ANGLE OF A POLYGON

Angles Of Polygon

The word polygon is a combination of two Greek words: "poly" which means many and "gon" which means angle. Together with its angles, a polygon as well has sides and vertices. "Tri" means "three," therefore, the simplest polygon is known as the triangle, due to the fact that it has three angles. It as well has three sides and three vertices. A triangle is always coplanar, which is not true of a lot of the other polygons.

A regular polygon is a polygon with all angles and all sides congruent, or equal. See a few regular polygons below:

We can make use of a formula to calculate the sum of the interior angles of any polygon. In this formula, the letter n stands for the number of sides, or angles, that the polygon has.

sum of angles = (n – 2)180°

Let's use the formula to calculate the sum of the interior angles of a triangle. Substitute 3 for n. We discover that the sum is 180 degrees. This is a significant fact to remember.

sum of angles = (n – 2)180°
= (3 – 2)180° = (1)180° = 180°

To calculate the sum of the interior angles of a quadrilateral, we can make use of the same formula again. This time, substitute 4 for n. We discover that the sum of the interior angles of a quadrilateral is 360 degrees.

sum of angles = (n – 2)180°
= (4 – 2)180° = (2)180° = 360°

Polygons can be divided into triangles by drawing all the diagonals that can be drawn from one single vertex. Let's showcase this with the quadrilateral as shown below. From vertex A, we can draw only one diagonal, to vertex D. A quadrilateral can therefore be separated into two triangles.

If you have a look again at the formula, you'll observe that n – 2 produces the number of triangles in the polygon, and that number is multiplied by 180, the sum of the measures of all the interior angles in a triangle. I suppose that you can now see where the "n – 2" comes from? It gives us the number of triangles in the polygon. How many triangles do you suppose that a 5-sided polygon will have?

The diagram above is a pentagon, a 5-sided polygon. From vertex A we can draw two diagonals which divides the pentagon into three triangles. We multiply 3 times 180 degrees to obtain the sum of all the interior angles of a pentagon, which is 540 degrees.

sum of angles = (n – 2)180°
= (5 – 2)180° = (3)180° = 540°

Angle Sum of Polygons

When you start with a polygon with four or more sides and draw all the diagonals possible from one vertex, the polygon then is divided into a lot of non overlapping triangles. The figure below illustrates this division using a seven‐sided polygon. The interior angle sum of this polygon can now be calculated by multiplying the number of triangles by 180°. When investigating, it is observed that the number of triangles is always two less than the number of sides. This fact is stated as a theorem.

Triangulation of a seven‐sided polygon to find the interior angle sum

Theorem:

If a convex polygon has n sides, then its interior angle sum is obtained by the following equation: S = ( n −2) × 180°.

An exterior angle of a polygon:

An exterior angle of a polygon is formed by extending just one of its sides. The non-straight angle adjacent to an interior angle is the exterior angle. Figure may suggest the following theorem:

The (nonstraight) exterior angles of a polygon

Theorem:

If a polygon is convex, then the sum of the degree measures of the exterior angles, one at each vertex, is 360°.

m∠1+m∠2+m∠3+m∠4+m∠5+m∠6=360

Example 1:

Find the interior angle sum of a decagon.

A decagon has 10 sides, therefore:

S=(10-2)×180°
S=1440°

Example 2:

Find the exterior angle sums, one exterior angle at each vertex, of a convex nonagon.
The sum of the exterior angles of any convex polygon is 360°.

Example 3:

An interior angle of a regular hexagon

Method 1:

Because the polygon is regular, all interior angles are equal, therefore you are only required to find the interior angle sum and divide by the number of angles.

S=(6-2)×180°
S=720°

There are six angles, therefore, 720 ÷ 6 = 120°.
Each interior angle of a regular hexagon has a measure of 120°.

Method 2:

Due to the fact that the polygon is regular and all its interior angles are equal, all its exterior angles are as well equal. Look at the Figure. This entails that:

m∠1=m∠2=m∠3=m∠4=m∠5=m∠6

Due to the fact that the sum of these angles will at all times be 360°, then each exterior angle would be 60° (360° ÷ 6 = 60°). If each exterior angle is 60°, then each interior angle is 120° (180° − 60° = 120°).

Polygon related angles 
Internal and External angles

• An angle that is part of a simple polygon is known as an interior angle if it lies on the inside of that simple polygon. A concave simple polygon has at least one interior angle that is a reflex angle.

In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or 1/2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex polygon with n sides add up to [(n − 2) × π] radians, or [(n − 2) × 180]°, (2n − 4) right angles, or (n/2 − 1) turn.

