**Polygons **are studied in middle-level mathematics in geometry. Polygons are the two-dimensional figures that have more than one angle. Polygons are made up of straight lines making a closed bounded shape. Mainly, there are two types of polygons - **regular **and **irregular **polygons. Regular polygons have the same measure of sides and angles. While, irregular polygons have different measures of sides and angles.

Polygons are also classified on the basis of their number of sides and angles. Polygon with 3 sides is better known as a **triangle. **The quadrilaterals such as **squares, rectangles, parallelogram **are examples of 4-sided polygons. The Polygon with 5 sides is known as a **pentagon**. The polygons with 6 sides are termed as **hexagon**, and the list goes so on.

The octagon is also a type of polygon that has 8 sides as well as 8 angles. The octagon is composed of two words - octa and gonia. In which, octa refers to eight and gonia means angles. Therefore, we can say that an octagon must have eight angles and eventually eight sides too. A regular octagon has all sides of the same length and its internal angles are also same. It has eight lines of reflective symmetry and rotational symmetry of order 8. An octagon may look like the following figure :

Let us learn more about octagons below.

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A polygon with 8 sides, 8 angles and 8 vertices is called as Octagon.

Octagon Angles = 8

Octagon Sides = 8

Picture of Octagon:

Given below are some important octagon facts:

A regular octagon is a 8 sided polygon where each side is of equal length. The formulas of a regular octagon with side length 'a' is

Area = 2(1 + $\sqrt{2}$)a^{2}

Perimeter = 8 a.

### Octagon Area Formula:

Perimeter = 8 a

Here, a = 3 (given)

=> Perimeter = 8 * 3 = 24 inches. An Octagon having all sides of equal length and all angles are congruent is called regular octagon.

The formula to find an interior angle of an n - sided polygon is $\frac{n - 2}{n}$ * 180 degrees.

Octagon sides = 8

=> Each interior angle for the decagon = $\frac{8 - 2}{8}$ * 180 degrees = 135$^o$.

An irregular octagon has 8 different sides. Irregular polygons are not symmetric and are not considered as having a centre.

A convex octagon is a polygon with interior angles less than 180$^o$. All the diagonals of a decagon lie within the interior of the polygon. Otherwise, it is concave octagon.

An Octagon is a polygon, which is a closed 2D shape constructed from straight lines. Let us construct a octagon with compass and straight edge.

**Step 1: **Draw a square and find its center by constructing the two diagonals.

Step 2: Construct four circles with the compass by considering each corner of the square as the center of one of the circles. The radius length should be equal half the length of one of the diagonals.

Step 3: Construct a point every where the circles intersect the original square and join all the points with straight edge.

Step 4: After Joining all the points, we have an octagon.

### Solved Examples

**Question 1: **The side length of the polygon is 8 m. Find the area and perimeter of an octagon.

** Solution: **

**Question 2: **Find the area of a regular octagon whose side is 50 cm.

** Solution: **

Octagon Angles = 8

Octagon Sides = 8

Picture of Octagon:

Given below are some important octagon facts:

- The number of octagon diagonals is 20.
- The sum of all interior angle = 1080 degrees.
- Since the sides for an regular octagon are equal, the interior angle at each vertex is = $\frac{1080}{8}$ = 135 degrees.
- The total exterior angle is 360 degrees.

A regular octagon is a 8 sided polygon where each side is of equal length. The formulas of a regular octagon with side length 'a' is

Area = 2(1 + $\sqrt{2}$)a

Perimeter = 8 a.

We can divide octagon into six triangles. So, the sum of the angles of each triangle is 180 degrees. Therefore,

Sum of the interior angles of an octagon = 6 * 180 = 1080 degrees

For octagon, the sum of the interior angles is 1080 degrees and there are eight sides. So, the measure of the interior angle of a regular octagon is as follows,

Interior Angle = $\frac{1080}{8}$

= 135 degrees

Let us make a circle in the middle of the octagon and measure the central angle of a regular octagon. A circle is 360 degrees around. And so, we can divide it by eight angles.

Therefore, Central Angle = $\frac{360}{8}$ = 45 degrees.

The total space inside the boundary of an octagon is called as the area of an octagon. Area is measured in terms of square unit.

The area of an regular octagon formula is as follows,

Area = ( 2 + 2 $\sqrt{2}$ )a^{2} = 2(1 + $\sqrt{2}$)a^{2} ^{ }

where, a is the length of the side.

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Perimeter of an Octagon is defined as the distance covered outside the boundary of the octagon. Therefore, perimeter is calculated by adding the length of the sides of the octagon.

The perimeter of a regular octagon formula is as follows,

Perimeter = 8a, a = length of an octagon side

Let us find the perimeter of the regular octagon with one side is 3 inches.

Perimeter = 8 a

Here, a = 3 (given)

=> Perimeter = 8 * 3 = 24 inches. An Octagon having all sides of equal length and all angles are congruent is called regular octagon.

The formula to find an interior angle of an n - sided polygon is $\frac{n - 2}{n}$ * 180 degrees.

Octagon sides = 8

=> Each interior angle for the decagon = $\frac{8 - 2}{8}$ * 180 degrees = 135$^o$.

An irregular octagon has 8 different sides. Irregular polygons are not symmetric and are not considered as having a centre.

A convex octagon is a polygon with interior angles less than 180$^o$. All the diagonals of a decagon lie within the interior of the polygon. Otherwise, it is concave octagon.

An Octagon is a polygon, which is a closed 2D shape constructed from straight lines. Let us construct a octagon with compass and straight edge.

Step 2:

Step 3:

Step 4:

Given below are some of the examples on area of a regular octagon.

Perimeter of an octagon is given as

Perimeter = 8 * a

= 8 * 8

= 64

Perimeter of an octagon = 64 m

Area = (2 + 2$\sqrt{2}$)a^{2}

= (2 + 2$\sqrt{2}$) * 8^{2}

= 4.828 * 64

= 308.992 m^{2}

Given that, side s = 50 cm.

We know that the area of a regular octagon = $\text{side}^2 \times (2 + 2(\sqrt{2}))$.

= $50^2 \times (2 + 2(1.414))$

= $2500 \times (2 + 2.828)$

= $2500 \times (4.828)$

= 12070 cm^{2}.

Therefore, the area of a regular octagon is 12070 cm^{2}.

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