Mathematics 7 December 30, 1899
Lesson 3: Measurement of Angles in Polygons
At the end of the lesson, the learners will be able to …
Acquisition:
Describe and explain the relationships between angle pairs (complementary, supplementary, linear pair,
vertical angles, adjacent angles) based on their measures.
Make Meaning:
Deduce the relationship between the exterior angle and the adjacent interior angle of a polygon.
Transfer:
Solve problems involving interior and exterior angles in practical applications, such as tiling patterns or
architecture.
Angles in a Polygon
Sum of the Interior Angles of a Convex Polygon
The sum of the measures of the angles of a polygon can be found by adding the measures of the angles of the
triangles formed when all possible diagonals are drawn from a vertex of the polygon. Knowing that the sum of the
measures of the angles in a triangle is 180 ° , we can easily determine the sum of the measures of the angles in any
polygon.
Consider the chart below to formulate a generalization in finding the sum of the measures of the angles in an n -gon.
Convex Polygon Number of Number of Triangles Sum of Angle Measures
Sides
Triangle 3 1 1(180)=180
Quadrilateral 4 2 2(180)=360
Pentagon 5 3 3(180)=540
Hexagon 6 4 4 (180)=720
Heptagon 7 5 5(180)=900
Octagon 8 6 6 (180)=1080
n -gon n n−2 (n−2)¿
The table shows that the number triangles is always two less than the number of sides of polygon. Since the sum of
the measures of the angles in a triangle is 180 ° , then the sum of the angles in a polygon is the number of triangles
formed from one vertex of the polygon multiplied by 180 .
In an n -gon, the sum of the angle measures can be determined using the formula S= ( n−2 ) 180, where S is the
sum of the interior angles in a polygon and n is the number of sides.
Mathematics 7 December 30, 1899
Example: Find the measure of each angle in a regular octagon.
Solution:
S= ( n−2 ) 180 Since all angles in a regular octagon are congruent.
S= ( 8−2 ) 180 1080°
Then the measure of each angle is or 135 °
S= ( 6 ) 180 8
S=1080
Example: Find the measure of the interior angle of a regular decagon
Solution: The sum of the angle measures of a decagon is
( 10−2 ) 180=8∗180=1440
1440
=144 °
10
Example: Three angels of a convex heptagon are congruent. Each of the other four angles has a measure thrice of
the other three. Find the measure of each angle.
Solution
A heptagon has 7 sides and 7 angles. Thus,
Let m ∠ A=m ∠ F=m∠ E=x S= x+ x + x +3 x+3 x +3 x
Let m ∠ B=m∠ C=m∠ D=3 x S=25 x
The sum of the measurements of all interior angles of 900=15 x
the heptagon is as follows:
x=60 °
S ( n−2 ) 180=( 7−2 ) 180=900
Each interior angle of a regular polygon
(n−2)×180
n
Example:
1. Find the measure of each interior angle of a regular octagon (8-sided polygon).
Solution
Formula
(8−2)×180 Substitute 8 to n 1080
8 8
135 Final answer
6 ×180 Simplify
8
Mathematics 7 December 30, 1899
2. Find the measure of each interior angle of a regular pentagon (5-sided polygon).
Formula
(5−2)×180 Substitute 5 to n 540
5 8
108 Final answer
3× 180 Simplify
5
Exterior Angles Of Polygon
The sum of all exterior angles of a polygon is 360
Example:
1. Find the measure of one exterior angle of a regular hexagon.
Solution:
A hexagon has 6 sides, therefore n=6
360°
=60 °
6
Answer: Each exterior angle of a regular hexagon is 60 ° .
2. A regular polygon has an exterior angle of 40 ° . How many sides does it have?
Solution:
360
n=
exterior angle
360
n= =9
40
Answer: The polygon has nine sides (a nonagon)
Learn more about this topic by clicking this link
https://youtu.be/ZlHe_I0-ZDo
https://youtu.be/1nSGOb_IipE
I. Solve the following.
1. Find the sum of the interior angles of the following polygons.
a. 7-sided
b. 14-sided
2. Solve for the measure of each interior angle of the following regular polygons.
a. 4 sided
b. 8-sided
c. 12-sided
Mathematics 7 December 30, 1899
3. Find the measure of each exterior angle of the following polygons.
a. triangle
b. hexagon
4. A regular polygon has an exterior angle of 30 ° . How many sides does it have?
Designing a Custom Garden Path Using Polygons
Objective: Apply your knowledge of interior and exterior angles by designing a garden path with
polygonal shapes, calculating the angles, and explaining how those angles affect the structure of
the path.
Introduction
You will begin by reviewing the formulas for interior and exterior angles of polygons.
You will also discuss how these angles are important in the real world, such as in designing
structures like garden paths, houses, or playgrounds.
You will design a garden path using polygons and calculate the interior and exterior angles for each
polygon in your design.
Designing the Path
You will sketch a garden path using 4 to 6 polygons of your choice (for example, a path with
squares, triangles, and hexagons). You will draw each polygon on the graph paper.
As you design, you will need to ensure that the corners (angles) of the polygons fit together. You will
calculate these angles using the interior and exterior angle formulas.
You will color the different polygons and label the interior and exterior angles on each side of the
polygons.
Calculating Angles
After designing the path, you will calculate the interior and exterior angle of each polygon in your
design.
Remember this
When solving for angles and understanding their purpose, we can reflect on how God designed everything with
intentionality, from the angles of leaves and branches to the angles in the buildings we live and work in. Just like the
Creator’s wisdom guides all creation, understanding the angles and structures of our world helps us build wisely.
Prepared by:
Teddie A. Dumalaoron Jr.
