A polygon is a two-dimensional (2-D) closed figure consisted of of right line segments. In geometry, the octagon is a polygon v 8 sides. If the lengths of all the sides and also the measure up of all the angles space equal, the octagon is dubbed a consistent octagon. In other words, the political parties of a constant octagon space congruent. Each of the interior angle and also the exterior angle measure 135° and 45° respectively, in a continuous octagon. Over there is a predefined set of formulas because that the calculation of perimeter, and also area the a constant octagon i m sorry is collectively called together octagon formula. Because that an octagon through the size of that edge together “a”, the formulas are noted below.
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|Area of one Octagon||2a2(1+√2)|
|Perimeter of an Octagon||8a|
Octagon formula helps us to compute the area and also perimeter the octagonal objects.
Derivation the Octagon Formulas:
Consider a consistent octagon v each next “a” units.
Formula for Area of an Octagon:
Area of one octagon is defined as the region occupied inside the boundary of an octagon.
In order to calculate the area of one octagon, we division it into small eight isosceles triangles. Calculation the area of among the triangles and then we have the right to multiply by 8 to uncover the full area that the polygon.
Take among the triangles and draw a heat from the apex come the midpoint of the basic to type a best angle. The basic of the triangle is a, the side length of the polygon and also OD is the height of the triangle.
Area of the octagon is provided as 8 x Area the 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 that ∆ AOB = (frac12 imes AB imes OD)= (frac12 imes a imes fraca2(1+sqrt2))= (fraca^24(1+sqrt2))Area of the octagon = 8 x Area that Triangle
Area that Octagon = (8 imes fraca^24(1+sqrt2))Area of an Octagon = (2a^2(1+sqrt2))
Formula for Perimeter of an Octagon:
Perimeter of one octagon is defined as the length of the border of the octagon. So perimeter will be the amount of the length of every sides. The formula because that perimeter of an octagon is given by:
Perimeter = length of 8 sides
So, the perimeter of one Octagon = 8a
Properties the a constant Octagon:It has actually eight sides and also eight angles.Lengths of every the sides and the measure up of every the angles room equal.The total number of diagonals in a regular octagon is 20.The amount of all inner angles is same to 1080 degrees, where each inner angle actions 135 degrees.The sum of every exterior angle is equal to 360 degrees, whereby each exterior angle steps 45 degrees.
Solved examples Using Octagon Formula:
Question 1: calculate the area and perimeter of a constant octagon whose next is 2.3 cm.
Solution: Given, next of the octagon = 2.3 cm
Area of one Octagon = (2a^2(1+sqrt2))Area of an Octagon = (2 imes 2.3^2(1+sqrt2)=25.54;cm^2)Perimeter the the octagon = 8a = 8 × 2.3 = 18.4 cm
Question 2: Perimeter of an octagonal avoid signboard is 32 cm. Discover the area the the signboard.
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Perimeter of the stop sign board = 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 stop sign plank = (2 imes 4^2(1+sqrt2)=77.248;cm^2)To solve more problems top top the topic, download BYJU’S – the learning App.