A polygon is a two-dimensional (2-D) close up door figure made up of right line segments. In geometry, the octagon is a polygon through 8 sides. If the lengths of all the sides and also the measurement of every the angles room equal, the octagon is called a consistent octagon. In various other words, the political parties of a continuous octagon room congruent. Each of the interior angle and also the exterior angle measure 135° and 45° respectively, in a continual octagon. There is a predefined set of formulas because that the calculate of perimeter, and area that a consistent octagon i beg your pardon is collectively called together octagon formula. For an octagon with the size of that is edge together “a”, the formulas are noted below.

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## Octagon Formulas

Formulas for Octagon
Area of an Octagon2a2(1+√2)
Perimeter of one Octagon8a

Octagon formula helps united state to compute the area and perimeter the octagonal objects.

## Derivation the Octagon Formulas:

Consider a continual octagon v each next “a” units.

### Formula because that Area of an Octagon:

Area of one octagon is defined as the an ar occupied within the border of one octagon.

In order to calculation the area of one octagon, we division it into tiny eight isosceles triangles. Calculation the area of among the triangles and then we have the right to multiply by 8 to find the total area that the polygon.

Take one of the triangles and draw a line from the apex come the midpoint of the base to form a ideal angle. The basic of the triangle is a, the side length of the polygon and also OD is the height of the triangle.

Area that the octagon is given as 8 x Area that Triangle.

2 sin²θ = 1- cos 2θ

2 cos²θ = 1+ cos 2θ

(tan^2 heta = frac1-cos2 heta1+cos2 heta\ tan^2(frac452)=frac1-cos451+cos45\ tan^2(frac452)=frac1-frac1sqrt21+frac1sqrt2\ tan^2(frac452)=fracsqrt2-1sqrt2+1=frac(sqrt2-1)^21\ tan(frac452)=sqrt2-1\ fracBDOD=sqrt2-1\ OD=fraca/2sqrt2-1=fraca2(1+sqrt2))

Area of ∆ AOB = (frac12 imes AB imes OD)= (frac12 imes a imes fraca2(1+sqrt2))= (fraca^24(1+sqrt2))Area that the octagon = 8 x Area that Triangle

Area of Octagon = (8 imes fraca^24(1+sqrt2))Area of an Octagon = (2a^2(1+sqrt2))

### Formula because that Perimeter of one Octagon:

Perimeter of one octagon is characterized as the size of the border of the octagon. So perimeter will certainly be the amount of the size of all sides. The formula because that perimeter of one octagon is given by:

Perimeter = size of 8 sides

So, the perimeter of an Octagon = 8a

### Properties the a continual Octagon:

It has eight sides and also eight angles.Lengths of every the sides and the measure up of all the angles are equal.The total number of diagonals in a continual octagon is 20.The amount of all inner angles is same to 1080 degrees, wherein each interior angle procedures 135 degrees.The amount of every exterior angle is equal to 360 degrees, where each exterior angle measures 45 degrees.

### Solved examples Using Octagon Formula:

Question 1: calculation the area and perimeter that a continuous octagon whose side is 2.3 cm.

Solution: Given, next of the octagon = 2.3 cm

Area of an Octagon = (2a^2(1+sqrt2))Area of an Octagon = (2 imes 2.3^2(1+sqrt2)=25.54;cm^2)Perimeter that the octagon = 8a = 8 × 2.3 = 18.4 cm

Question 2: Perimeter of an octagonal protect against signboard is 32 cm. Find the area that the signboard.

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Solution: Given,

Perimeter of the prevent sign plank = 32 cm

Perimeter of one Octagon = 8a

32 centimeter = 8a

a = 32/8 = 4 cm

Area of one Octagon = (2a^2(1+sqrt2))Area of the protect against sign plank = (2 imes 4^2(1+sqrt2)=77.248;cm^2)To solve much more problems ~ above the topic, download BYJU’S – the discovering App.