay so the problem gives me a regular hexagon with the radii measuring 8 units.

i know that the formula to find the area of a regular polygon is area=1/2(apothme)(perimeter)

my question is how do i find the apothem so i acn solve for the area then for the perimeter and also how do i find the perimeter of it once i know the area.
 
Re: finding the apothem, area, and perimeter of regular poly

Hello, 472flj!

Given: regular hexagon with radius 8 units.
Find the area of the hexagon.

Make a sketch . . .

You'll find that the regular hexgon is comprised of six equilateral triangles,
. . each with side 8.

The apothem is the altitude of the triangle.

So we have:
Code:
            *
           /|\
          / | \
       8 /  |a \
        /   |   \
       /    |    \
      * - - + - - *
         4

Use Pythagorus and solve for \(\displaystyle a\).

Got it?

 
Re: finding the apothem, area, and perimeter of regular poly

472flj said:
okay so the problem gives me a regular hexagon with the radii measuring 8 units.

i know that the formula to find the area of a regular polygon is area=1/2(apothme)(perimeter)

my question is how do i find the apothem so i acn solve for the area then for the perimeter and also how do i find the perimeter of it once i know the area.

Polygons

A polygon is a plane figure with three or more line segments and angles that are joined end to end so as to completely enclose an area without any of the line segments intersecting.

A convex polygon is one where the line segments joining any two points of the polygon remain totally inside the polygon, each interior angle being less than 180º.

A concave polygon is one where one or more line segments joining any two points of the polygon are outside of the polygon and one or more of the interior angles is greater than 180º. The inward pointing angle of a concave polygon is referred to as a reentrant angle. The angles less than 180º are called salient angles.

A regular polygon is one where all the sides have the same length and all the interior angles are equal.

A diagonal is a straight line connecting any two opposite vertices of the polygon.

Polygons are classified by the number of sides they have.

No. of sides.........Polygon Name
......3.....................Triangle
......4..................Quadrilateral
......5....................Pentogon
......6....................Hexagon
......7....................Heptagon
.....8......................Octagon
.....9......................Nonagon
....10.....................Decagon
....11....................Undecagon
....12....................Dodecagon
....13....................Tridecagon
....14....................Tetradecagon
....15....................Pentadecagon
......n........................n-gon

Regular Polygon Terminology

n = the number of sides

v = angle subtended at the center by one side = 360/n

s = the length of one side = R[2sin(v/2)] = r[2tan(v/2)]

R = the radius of the circumscribed circle = s[csc(v/2]/2 = r[sec(v/2)]

r = the radius of the inscribed circle = R[cos(v/2)] = s[cot(v/2)]/2

a = apothem = the perpendicular distance from the center to a side (the radius of the inscribed circle)

p = the perimeter = ns

Area = s^2[ncot(v/2)]/4 = R^2[nsin(v)]/2 = r^2[ntan(v/2)]

The formula for the area of a regular polygon is also A = (1/2 )ap = (1/2)ans, where a is the apothem, p is the perimeter, s is the side length and n is the number of sides..

The sum of all the interior angles in a polygon is 180(n - 2)

The sum of the exterior angles in a polygon is 360º.

The internal angle between two adjacent sides of a regular polygon is given by 180(n - 2)/n

The external angle between any side and the extended adjacent side of a regular polygon is given by 360/n.

You might be interested in why the sum of all the interior angles of a polygon is 180(n - 2).
Consider first the square, rectangle and trapazoid. Draw one ofthe diagonals in each of these figures.
What is created is two triangles within each figure.
The sum of the interior angles of any triangle is 180 deg.
Therefore, the sum of the interior angles of each of these 4 sided figues is 360 Deg.
Now consider a pentagon with 5 sides that can be divided up into 3 triangles.
Therefore, the sum of the interior angles of a pentagon is 540 Deg.
What about a hexagon. I tink you will soonsee that the sum of the interior angles is 720 Deg.
Do you notice anything?
n = number of sides........3........4........5........6
Sum of Int. Angles.........180....360....540....720
The sum of the interior angles is representable by 180(n - 2).

Consider also the sum of the exterior angles.
Each exterior angle is 180 - 180(n - 2)/n = (180 - 180n + 360)/n = 360/n.
Therefore, the sum of the exterior angles is 360n/n or 360 Deg.
 
is there any possibility you can make this more of a step by step help please im still a bit confused
 
472flj said:
is there any possibility you can make this more of a step by step help please im still a bit confused
You've been given a description (that is, an explanation) of the reasoning and a picture of the appropriate triangle, and have been told the formula to use and the values to plug into that formula. Please clarify which part of the Pythagorean Theorem you are needing broken down into further steps.

When you reply, please show all of your work, so we can see where you are bogging down. Thank you! :D

Eliz.
 
Top