Angles in polygons

A polygon is any closed figure made of straight sides. Triangles have $3$, quadrilaterals have $4$, pentagons have $5$, and so on. Once you know the angle-sum trick for quadrilaterals, you can extend it to any polygon , and you will discover one fact that holds for all of them combined.

Concept

Interior angle sum. A polygon with $n$ sides can be cut from one vertex into $n - 2$ triangles. Each triangle's angles sum to $180^\circ$, so the polygon's interior angles sum to

$(n - 2) \times 180^\circ.$

Examples:

• Triangle ($n=3$): $1 \times 180^\circ = 180^\circ$. ✓
• Quadrilateral ($n=4$): $2 \times 180^\circ = 360^\circ$. ✓
• Pentagon ($n=5$): $3 \times 180^\circ = 540^\circ$.
• Hexagon ($n=6$): $4 \times 180^\circ = 720^\circ$.
• Decagon ($n=10$): $8 \times 180^\circ = 1440^\circ$.

Exterior angle sum. At each vertex, extend one side; the angle between the extension and the next side is an exterior angle. The remarkable fact:

$\text{Sum of exterior angles of any polygon} = 360^\circ.$

The polygon could be a triangle or a thousand-sided figure , the sum is always $360^\circ$. Why? Imagine walking once around the polygon. Each time you turn a corner you turn through the exterior angle. By the time you are back where you started, you have turned a full circle: $360^\circ$.

Regular polygons. A polygon is regular if all its sides are equal and all its angles are equal.

For a regular $n$-gon:

• Each interior angle $= \dfrac{(n-2) \times 180^\circ}{n}$.
• Each exterior angle $= \dfrac{360^\circ}{n}$.

A quick way to compute: interior $+$ exterior $= 180^\circ$ at any vertex.

These angles appear everywhere from honeycomb hexagons to the octagonal STOP sign.

Worked examples

Example 1. Find the sum of interior angles of a $12$-sided polygon (a dodecagon).

• $(12 - 2) \times 180^\circ = 10 \times 180^\circ = 1800^\circ$.

Example 2. Each interior angle of a regular polygon is $150^\circ$. How many sides does it have?

• Exterior angle $= 180 - 150 = 30^\circ$.
• Number of sides $= 360 / 30 = 12$.

Example 3. Each exterior angle of a regular polygon is $40^\circ$. Find its interior angle and number of sides.

• Number of sides $= 360 / 40 = 9$.
• Interior $= 180 - 40 = 140^\circ$.

Example 4. A pentagon has four of its angles equal to $100^\circ, 110^\circ, 120^\circ, 130^\circ$. Find the fifth.

• Sum of interior angles of pentagon $= 540^\circ$.
• Known sum $= 100 + 110 + 120 + 130 = 460$.
• Fifth angle $= 540 - 460 = 80^\circ$.

Try it yourself

1. Find the sum of interior angles of a heptagon ($7$ sides).
2. Each interior angle of a regular polygon is $108^\circ$. How many sides?
3. Find each interior angle of a regular $20$-gon.
4. Find each exterior angle of a regular $12$-gon.
5. A polygon has interior angle sum $1080^\circ$. How many sides?
6. Can each exterior angle of a regular polygon be $7^\circ$? Justify.
7. The five angles of a pentagon are $x, 2x, 2x, 3x, 4x$. Find $x$.
8. Why is the sum of exterior angles always $360^\circ$ regardless of $n$? Give the "walking around" argument in your own words.

Activity

Tile with regular polygons. Cut out many copies of regular triangles, squares, pentagons, hexagons, and octagons. Try to tile your desk with copies of just one shape (no gaps, no overlaps). You will find this works for triangles, squares, and hexagons but not for pentagons or heptagons. The reason is that to tile flatly, the angles meeting at a point must add to $360^\circ$ , and only certain regular angles ($60^\circ, 90^\circ, 120^\circ$) divide $360^\circ$ evenly. This is the geometric reason honeycomb cells are hexagonal.