Octagon
What Is an Octagon?
An octagon is a geometric shape with eight sides, eight corners, and eight angles inside it. Like other polygons, octagons come in different types depending on their sides and angles. Some might have sides of different lengths, while others have all sides equal. Each type has its own set of properties and characteristics. You might have seen an octagon before—it's like the shape of a stop sign you see on the road. That sign is a perfect example of an octagon, with its eight sides and eight corners.
Shape of an Octagon
The word octagon comes from two parts: "octa," meaning eight, and "gon," which means sides. Every shape has its unique form that helps us recognize it.
An octagon is one of many geometric figures which is `2D` shape of eight straight lines forming eight interior angles. When we say it's a closed shape, we mean that all the lines connect to create a complete figure. But these lines must be straight; if there's any curve, it's not an octagon
Types of Octagon
An octagon can be categorized based on its sides and angles into the following types:
- Regular and Irregular Octagon
- Convex and Concave Octagon
Regular Octagon
A regular octagon is a `8` sided shape where all its eight sides and eight angles are equal. All the sides of a regular octagon are the same length and all the octagon angles inside it have the same measure. When you add up all the angles inside a regular octagon, they sum up to `1080` degrees. Moreover, a regular octagon has an interior angle at each vertex of `135°` and the central angle is `45°`. It consists of `8` lines of symmetry
Irregular Octagon
An irregular octagon shape doesn't have all its sides and angles equal. This means some sides might be longer or shorter than others, and some angles might be bigger or smaller. However, even in an irregular octagon all the angles add up to`1080°`.
Convex Octagon
A convex octagon is one where all the angles point outward, away from the center. None of the angles inside a convex octagon are greater than `180°`. Picture it like a shape that bulges outward evenly all around.
Concave Octagon
The concave octagon looks a bit different. It has at least one angle that points inward towards the center of the shape. This inward angle creates an indentation or a recess in the shape. At least one angle in a concave octagon is larger than `180` degrees, giving it a different look from a convex one.
Properties of Octagon
- An octagon typically refers to a polygon with eight sides and eight angles.
- The total sum of interior angles of an octagon adds up to `1080` degrees.
- A regular octagon has all the sides and angles of equal measure. Each interior angle measures `135` degrees.
- For exterior angles, the total sum is `360` degrees, with each exterior angle measuring `45` degrees.
- In a regular octagon, there are `20` diagonals.
- By connecting the diagonals of a regular octagon from a common vertex, you can create six triangles. These diagonals come in three different lengths depending on which vertices they connect.
Diagonals of Octagon
A diagonal in an octagon is a line segment that connects two vertices (corners) of the octagon but is not adjacent to each other. In simpler terms, it's like drawing a line from one corner of the octagon to another but skipping over the corners in between.
To find the number of diagonals in any polygon, including an octagon, we use a formula:
`{n(n – 3)}/2` where, 'n' represents the number of sides of the polygon.
For an octagon, which has eight sides, we substitute '`n`' as `8` into the formula:
Number of diagonals `= {8(8 – 3)}/2`
Solving this equation, we get
Number of diagonals `= {8(5)}/2 = 40/2 = 20` diagonals.
Diagonal Length of Octagon
Shown below is a regular octagon with each side measuring a units.
In the diagram provided, you'll notice three types of diagonals: the longest diagonal `(o)`, the shorter diagonal `(n)`, and the shortest diagonal `(m)`. To calculate their lengths, we use the following formulas:
- The length of the shortest diagonal `(m)` is given by `BH = m = asqrt{(1+sqrt2)}`.
- The length of the shorter diagonal `(n)` is represented by `CH = n = asqrt{(2+sqrt2)}`.
- Lastly, the length of the longest diagonal `(o)` is denoted by `DG = o = asqrt(4+2sqrt2)`.
Perimeter of Octagon
The perimeter of an octagon refers to the total length of all its sides or boundaries, which form a closed shape.
For any octagon, whether regular or irregular, you can find its perimeter by adding up the lengths of all its sides. In the case of a regular octagon, where all sides are of equal length, this becomes simpler.
So, for a regular octagon, the formula to calculate its perimeter is:
Perimeter `=` Sum of all Sides `= 8a`
Here, '`a`' represents the length of one side of the octagon.
Area of Octagon Formula
The area of any shape represents the amount of space it occupies in a plane. Similarly, the area of an octagon is the space enclosed within its eight sides. The formula to compute the area of a regular octagon is:
Area of a regular octagon `= 2a^2(1 + sqrt2)`
Here, '`a`' denotes the length of one side of the given octagon. The resulting area is expressed in square units.
Angles of an Octagon
An octagon has both interior and exterior angles. There are `8` of each in total. In a regular octagon, each interior angle measures `135°`. The total sum of the interior angles in an octagon is `1080°`. Each exterior angle of a regular octagon measures `45°`, and the sum of all exterior angles is `360°`.
To find the sum of the interior angles of any polygon, including an octagon, you can use the formula:
The sum of interior angles `= (n - 2) × 180°`, where '`n`' represents the number of sides.
For an octagon, with `8` sides, the sum would be `(8 - 2) × 180°`, which equals `1080°`.
