how to find area of a polygon

Table of Contents

Regular polygons have all straight sides equal in length and all interior angles equal. Finding the area of any regular polygon (the space of the interior) is easy if y'all know what an apothem is. Read, sentinel, and learn!

1. Area of a Regular Polygon
• Surface area of a Regular Polygon Formula
• How To Observe The Apothem
• Apothem Area Formula
2. How to Discover the Surface area of a Regular Polygon
• Area of a Regular Polygon Examples

Expanse of a Regular Polygon

The area of any closed shape is the interior infinite formed by the shape'southward sides. Area is always expressed in square units, such equally $c{\mathrm{chiliad}}^2,f{t}^2,i{n}^{\mathrm{ii}}$.

Regular polygons apply line segments that form sides enclosing a space (the polygon's interior). For regular polygons, you demand to know the length of only 1 side, $s$, and the number of sides, $\mathrm{north}$. To work with the apothem of the polygon, you must know the length of a side.

Area of a Regular Polygon Formula

Combine the number of sides, $n$, and the mensurate of one side, $s$, with the apothem, $a$, to find the area, $A$, of any regular polygon.

Let's dive into the details:

How To Find The Apothem

This may be a new word to you lot, just the apothem (pronounce it like APP-uh-them) is the distance of a perpendicular line from any side of the polygon to its center.

Regular polygons are the only geometric figures that have apothems. The apothem is likewise the radius of a circle that tin can be drawn completely inside the regular polygon. That circle is besides called the incircle, and its incenter is the center of the regular polygon.

Finding the Centre

To find the center or incenter of a regular polygon, connect opposite vertices using diagonals. Any two crossing diagonals volition locate the center, but you can triple-check past drawing in boosted diagonals. Here is a decagon or 10-gon with all five diagonals drawn in:

Notice all five diagonals create 10 small triangles. Cartoon a line from the center or incenter to any side of the regular polygon gives you the apothem. It is as well the distance or height of all those triangles.

Apothem Expanse Formula

You must know these three facts nearly your regular polygon:

1. The number of sides, $n$
2. The length of the apothem, $a$
3. The length of any one side, $s$

If you know all three numbers, you can find the area, $A$, past applying this formula:

How to Find the Area of a Regular Polygon

Let's say you have that regular decagon (10 sides; $n=10$) with sides, $\mathrm{south}$, 8 meters in length and an apothem, $a$, of 12.31 meters.

Permit's put those numbers into the formula:

$A=\frac{(\mathrm{ten}\times 8\times 12.31)}{\mathrm{two}}$

$A=\frac{(80\times 12.31)}{2}$

$A=\frac{984.8}{\mathrm{ii}}$

$A=492.4$

The surface area of our decagon is $\mathrm{492.iv}square\mathrm{1000}\mathrm{eastward}ter\mathrm{south}$, or $\mathrm{492.four}{m}^2$.

Surface area of a Regular Polygon Examples

Hither is an easier shape to piece of work with. Consider a regular octagon (8 sides; $n=8$) with sides xx centimeters in length. The apothem is 24.142 centimeters. What is the area? Try it yourself before looking at the steps below.

$A=\frac{(n\times \mathrm{south}\times a)}{2}$

$A=\frac{(\mathrm{viii}\times 20\times 24.142)}{\mathrm{two}}$

$A=\frac{(160\times 24.142)}{2}$

$A=\frac{\mathrm{3,862.72}}{2}$

$A=\mathrm{1,931.36}$

Did you become the area of $\mathrm{i,931.36}squar\mathrm{eastward}c\mathrm{due\; east}ntimeters$, or $\mathrm{ane,931.36}c{\mathrm{yard}}^2$?

Lesson Summary

You have learned to define and identify a regular polygon, including its parts such as sides and area. Y'all learned what an apothem is, and how to detect it on whatsoever regular polygon. Y'all also learned the formula for finding the surface area of whatever regular polygon if you lot know the length of i side and the apothem: $A=\frac{(\mathrm{due\; north}\times \mathrm{due\; south}\times a)}{2}$, where $n$ is the number of sides, $s$ is the length of one side, and $a$ is the apothem.

Next Lesson:

How to Discover the Bending of a Triangle

Source: https://tutors.com/math-tutors/geometry-help/how-to-find-the-area-of-regular-polygons

Posted by: amadorhagerre1998.blogspot.com

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