3.2.4 Angles in polygon
A polygon is a two-dimensional geometric shape that is made up of straight lines and angles. Specifically, it is a closed shape with three or more sides, where each side intersects with exactly two other sides at their endpoints, and the line segments that make up the sides do not cross each other.
Polygons can have different numbers of sides, which gives them different names.
For example, a polygon with three sides is called a triangle, a polygon with four sides is called a quadrilateral, and a polygon with five sides is called a pentagon.
There are also many other names for polygons with different numbers of sides, such as hexagon (six sides), heptagon (seven sides), octagon (eight sides), and so on.
Sums of angles in polygons:
The sum of angles in a polygon is the total measure of all the angles inside the polygon.
The sum of angles in a polygon depends on the number of sides the polygon has.
To find the sum of angles in a polygon with n sides, we can use the formula:
\( sum \ of \ angles \ = \ (n – 2) \times 180° \)
For example, consider a triangle, which is a polygon with three sides.
Using the formula, we get:
\( sum \ of \ angles \ = \ (3 – 2)\times 180° = 1\times 180°= 180° \)
So, the sum of angles in a triangle is always \( 180° \).
Similarly, for a quadrilateral (a polygon with four sides), we have:
\( sum \ of \ angles \ = (4 – 2)\times 180°= 2\times 180°=360° \)
So, the sum of angles in a quadrilateral is always \( 360° \)
This formula works for any polygon, regardless of the number of sides.
Tip:
It is important to note that the sum of angles in a polygon only applies to polygons with flat, planar surfaces.
It does not apply to three-dimensional shapes such as pyramids or cones.
Interior and exterior angles of a polygon
An interior angle of a polygon is an angle formed by two adjacent sides inside the polygon.
An exterior angle of a polygon is an angle formed by one side of the polygon and the extension of an adjacent side outside the polygon.
The sum of an interior angle and its adjacent exterior angle is always 180 degrees. This is known as the Exterior Angle Theorem.
The measure of an interior angle of a regular polygon with n sides is given by
\( interior \ angle = \frac{n-2}{n}\times 180° \)
The measure of an exterior angle of a regular polygon with n sides is given by
\( exterior \ angle=\frac{360°}{n} \)
These formulas work for regular polygons, where all sides and angles are equal.
For irregular polygons, the measures of the interior and exterior angles can vary depending on the shape of the polygon.
Test yourself
Question 1:
The interior angle of a regular polygon with \( n \) sides is \( 150° \).
Calculate the value of \( n \) .
[2]
Question 2:
A regular polygon has \( 72 \) sides.
Find the size of an interior angle.
[3]
Question 3:
The angles of a quadrilateral are \( x° , x+5°, 2x-25° \) and \( x+10° \).
Find the value of \( x \).
[3]
Question 4:
The diagram shows a triangle and a quadrilateral. All angles are in degrees.
For the triangle, show that \( 3a+5b=170 \).
[1]
For the quadrilateral, show that \( 9a+7b=310 \).
[1]
Solve the simultaneous equations.
Show all your working.
[3]
Find the size of the smallest angle in the triangle.
[1]
Question 5:
\( PQRST \) is a regular pentagon.
\( R, U \) and \( T \)are points on a circle, center \( O \).
\( QR \) and \( PT \) are tangents to the circle.
\( RSU \) is a straight line.
Prove that \( ST=UT \)
[5]