# AREA OF REGULAR POLYGON

## Area of an Equilateral Triangle

The area of any triangle with base length b and height h is given by A  =  1/2⋅bh. The following formula for equilateral triangles, however, uses only the side length.

Theorem :

The area of an equilateral triangle is one fourth the square of the length of the side times √3.

Area of the equilateral triangle shown above is

=  √3/4 ⋅ s2

Proof :

Given : ΔABC is equilateral

To prove : Area of ΔABC is A  =  1/4 ⋅ √3 ⋅ s2

In ΔABC, draw the altitude from B to side AC.

Then ΔABD is a 30°-60°-90° triangle.

To find the length of the altitude BD, use Pythagorean theorem in right ΔBDC.

So, we have

BD2 + DC2  =  BC2

Substitute.

BD2 + (s/2)2  =  s2

BD2 + s2/4  =  s2

Subtract s2/4 from each side.

BD2  =  s2 s2/4

Simplify.

BD2  =  3s2/4

Take square root on each side.

BD  =  3s/2

Using the formula for the area of a triangle.

A  =  1/2 ⋅ bh

Substitute.

A  =  1/2 ⋅ s(3s/2)

A  =  √3/4 ⋅ s2

## Area of a Regular Polygon

Theorem :

The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P.

So,

A  =  1/2 ⋅ aP     or     A  =  1/2 ⋅ a ⋅ (ns)

Proof :

Think of the hexagon ABCDEF as inscribed in a circle.

The center of the polygon and radius of the polygon are the center and radius of its circumscribed circle, respectively.

The distance from the center to any side of the polygon is called the apothem of the polygon. The apothem is the height of a triangle between the center and two consecutive vertices of the polygon.

We can find the area of any regular n-gon by dividing the polygon into congruent triangles.

A  =  area of one triangle ⋅ number of triangles

A  =  (1/2 ⋅ apothem ⋅ side length s) ⋅ number of sides

A  =  1/2 ⋅ apothem ⋅ (number of sides ⋅ side length s)

A  =  1/2 ⋅ apothem ⋅ perimeter of polygon

This approach can be used to find the area of any regular polygon.

## Central Angle of a Regular Polygon

A central angle of a regular polygon is an angle whose vertex is the center and whose sides contain two consecutive vertices of the polygon. We can divide 360° by the number of sides to find the measure of each central angle of the polygon.

## Finding the Area of an Equilateral Triangle

Example :

Find the area of an equilateral triangle with side length 4 inches.

Solution :

The formula for area of an equilateral triangle is given by

A  =  √3/4 ⋅ s2

Plug s  =  4

A  =  √3/4 ⋅ 42

Simplify.

A  =  √3/4 ⋅ 16

A  =  √3 ⋅ 4

A  =  4√3

So, the area of the equilateral triangle is 4√3 square inches.

## Finding the Area of a Regular Polygon

Example :

Find the area of the regular pentagon shown below.

Solution :

To find area of any regular polygon, we need to know the length of apothem and perimeter.

In the above regular pentagon,

Apothem  =  15 units

Pentagon has 5 sides. So, the perimeter is

P  =  side length ⋅ no. of sides

P  =  18 ⋅ 5

P  =  90 units

The formula for area of a n-gon is given by

A  =  1/2 ⋅ apothem ⋅ perimeter of polygon

Substitute.

A  =  1/2 ⋅ 15 ⋅ 90

Simplify.

A  =  675

So, the area of the regular pentagon is 675 square units.

To get more problems on area of a regular polygon,

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

We always appreciate your feedback.

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6