Angle Of N Sided Polygon

Angles, lines and polygons

Polygons are multi-sided shapes with different properties. Shapes have symmetrical properties and some can tessellate.

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Polygons

A polygon is a second shape with at to the lowest degree three sides.

Types of polygon

Polygons can be regular or irregular. If the angles are all equal and all the sides are equal length it is a regular polygon.

Interior angles of polygons

To detect the sum of interior angles in a polygon divide the polygon into triangles.

The sum of interior angles in a triangle is 180°. To discover the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.

Example

Calculate the sum of interior angles in a pentagon.

A pentagon contains 3 triangles. The sum of the interior angles is:

\[180 \times iii = 540^\circ\]

The number of triangles in each polygon is two less than the number of sides.

The formula for calculating the sum of interior angles is:

\((n - 2) \times 180^\circ\) (where \(north\) is the number of sides)

Question

Calculate the sum of interior angles in an octagon.

Using \((n - two) \times 180^\circ\) where \(northward\) is the number of sides:

\[(8 - 2) \times 180 = 1,080^\circ\]

Computing the interior angles of regular polygons

All the interior angles in a regular polygon are equal. The formula for computing the size of an interior angle is:

\[\text{interior angle of a polygon} = \text{sum of interior angles} \div \text{number of sides}\]

Question

Summate the size of the interior angle of a regular hexagon .

Hexagon with all internal angles highlighted

The sum of interior angles is \((6 - 2) \times 180 = 720^\circ\) .

One interior angle is \(720 \div vi = 120^\circ\) .

Exterior angles of polygons

If the side of a polygon is extended, the bending formed outside the polygon is the exterior angle.

The sum of the exterior angles of a polygon is 360°.

External angles produced along the sides of a pentagon equal 360 degrees

Calculating the exterior angles of regular polygons

The formula for calculating the size of an exterior bending is:

\[\text{exterior angle of a polygon} = 360 \div \text{number of sides}\]

Call up the interior and exterior bending add together up to 180°.

Question

Calculate the size of the exterior and interior bending in a regular pentagon .

Pentagon with internal and external angles highlighted

Method one

The sum of exterior angles is 360°.

The outside angle is \(360 \div 5 = 72^\circ\) .

The interior and exterior angles add up to 180°.

The interior bending is \(180 - 72 = 108^\circ\) .

Method 2

The sum of interior angles is \((5 - 2) \times 180 = 540^\circ\) .

The interior bending is \(540 \div five = 108^\circ\) .

The interior and exterior angles add up to 180°.

The exterior angle is \(180 - 108 = 72^\circ\) .

The sum of interior angles in a triangle is 180°. To observe the sum of interior angles of a polygon, multiply the number of triangles in the polygon past 180°.
The formula for computing the sum of interior angles is \((north - 2) \times 180^\circ\) where \(n\) is the number of sides.
All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is: interior bending of a polygon = sum of interior angles ÷ number of sides.
The sum of outside angles of a polygon is 360°.
The formula for computing the size of an exterior bending is: exterior angle of a polygon = 360 ÷ number of sides.