• The supplement of an interior angle is known as an exterior angle, that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical angles and therefore are equal. An exterior angle measures the amount of rotation one has to make at a vertex to trace out the polygon. If the equivalent interior angle is a reflex angle, the exterior angle ought to be considered negative. Even in a non-simple polygon it may be possible to define the exterior angle, but one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.

In Euclidean geometry, the sum of the exterior angles of a simple convex polygon will be one full turn (360°). The exterior angle here could be referred to as a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle Geometry when drawing regular polygons.

• In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).

• In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.

• In a triangle, three intersection points, two of them between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended, are collinear.

• A few authors make use of the name exterior angle of a simple polygon to simply mean the implement exterior angle (not supplement!) of the interior angle. This conflicts with the above usage.

Plane related angles

• The angle between two planes (like two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.

• The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that passes through the point of intersection and is normal to the plane.

Calculating the interior and exterior angles of regular polygons Calculating the interior angle

We already know the way to calculate the sum of the interior angles of a polygon making use of the formula

We as well know that all the interior angles of a regular polygon are equal.

Interior angle of a regular polygon = sum of interior angles ÷ number of sides

Question

Find the interior angle of a regular hexagon.

Calculating the exterior angle

We know that the exterior angles of a regular polygon always add up to 360°, therefore, the exterior angle of a regular hexagon is 360/6=60°

Bear In mind : 
The interior angle and its corresponding exterior angle at all times add up to 180° (for a hexagon, 120° + 60° = 180°).

Question

Calculate the exterior angle of a regular octagon, and write down the value of the interior angle.

Regular and irregular polygons

The simplest polygon is a triangle (a 3-sided shape). Polygons of all types can be regular or irregular. A regular polygon has sides of equal length, and all its interior angles are of equal size. Irregular polygons can have sides of any length and angles of any size.

Here are the names of some common polygons:

Number of sides	Name of polygon	Shape
3	triangle
4	quadrilateral
5	pentagon
6	hexagon
8	octagon
10	decagon

Angle properties of polygons

In your exam, you may be asked to calculate angles of polygons.
The formula for calculating the sum of the interior angles of a regular polygon is: (n - 2) × 180° where n is the number of sides of the polygon.
This formula comes from dividing the polygon up into triangles using full diagonals.
We are already aware of the fact that the interior angles of a triangle add up to 180°. For any polygon, count up the number of triangles it can be split into. Then multiply the number of triangles by 180.

This quadrilateral has been divided into two triangles, therefore the interior angles add up to 2 × 180 = 360°.

This pentagon has been divided into three triangles, therefore the interior angles add up to 3 × 180 = 540°.

In an equivalent way, a hexagon can be divided into 4 triangles, a 7-sided polygon into 5 triangles etc.

Can you observe the pattern forming? The number of triangles is equal to the number of sides minus 2.

Question

Question

What is the sum of the interior angles of an octagon?

Answer

Did you get the answer 1080°? You calculate 6 × 180 = 1080°.
Bear in mind that the formula for the sum of interior angles is (n - 2) × 180. An octagon has 8 sides. Therefore, n = 8, and n - 2 = 6.
If you find it difficult to memorize formulae, merely add 180° each time in the following way:

Number of sides	Name of polygon	Shape
3	180°
4	180° + 180° = 360°
5	360° + 180° = 540°
6	540° + 180° = 720°
8	900° + 180° = 1080°
10	1260° + 180° = 1440°

The exterior angle of a polygon and its corresponding interior angle at all times add up to 180° (because they make a straight line).

For any polygon, the sum of its exterior angles is 360°.

Determining the number of sides in a regular polygon, given the interior angle

We are already aware of the following facts about polygons:

• The interior and exterior angles add up to 180° (a straight line - eg, a + f = 180°), and
• The sum of the exterior angles is 360° (a + b + c + d + e = 360°).

Question

The interior angles of a regular polygon are each 120°. Calculate the number of sides.

Answer

The interior angles are 120°, therefore the exterior angles are 180° - 120° = 60°. The exterior angles add up to 360°, therefore, if we divide 360° by 60° we find there are 6 exterior angles. Thus, there are 6 sides (it is a hexagon).

Question

The interior angles of a regular polygon are each 150°. Calculate the number of sides.

Answer

The answer is 12. Did you get that correct? If so - well done!
Remember that the exterior angles are each 30° (180° - 150°).
360 ÷ 30 = 12, 
Therefore there are 12 exterior angles, and 12 sides

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