Polygons & Angle Sums
By the end of this lesson, you’ll be able to:
- Identify and classify polygons.
- Compute the sum of interior angles of any polygon.
- Find the measure of each interior angle in a regular polygon.
- Use the exterior angle sum and find each exterior angle of a regular polygon.
- Compute the number of diagonals in a polygon.
- Solve angle-chase problems involving polygons.
Key Ideas
A polygon is a closed figure made of straight line segments.
Examples include triangles, quadrilaterals, pentagons, hexagons, etc.
Polygons can be:
- Regular: all sides and angles equal
- Irregular: sides/angles not all equal
- Convex: all interior angles less than 180°
- Concave: one or more interior angles greater than 180°
1. Interior Angle Sum
For any polygon with \(n\) sides:
🔹 Interior angle sum formula
\[ \text{Sum of interior angles} = (n - 2) \cdot 180^\circ \]
This is because the polygon can be divided into \((n - 2)\) triangles.
🔹 Example
For an octagon (\(n = 8\)):
\[ (8 - 2) \cdot 180^\circ = 6 \cdot 180^\circ = 1080^\circ \]
2. Interior Angle of a Regular Polygon
If all angles are equal:
\[ \text{Interior angle} = \frac{(n - 2) \cdot 180^\circ}{n} \]
Example
A regular pentagon (\(n = 5\)):
\[ \frac{(5 - 2)180^\circ}{5} = \frac{540^\circ}{5} = 108^\circ \]
3. Exterior Angles
🔹 Exterior angle sum rule
For any polygon:
\[ \text{Sum of exterior angles} = 360^\circ \]
🔹 Exterior angle of a regular polygon
\[ \text{Exterior angle} = \frac{360^\circ}{n} \]
Interior + exterior angle at each vertex = 180°.
4. Number of Diagonals
A diagonal is a segment connecting two non-adjacent vertices.
🔹 Formula
\[ \text{Number of diagonals} = \frac{n(n - 3)}{2} \]
Example
$ n = 7$ (heptagon):
\[ \frac{7(7 - 3)}{2} = \frac{28}{2} = 14 \]
Common Problem Types
1. Find the Sum of Interior Angles
Example:
Find the sum of interior angles of a 12-sided polygon.
Solution:
\[
(12 - 2)180^\circ = 10 \cdot 180^\circ = 1800^\circ
\]
2. Find the Measure of Each Interior Angle (Regular)
Example:
Regular decagon (\(n=10\)):
\[ \frac{(10 - 2)180^\circ}{10} = 144^\circ \]
3. Find the Measure of Each Exterior Angle (Regular)
Example:
Regular 18-gon:
\[ \frac{360^\circ}{18} = 20^\circ \]
4. Find the Number of Sides From an Angle
Example:
A regular polygon has interior angles of \(135^\circ\). How many sides?
Use interior angle formula:
\[ 135 = \frac{(n-2)180}{n} \]
Solve:
Multiply both sides by \(n\):
\[ 135n = 180n - 360 \]
\[ 45n = 360 \]
\[ n = 8 \]
It’s a regular octagon.
Strategies
- If the polygon is regular, each angle is simply (sum)/n.
- Exterior angles of regular polygons are often the fastest way to find \(n\).
- Use diagonals formula when asked about “how many segments” in a polygon diagram.
- Draw and label when in doubt—polygons are very visual.
Worked Examples
Example 1
What is the sum of interior angles of a 15-sided polygon?
\[ (15 - 2)180 = 13 \cdot 180 = 2340^\circ \]
Example 2
A regular polygon has exterior angles of \(24^\circ\). How many sides?
\[ \frac{360}{24} = 15 \]
Answer: 15-gon
Example 3
How many diagonals does a 20-gon have?
\[ \frac{20(17)}{2} = 170 \]
Practice Problems
- Find the sum of interior angles of a 9-gon.
- Find the measure of each interior angle of a regular 12-gon.
- Find the exterior angle of a regular 30-gon.
- A regular polygon has interior angles of 160°. How many sides?
- How many diagonals are in a decagon?
1. \((9 - 2)180 = 1260^\circ\)
2. \(\frac{(12 - 2)180}{12} = 150^\circ\)
3. \(360/30 = 12^\circ\)
4. Solve
\(160 = \frac{(n - 2)180}{n}\)
\(160n = 180n - 360\)
\(20n = 360\)
\(n = 18\)
5. $ = 35$
Summary
- Interior angle sum formula: \((n - 2)180^\circ\)
- Regular polygon interior angle: \(\frac{(n - 2)180^\circ}{n}\)
- Exterior angle sum: \(360^\circ\)
- Regular polygon exterior angle: \(\frac{360^\circ}{n}\)
- Diagonals: \(\frac{n(n - 3)}{2}\)
- Exterior angles are the quickest way to find the number of sides.
- Use diagonals formula for complex shape counting problems.
- Regular polygons = equal sides = equal angles.