What is the number of diagonals of an octagon?
20
Understanding Polygon Diagonals
A polygon is a closed shape made up of straight line segments. A diagonal is a line segment that connects two non-adjacent vertices of a polygon. The question asks for the number of diagonals in an octagon.
What is an Octagon?
An octagon is a polygon with 8 sides. It also has 8 vertices. Examples of octagons include a stop sign.
Formula for Number of Diagonals
To find the number of diagonals in any polygon, we can use a specific formula. For a polygon with 'n' sides (and hence 'n' vertices), the number of diagonals is given by:
Number of diagonals \(= \frac{n(n-3)}{2}\)
Let's break down this formula:
- From each vertex, we can draw a line segment to every other vertex. There are \((n-1)\) other vertices.
- If we connect a vertex to itself, it's not a line segment.
- If we connect a vertex to its two adjacent vertices, these are the sides of the polygon, not diagonals.
- So, from each vertex, we can draw \((n-1) - 2 = (n-3)\) diagonals.
- Since there are 'n' vertices, we might think the total number is \(n(n-3)\). However, this counts each diagonal twice (once from each endpoint).
- Therefore, we must divide by 2 to get the unique number of diagonals.
Calculating Octagon Diagonals
For an octagon, the number of sides, 'n', is 8.
Using the formula for the number of diagonals:
Number of diagonals of an octagon \(= \frac{n(n-3)}{2}\)
Substitute \(n=8\) into the formula:
Number of diagonals \(= \frac{8(8-3)}{2}\)
Calculate the value inside the parenthesis:
Number of diagonals \(= \frac{8(5)}{2}\)
Multiply the numbers in the numerator:
Number of diagonals \(= \frac{40}{2}\)
Divide to find the final number:
Number of diagonals = 20
So, an octagon has 20 diagonals.
Understanding the Result
The calculation shows that for a polygon with 8 vertices, there are 20 unique line segments connecting non-adjacent vertices. This is significantly more than the number of sides (8).
Revision Table: Polygons and Diagonals
| Polygon Name | Number of Sides (n) | Formula: \(\frac{n(n-3)}{2}\) | Number of Diagonals |
|---|---|---|---|
| Triangle | 3 | \(\frac{3(3-3)}{2} = \frac{3(0)}{2}\) | 0 |
| Quadrilateral | 4 | \(\frac{4(4-3)}{2} = \frac{4(1)}{2}\) | 2 |
| Pentagon | 5 | \(\frac{5(5-3)}{2} = \frac{5(2)}{2}\) | 5 |
| Hexagon | 6 | \(\frac{6(6-3)}{2} = \frac{6(3)}{2}\) | 9 |
| Heptagon | 7 | \(\frac{7(7-3)}{2} = \frac{7(4)}{2}\) | 14 |
| Octagon | 8 | \(\frac{8(8-3)}{2} = \frac{8(5)}{2}\) | 20 |
| Nonagon | 9 | \(\frac{9(9-3)}{2} = \frac{9(6)}{2}\) | 27 |
| Decagon | 10 | \(\frac{10(10-3)}{2} = \frac{10(7)}{2}\) | 35 |
Additional Information about Octagons and Diagonals
- A regular octagon has all sides equal in length and all interior angles equal (135 degrees each).
- The diagonals of an octagon can have different lengths depending on whether the octagon is regular or irregular, and which vertices they connect.
- The formula \(\frac{n(n-3)}{2}\) works for any simple polygon (a polygon that does not intersect itself), whether it is regular or irregular.
- Understanding diagonals is important in geometry for studying properties of polygons, such as triangulating a polygon (dividing it into triangles by drawing non-intersecting diagonals from a single vertex).
Similar Questions
What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ?
What is the interior angle of a regular octagon of side length 2 cm?
Consider a regular polygon with 10 sides. What is the number of triangles that can be formed by joining the vertices which have no common side with any of the sides of the polygon?
Important Questions from Polygons
One angle of a pentagon is 140° and the remaining angles are in the ratio 1 : 2 : 3 : 4. The largest angle of the pentagon is equal to :
The sum of interior angles of a polygon is 2160°. The number of sides of the polygon is:
If a twelve sided regular polygon is inscribed in a circle of radius 3 centimeters, then the length of each side of the polygon is
Which is a more appropriate advantage of a histogram over a polygon?
- In a pentagon ABCDE, $\angle A = (2x + 9^\circ)$, $\angle B = (2x + 1^\circ)$, $\angle C = (2x-1^\circ)$, $\angle D = (2x +5^\circ)$ and $\angle E = (2x-4^\circ)$. Then, value of $(2x+10^\circ)$ is :
