Polygon Angle Sums

🔷 Polygon Angle Sums

Part 1 of 5 — Polygons & the Triangle Foundation

Topics in This Part

🔑 Key Concept: The entire topic rests on one fact you already know — the angles of a triangle add up to $180°$. Every polygon can be sliced into triangles, so its angle sum is just a count of triangles times $180°$. Part 1 sets up that idea.

What Is a Polygon?

A polygon is a closed, flat shape made of straight line segments (its sides) joined end to end. The corners where two sides meet are vertices, and the angles inside the shape are its interior angles.

We name a polygon by its number of sides $n$:

💡 Rule of thumb: A polygon always has the same number of vertices, sides, and interior angles. A hexagon has $6$ of each.

Name That Polygon 🔽

Match each polygon to its number of sides.

Convex vs. Concave

• A convex polygon has no dents: every interior angle is less than $180°$, and no side, if extended, cuts through the shape. A stop sign (octagon) is convex.
• A concave polygon has at least one "caved-in" vertex — an interior angle greater than $180°$ (a reflex angle). An arrowhead or a star outline is concave.

A polygon is regular when it is both equilateral (all sides equal) and equiangular (all angles equal). A regular polygon is always convex — think of a square or a regular hexagon.

⚠️ Watch out: All the simple angle-sum formulas in this lesson assume a convex polygon. They still work for any simple (non-self-crossing) polygon, but the cleanest reasoning is convex.

The Triangle: $180°$

The cornerstone fact:

$\text{interior angles of any triangle} = 180°$

This is true for every triangle — tall, skinny, right, or obtuse. If you tear off the three corners of a paper triangle and lay them side by side, they form a straight line, which measures $180°$.

Quick Example

A triangle has angles $70°$ and $55°$. The third angle is:

$180° - 70° - 55° = 55°$

🔑 Why this matters: A quadrilateral can be split into $2$ triangles, a pentagon into $3$, and so on. Each triangle contributes $180°$ — that single idea generates every formula in Part 2.

Find the Missing Angle 🧮

Use "the three angles add to $180°$." Enter the missing angle in degrees (number only).

1) A triangle has angles $90°$, $40°$, and $\,?$ 2) A triangle has angles $60°$, $60°$, and $\,?$ (equilateral) 3) A triangle has angles $25°$, $35°$, and $\,?$

Concept Check 🎯

🔷 Polygon Angle Sums

Part 2 of 5 — The Interior Angle Sum Formula

🔑 The Idea: From any one vertex of a convex polygon, draw all the diagonals. The polygon splits into triangles — exactly $(n-2)$ of them. Since each triangle is $180°$, the total interior angle sum is $(n-2)\cdot 180°$.

Slicing a Polygon into Triangles

Pick one vertex and draw a diagonal to every non-adjacent vertex. The polygon fans out into triangles:

Notice the number of triangles is always two less than the number of sides. That gives the master formula:

$S = (n - 2)\cdot 180°$

where $S$ is the sum of the interior angles and $n$ is the number of sides.

💡 Why $n-2$? Two of the polygon's sides meet at the vertex you chose, so they can't start a diagonal. Every diagonal you can draw carves off one triangle, and you always end up with $n-2$ of them.

Worked Examples

Example: Heptagon ($n = 7$)

$S = (7 - 2)\cdot 180° = 5 \cdot 180° = 900°$

Example: Octagon ($n = 8$)

$S = (8 - 2)\cdot 180° = 6 \cdot 180° = 1080°$

Example: Decagon ($n = 10$)

$S = (10 - 2)\cdot 180° = 8 \cdot 180° = 1440°$

✅ Check the pattern: each time you add one side, the sum grows by exactly $180°$ — one more triangle.

Concept Check 🎯

Apply the Formula 🧮

Use $S = (n-2)\cdot 180°$. Enter each interior angle sum in degrees (number only).

1) Hexagon ($n = 6$): $S = \,?$ 2) Nonagon ($n = 9$): $S = \,?$ 3) Dodecagon ($n = 12$): $S = \,?$

Work Backwards 🧮

Given the interior angle sum, find the number of sides $n$. (Solve $S = (n-2)\cdot 180$, so $n = \dfrac{S}{180} + 2$.)