Octagon Lines of Symmetry
A line of symmetry is like a mirror that splits a shape into two equal parts. A regular octagon has eight lines of symmetry - `4` through opposite vertices and `4` through midpoints of parallel sides. These lines split any regular octagon into identical halves. These lines of symmetry are evenly spaced around the shape, dividing it into eight equal sections.
Solved Examples
Example `1`: Find the perimeter and area of a regular octagon if the length of one side is `5` `\text{cm}`.
Solution:
Given, the length of one side ` a = 5 \ \text{cm} `.
Perimeter:
The perimeter of a regular octagon is calculated by multiplying the length of one side by `8`, since all sides are equal.
\( \text{Perimeter} = 8a = 8 \times 5 = 40 \, \text{cm} \)
Area:
The area of a regular octagon is calculated using the formula ` 2a^2(1+\sqrt{2}) `.
\( \text{Area} = 2 \times 5^2(1+\sqrt{2}) = 2 \times 25 \times (1+\sqrt{2}) \)
\( = 50 + 50\sqrt{2} \approx 120.71 \, \text{cm}^2 \)
Example `2`: Calculate the area of a regular octagon with a side length of `7` `\text{cm}`.
Solution:
Given, the length of one side ` a = 7 \ \text{cm} `.
Area:
\( \text{Area} = 2 \times 7^2(1+\sqrt{2}) = 2 \times 49 \times (1+\sqrt{2}) \)
\( = 98 + 98\sqrt{2} \approx 236.6 \, \text{sq.cm.} \)
Example `3`: Determine the length of the longest diagonal of a regular octagon with a side length of `10` `\text{cm}`.
Solution:
Given, the side length of the regular octagon `a = 10 \ \text{cm}`.
Length of the longest diagonal:
Using the formula ` L = a\sqrt{4 + 2\sqrt{2}} `:
\( L = 10\sqrt{4 + 2\sqrt{2}} \)
\( L = 10\sqrt{6.828} \)
\( L \approx 10 \times 2.613 \)
\( L \approx 26.13 \, \text{cm} \)
Example `4`: The interior angles of a regular hexagon are each `135` degrees. What is the sum of all interior angles of the hexagon?
Solution:
Each interior angle of the regular hexagon `= 135` degrees
To find the sum of all interior angles, we multiply the measure of one interior angle by the number of angles in the hexagon.
Sum of interior angles `= 135` degrees `* 8 = 1080` degrees
Therefore, the sum of all interior angles of the regular hexagon is `1080` degrees.
Example `5`: If the perimeter of a regular octagon is `48` `\text{cm}`, find the length of each side and the area of the octagon.
Solution:
Given, the perimeter of the regular octagon `= 48 \ \text{cm}`.
Length of each side:
`a = \frac{48}{8} = 6 \ \text{cm}`
Area:
\( \text{Area} = 2a^2(1+\sqrt{2}) \)
\( = 2 \times 6^2 (1+\sqrt{2}) \)
\( = 72 \times (1+\sqrt{2}) \approx 295.3 \, \text{sq. cm} \)
Practice Problems
Q`1`. A regular octagon has a side length of `12` `\text{cm}`. What is the perimeter of the octagon?
- `96` `\text{cm}`
- `108` `\text{cm}`
- `120` `\text{cm}`
- `132` `\text{cm}`
Answers: a
Q`2`. How many diagonals does an octagon have?
- `10`
- `15`
- `20`
- `30`
Answers: c
Q`3`. The area of a regular octagon is ` 98(1 + \sqrt{2}) \ \text{cm}^2 `. What is the length of each side?
- `7` `\text{cm}`
- `10` `\text{cm}`
- `15` `\text{cm}`
- `12` `\text{cm}`
Answers: a
Q`4`. Find the area of a regular octagon if the length of each side is ` 6 \ \text{cm} `.
- ` 154.5 \ \text{cm}^2 `
- ` 173.8 \ \text{cm}^2 `
- ` 200 \ \text{cm}^2 `
- ` 203.8 \ \text{cm}^2 `
Answers: b
Q`5`. The perimeter of a regular octagon is `80` `\text{cm}`. What is the area of the octagon?
- ` 100\sqrt{2} \ \text{cm}^2 `
- ` 100 \ \text{cm}^2 `
- ` 200 \ \text{cm}^2 `
- ` 200(1 + \sqrt{2}) \ \text{cm}^2 `
Answers: d
Frequently Asked Questions
Q`1`. How many sides does an octagon have?
Answer: An octagon has `8` sides.
Q`2`. What is a octagon?
Answer: An octagon is a polygon with eight sides and eight angles.
Q`3`. How many diagonals does an octagon have?
Answer: An octagon has `20` diagonals.
Q`4`. What is the sum of the interior angles of an octagon?
Answer: The sum of the interior angles of an octagon is `1080` degrees.
Q`5`. How many lines of symmetry does an octagon have?
Answer: An octagon has `8` lines of symmetry.
Q`6`. What is the difference between a regular octagon and an irregular octagon?
Answer: A regular octagon has all sides and angles equal, while an irregular octagon has unequal sides and angles.
Q`7`. How many equilateral triangles are there in a regular octagon?
Answer: A regular octagon has a total of `8` equilateral triangles.