The area of an octagon is a bit complex compared to lower sided shapes. Area of a regular octagon(A) = **4as**. Where ‘a’ is the apothem (straight line from the centre of the pentagon to the side) and ‘**s**‘ is the side of the octagon.

In geometry, an **octagon** (from the Greek **ὀκτάγωνον oktágōnon**, which means “eight angles”) is an eight-sided polygon.

Sum of all the internal angles is equal to 1080𝆩(135*8)

An octagon is a convex equilateral polygon and also a isogonal and a isotoxal figure.

In a regular octagon each side makes an interior angle 45° at the center of octagon and the angle between two adjacent sides(exterior angle) is 135°.

Index

**Finding** **Area of Octagon**

Octagon’s area can be found in 3 ways.

**Area of Octagon from the Length of Side and Apothem**

The apothem is a straight line from the centre of the pentagon to the side, intersecting the side at a 90º right angle. For octagon, apothem is its height.

**Step 1:** Draw five lines from the center of the pentagon to each vertex (corner).

Now we have eight equilateral triangles.

**Step 2:** Calculate the area of the triangle using the formula ½ bh.

**Step 3:** Multiply the area of 1triangle by 8 so that we can get the total octagon’s area.

Consider a triangle AOB.

We have,

2 sin²θ = 1- cos 2θ

2 cos²θ = 1+ cos 2θ

tan^{2}θ = (1- cos 2θ)/ (1+ cos 2θ)

From the triangle AOB.

tan^{2}(45/2) = (1- cos(45))/(1+cos(45))

= (1-(1/√2))/(1+(1/√2))

= (√2-1)/(√2+1)

= (√2-1)^{2}/1

tan^{2} (45/2) = (√2-1)^{2}

tan(45/2)= √2-1

BD/OD = √2−1

OD = (a/2)/(√2−1) = (1+√2)a/2

Area of ∆ AOB = ½ ×AB×OD

= ½× a×(1+√2)a/2

= a^{2}(1+√2)/4

**Area of octagon** = 8 × area of 1 triangle

= 8 × a^{2}(1+√2)/4 = **2a ^{2}(1+√2) **

**Dividing into Simple Polygons**

We can also find octagon’s area by dividing it into simple polygons.

**Step 1:** Divide the octagon into squares and triangles.

**Step 2:** We end up in getting 5 squares with same side lengths and 4 triangles with same side lengths.

**Step 3:** Calculate the area of a square and a triangle.

**Step 4:** Area of the octagon = 5×area of a square + 4×area of triangle.

**Using Formula**

Area of a regular polygon(A) = *pa*/2.

Area of a regular pentagon(A) = pa/2 = 8sa/2 = 4as.

where,*p* = perimeter of the octagon,*a* = apothem of the octagon,*s* = side length of the octagon.

**Examples**

**Question 1.** Suraj was given an octagon of the area 68.98 units square. Help him in finding the length of the side of the octagon.

**Solution.** Given that the,

Area of the octagon(A) = 68.98 units square.

We know that the` area of the octagon,

A = 2a^{2}(1+√2)

68.98 = 2 × a^{2}(1+√2)

a = 3.78units

The length of the side of the octagon whose area is 68.98 units square is given by 3.78 units.

**Question 2.** Perimeter of an octagonal clock is 64 cm. Find the area of the clock.

**Solution.** Given that the,

Perimeter of the clock = 32 cm

Perimeter of an Octagon = 8a

Where a is side of octagon

8a = 64cm

a = 64/8 = 8 cm

Area of an Octagon = 2a^{2}(1+√2)

Area of the clock = 2×8^{2}(1+√2)

Area of the clock with perimeter 64 cm is 309.019cm^{2}.

**Question 3.** Find the apothem of the octagon whose area is 300cm^{2}.

**Solution.** Given that,

Area of the octagon (A) = 300cm^{2}

We know that the area of octagon = 2s^{2}(1+√2) = 4sa

Here s is side of octagon a is apothem.

2s^{2}(1+√2) = 300

s = 7.9

A = 4as

300 = 4 × 7.9 × a

Apothem (a) = 9.5cm

**FAQs**

**What are the formulas for finding the area of the octagon?**Area of octagon = 2s^{2}(1+√2) = 4sa.

where s is side of octagon and a is apothem.

**What is the possible number of diagonals formed in an octagon?**The total number of diagonals in a regular octagon are 20.

**What does octagon symbolize?**Occasionally, the octagon is viewed as the symbol for infinity.

It was suggested that the octagon is a circle attempting to become a square, and a square attempting to become a circle.

**What is the length of the longest diagonal in an octagon**?The length of the longest diagonal in an octagon = s*√(4 + 2√2).

Where s is the side of the octagon.