1) $S = 1080°$, so $n = \,?$ 2) $S = 540°$, so $n = \,?$

🔷 Polygon Angle Sums

Part 3 of 5 — Each Angle of a Regular Polygon

🔑 The Idea: In a regular polygon all $n$ interior angles are equal, so each one is just the total sum shared equally among the vertices: $\text{each interior angle} = \frac{(n-2)\cdot 180°}{n}$

Splitting the Sum Evenly

Because a regular polygon is equiangular, divide the total interior sum by the number of angles $n$:

$\text{each interior angle} = \frac{(n-2)\cdot 180°}{n}$

Example: Regular Pentagon ($n = 5$)

$\frac{(5-2)\cdot 180°}{5} = \frac{540°}{5} = 108°$

Example: Regular Hexagon ($n = 6$)

$\frac{(6-2)\cdot 180°}{6} = \frac{720°}{6} = 120°$

💡 Sanity check: Each interior angle of a convex polygon must be less than $180°$. All the values above pass — and as $n$ grows, the angle creeps toward (but never reaches) $180°$.

Concept Check 🎯

Each Interior Angle 🧮

Find each interior angle of a regular polygon. Enter the value in degrees (number only).

1) Regular triangle ($n = 3$): $\,?$ 2) Regular quadrilateral / square ($n = 4$): $\,?$ 3) Regular decagon ($n = 10$): $\,?$

Build the Calculation 🔽

You want each interior angle of a regular hexagon ($n = 6$). Choose each step.

🔷 Polygon Angle Sums

Part 4 of 5 — Exterior Angles & the $360°$ Rule

🔑 The Surprise: No matter how many sides a convex polygon has, its exterior angles always add up to $360°$. Not $n$-dependent — always exactly one full turn.

What Is an Exterior Angle?

At each vertex, extend one side. The angle between that extension and the next side is the exterior angle. At any vertex, the interior and exterior angles form a straight line:

$\text{interior} + \text{exterior} = 180°$

So they are supplementary — knowing one gives the other.

The Big Fact

$\text{sum of exterior angles (one per vertex)} = 360°$

Why? Walk all the way around the polygon's edge. At each corner you turn by the exterior angle, and by the time you return to the start you've made exactly one full $360°$ turn.

⚠️ Careful: This $360°$ rule uses one exterior angle per vertex. It holds for every convex polygon, regular or not.

Regular Polygons: Each Exterior Angle

Since a regular polygon's exterior angles are all equal and sum to $360°$:

$\text{each exterior angle} = \frac{360°}{n}$

This is often the fastest route to the interior angle, too:

$\text{each interior angle} = 180° - \frac{360°}{n}$

Example: Regular Pentagon ($n = 5$)

$\text{exterior} = \frac{360°}{5} = 72°, \qquad \text{interior} = 180° - 72° = 108°$

Example: Regular Octagon ($n = 8$)

$\text{exterior} = \frac{360°}{8} = 45°, \qquad \text{interior} = 180° - 45° = 135°$

💡 Both interior values ($108°$ and $135°$) match what we found in Part 3 — two roads, same destination.

Concept Check 🎯

Interior ↔ Exterior 🔽

Use "interior $+$ exterior $= 180°$" and "each exterior $= 360°/n$".

Find the Number of Sides 🧮

Use each exterior angle $= \dfrac{360°}{n}$, so $n = \dfrac{360°}{\text{exterior}}$. Enter $n$ (number only).

1) Each exterior angle is $40°$, so $n = \,?$ 2) Each exterior angle is $24°$, so $n = \,?$ 3) Each interior angle is $160°$, so $n = \,?$ (find the exterior first)

🔷 Polygon Angle Sums

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name polygons and find a triangle's missing angle, (2) compute any interior angle sum with $(n-2)\cdot 180°$, (3) find each angle of a regular polygon, and (4) use the exterior-angle $360°$ rule. Let's put it all together.

Quick Reference

⚠️ Don't mix them up: the interior sum grows with $n$, but the exterior sum is always $360°$. For a single angle of a regular polygon, dividing the exterior by $360°/n$ is usually the fastest path.

Mixed Practice 🎯

Mixed Drill 🧮

Enter each answer in degrees or as a count (number only).

1) Interior angle sum of a heptagon ($n = 7$): $\,?$ 2) Each interior angle of a regular dodecagon ($n = 12$): $\,?$ 3) A regular polygon has each exterior angle $= 30°$. Number of sides $n = \,?$

Exit Quiz ✅

Answer all three to finish the lesson.