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 ⋅ s^{2}

**Proof : **

Given : ΔABC is equilateral

To prove : Area of ΔABC is A = 1/4 ⋅ √3 ⋅ s^{2}

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

BD^{2} + DC^{2} = BC^{2}

Substitute.

BD^{2} + (s/2)^{2} = s^{2}

BD^{2} + s^{2}/4 = s^{2}

Subtract s^{2}/4 from each side.

BD^{2} = s^{2 }- s^{2}/4

Simplify.

BD^{2} = 3s^{2}/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 ⋅ s^{2}

**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.

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.

**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 ⋅ s^{2}

Plug s = 4

A = √3/4 ⋅ 4^{2}

Simplify.

A = √3/4 ⋅ 16

A = √3 ⋅ 4

A = 4√3

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

**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.

HTML Comment Box is loading comments...

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**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**