find the angle of rotation of a regular octagon mention its formula also
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Answers
Answer:
360÷8=45
Step-by-step explanation:
so the answer is 45 degree.
find the angle of rotation of a regular octagon mention its formula ❓❓❓
Octagon is a polygon in geometry, which has 8 sides and 8 angles. That means the number of vertices is 8 and the number of edges is 8. All the sides are joined with each other end-to-end to form a shape. These sides are in a straight line form; they are not curved or disjoint with each other. Each interior angle of a regular octagon is 135°. Therefore, the measure of exterior angle becomes 180° – 135° = 45°. The sum of the interior angles of the octagon is 135 × 8 = 1080°. In this article, let us discuss the octagon shape, its formulas, properties, and examples in detail.
★Shape of Octagon
Octagon is a geometrical shape in a two-dimensional plane. Like the other polygon shapes, which we have studied in geometry, such as triangle, square, pentagon, hexagon, rectangle, etc., the octagon is also a polygon. The points which define it different from other geometrical shapes is that it has 8 sides and 8 angles.
If squares are built internally or externally on all the sides of an octagon, then the midpoints of the sections joining the centres of opposite squares form a quadrilateral: equi-diagonal and orthodiagonal ( whose diagonals length are equal and they bisect each other at 90 degrees).
★You can see in the above figure, there are 8 sides of the polygon and eight vertices as well. This is a regular octagon because all the angles and sides here are equal. In the same way, based on sides and angles, there are many types of polygons, such as:
★Triangle
★Quadrilateral
★Pentagon
★Hexagon
★Heptagon
★Nonagon
★Decagon
★We might have observed that different objects that we use in our everyday life contain an octagonal shape. Some of the examples include the following:
Outline of an umbrella with 8 ribs
Stop sign board at the signals
A wall clock with 8 edges
A regular octagon is a closed shape with sides of equal length and interior angles of the same measurement. It has eight symmetric lines and rotational equilibrium of order 8. The interior angle at each vertex of a regular octagon is 135°. The central angle is 45°.
Convex and Concave Octagon
The octagon which has all its angles pointing outside or no angles pointing inwards, is a convex octagon. The angles of the convex octagon are not more than 180°. And the octagon, with one of its angles pointing inward, is a concave-shaped octagon.
In the above figure, you can see, the convex octagon has all its angles pointing outside from the center or origin point. Whereas on the right side, the concave octagon has one of the angles pointing
towards the inside of the polygon.
Octagon diagonals
For any n-sided polygon, we can find the number of diagonals using the formula n(n – 3)/2.
Similarly, we can find the number of diagonals in an octagon.
For octagon, n = 8
Substituting n – 8 in the required formula, we get;
n(n – 3)/2 = 8(8 – 3)/2 = 4(5) = 20
Therefore, an octagon contains a total of 20 diagonals. These can be drawn as shown in the below figure.
Octagon 5
Length of the Diagonal of Octagon
If we join the opposite vertices of a regular octagon, then the diagonals formed have the length equal to:
L = a√(4 + 2√2)
where a is the side of the octagon.
Perimeter of Octagon
The perimeter of the octagon is the length of the sides or boundaries of the octagon, which forms a closed shape.
Therefore,
Perimeter = Sum of all Sides = 8a
Where a is the length of one side of the octagon.
Area of Regular Octagon
Area of the octagon is the region covered by the sides of the octagon. The formula for the area of a regular octagon which has 8 equal sides and all its interior angles are equal to 135°, is given by:
Area = 2a2(1 + √2)
This is the octagon area formula incase of equal sides.
Properties of Octagon
In the case of properties, we usually consider regular octagons.
These have eight sides and eight angles.
All the sides and all the angles are equal, respectively.
There are a total of 20 diagonals in a regular octagon.
The total sum of the interior angles is 1080°, where each angle is equal to 135°(135×8 = 1080)
Sum of all the exterior angles of the octagon is 360°, and each angle is 45°(45×8=360).