Geometry is a mathematical branch that is about the study of shapes. The category of shapes is divided into two viz. flat shapes and solid shapes. Geometry deals with the study of the area, perimeter, volume, and other parameters of these shapes by giving standard formulas.
The article explains the octagon formula which gives the formula of area and perimeter of an octagon. It also comprises sample numerical problems for better understanding.
Octagon
An octagon is a plane shape having eight sides and eight angles. It is a regular polygon of eight sides. Each interior angle of the octagon measures 135° and the sum of all the interior angles of an octagon equals 108°. Similarly, the exterior angle of an octagon is 45 degrees and the sum of all the exterior angles equals 360°.
The octagon consists of 20 diagonals that meet at the center of the figure. All these diagonals have the same length.
Regular octagonOctagon Formulas
The geometry provides separately derived formulas for the calculation of the perimeter, area, and diagonals of a regular octagon. The perimeter, area, and diagonal formula of an octagon is collectively known as the Octagon Formula.
To find the number of diagonals of an octagon we use the given formula.
Number of Diagonals = n(n - 3)/2Â
8(8 - 3)/2Â
20
Where s denotes  side length
And, n denotes the number of sides
A regular polygon generally consists of 20 diagonals. So, the octagon formula is mostly used to calculate the area and perimeter of an octagon. These calculations are carried out by using the length of a side of the octagon.
The area formula of  an octagon is given by
Area of octagon(A) = 2s2(1 + √2)
Where s is the length of a side
The perimeter formula of an octagon is given by,
The perimeter of the octagon(P) = 8s
Where s is the length of a side.
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Sample Problems - Octagon Formulas
Question 1: Find the area and perimeter of an octagon having a side 2cm using the octagon formula.
Solution:
Given:
length of a side of the octagon is 2cm
By using the octagon formula for the area
Area of octagon(A) = 2s2(1 + √2)
A = 2(2)2(1 + √2)
A = 19.31cm2
By using the octagon formula for the perimeter,
Perimeter of the octagon (P) = 8s
P = 8 × 2
P = 16cm
Hence, the area and perimeter of the given octagon are 19.31cm2 and 16cm respectively.
Question 2: Find the area and perimeter of an octagon having a side of 4cm using the octagon formula.
Solution:
Given:
length of a side of the octagon is 4cm
By using the octagon formula for the area
Area of octagon(A) = 2s2(1 + √2)
A = 2(4)2(1 + √2)
A = 77.25cm2
By using the octagon formula for the perimeter
The perimeter of the octagon (P) = 8s
P = 8 × 4
P = 32cm
Hence, the area and perimeter of the given octagon are 77.25cm2 and 32cm respectively.
Question 3: Find the area and perimeter of an octagon having a side of 2.5cm using the octagon formula.
Solution:
Given:
length of a side of the octagon is 2.5cm
By using the octagon formula for the area
Area of octagon(A) = 2s2(1 + √2)
A = 2(2.5)2(1 + √2)
A = 30.17cm2
By using the octagon formula for the perimeter
The perimeter of the octagon (P)=8s
P = 8 × 2.5
P = 20cm
Hence, the area and perimeter of the given octagon are 30.17cm2 and 20cm respectively.
Question 4: A regular octagon is given which has a perimeter equal to 32cm. Find its area using the octagon formula.
Solution:
Given:
The perimeter of the octagon is 32cm.
The perimeter of the octagon(P) = 8s
32 = 8s
s = 4cm
By using the octagon formula for the area
Area of octagon(A) = 2s2(1 + √2)
A = 2(4)2(1 + √2)
A = 77.25cm2
Question 5: A regular octagon is given which has a perimeter of 48cm. Find its area using the octagon formula.
Solution:
Given:
The perimeter of the octagon is 48cm.
The perimeter of the octagon(P) = 8s
48 = 8s
s = 6cm
By using the octagon formula for the area
Area of octagon(A) = 2s2(1 + √2)
A = 2(6)2(1 + √2)
A = 173.82cm2
Question 6: If the perimeter of an octagon is given which is equal to 40cm. Calculate the area of the given octagon.
Solution:
Given:
The perimeter of the octagon is 40cm.
The perimeter of the octagon(P) = 8s
40 = 8s
s = 5cm
By using the octagon formula for the areaÂ
Area of octagon(A) = 2s2(1 + √2)
A = 2(5)2(1 + √2)
A = 120.71cm2
Question 7: If an octagon is given having length of 3cm, its area and perimeter be, calculated using the octagon formula?
Solution:
Given:
The side of the octagon is 3cm
By using the octagon formula for the areaÂ
Area of octagon(A) = 2s2(1 + √2)
A = 2(3)2(1 + √2)
A = 43.45cm2
By using the octagon formula for the perimeter
The perimeter of the octagon (P) = 8s
P = 8 × 3
P = 24cm
Hence, the area and perimeter of the given octagon are 43.45cm2 and 24cm respectively.
Practice Problems
Problem 1: Find the interior angle of a regular octagon.
Problem 2: Calculate the area of a regular octagon with the side length of 10 units.
Problem 3: Determine the perimeter of a regular octagon where each side measures 12 units.
Problem 4: Given a regular octagon with an apothem of 7 units calculate the area of the octagon.
Problem 5: Calculate the length of the diagonal connecting two non-adjacent vertices in the regular octagon with the side length 8 units.
Problem 6: Find the radius of the circumcircle of a regular octagon with the side length 6 units.
Problem 7: Determine the total interior angle sum of an octagon.
Problem 8: Compute the length of the diagonal connecting vertices with the two sides in between in a regular octagon with the side length 9 units.
Problem 9: Calculate the area of an irregular octagon with the side lengths 4,5,4,5,6,7,6 and 7 units and one of the interior angles measuring 135∘.
Problem 10: Given a regular octagon inscribed in a circle with the radius 10 units find the length of each side of the octagon.
Summary
The Octagons are polygons with the eight sides and angles and they have specific properties and formulas for both the regular and irregular cases. Key formulas include the interior angle calculation, area, perimeter and diagonal lengths. Understanding these formulas helps in solving the geometric problems related to the octagons and applying these concepts in the practical scenarios. The Mastery of octagon properties enhances spatial reasoning and problem-solving the skills in geometry.
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