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Lesson 13 ~25 min Unit 3 · Geometry +85 XP

Angles in Polygons

Every polygon hides a number pattern. The interior angles add to $(n - 2) \times 180^{\circ}$, the exterior angles always sum to $360^{\circ}$, and for a regular polygon each interior angle is $\dfrac{180^{\circ}(n - 2)}{n}$.

Today's hook: A pentagon's interior angles sum to $540^{\circ}$. A hexagon's to $720^{\circ}$. Same trick — cut into triangles.
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Printable Worksheets

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Think First
warm-up

You already know a triangle's angles add to $180^{\circ}$. A quadrilateral's add to $360^{\circ}$. Predict: what should a pentagon (5 sides) add to? A hexagon (6 sides)? Without looking it up, what's the pattern in the numbers?

Record your answer in your workbook.
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Interior Angle Sum
+5 XP

Pick one vertex of a polygon and draw every possible diagonal from it. An $n$-sided polygon splits into exactly $(n - 2)$ triangles. Since each triangle contributes $180^{\circ}$, the interior angle sum of any $n$-gon is $(n - 2) \times 180^{\circ}$.

The formula $S = (n - 2) \times 180^{\circ}$ works for ANY polygon (convex or not, as long as it's simple). For $n = 3$: $S = 180^{\circ}$. For $n = 4$: $S = 360^{\circ}$. For $n = 5$ (pentagon): $S = 540^{\circ}$. For $n = 6$ (hexagon): $S = 720^{\circ}$. For $n = 8$ (octagon): $S = 1080^{\circ}$.

Pentagon splits into 3 triangles -> 540 degrees A B C D E 3 triangles x 180 = 540
Interior angle sum: $S = (n - 2) \times 180^{\circ}$
Always TWO less
The triangle count is $n - 2$, never $n$. Two vertices "anchor" the triangles.
Reason phrase
"$\angle$ sum of polygon" or "$S = (n - 2) \times 180^{\circ}$".
Works for ANY $n \geq 3$
No matter how many sides — just count $n$ and subtract 2.
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What You'll Master
objectives

Know

  • Interior angle sum formula $S = (n - 2) \times 180^{\circ}$
  • Exterior angle sum is always $360^{\circ}$ for any convex polygon
  • Each interior angle of a regular $n$-gon is $\frac{180^{\circ}(n - 2)}{n}$

Understand

  • Why an $n$-gon splits into $n - 2$ triangles
  • Why exterior angles sum to $360^{\circ}$ no matter how many sides
  • How interior and exterior angles at one vertex add to $180^{\circ}$

Can Do

  • Compute the interior angle sum for any $n$-gon
  • Find a missing angle given the others
  • Find each interior or exterior angle of a regular polygon
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Words You Need
vocabulary
PolygonA closed figure made of straight line segments meeting at vertices.
Interior angleAn angle INSIDE the polygon at a vertex.
Exterior angleThe angle between one side and the extension of the next side.
Regular polygonAll sides equal AND all interior angles equal.
ConvexNo interior angle greater than $180^{\circ}$; no "dents".
Pentagon5-sided polygon. Interior sum $= 540^{\circ}$.
Hexagon6-sided polygon. Interior sum $= 720^{\circ}$.
Octagon8-sided polygon. Interior sum $= 1080^{\circ}$.
4
The Exterior Angle Rule
+5 XP

Walk around the boundary of any convex polygon. At each corner, you turn through the exterior angle. By the time you return to your starting point, you've turned through a full revolution. So the exterior angles of any convex polygon sum to $360^{\circ}$ — no matter how many sides.

At each vertex, the interior and exterior angles sit on a straight line, so they add to $180^{\circ}$: $\text{int} + \text{ext} = 180^{\circ}$. For ALL $n$ vertices of a convex polygon, the exterior angles together sum to $360^{\circ}$. For a regular polygon, each exterior angle is just $\dfrac{360^{\circ}}{n}$ — an instant way to find one interior angle ($180^{\circ}$ minus exterior).

Exterior angles of a polygon sum to 360 ext ext Sum of exts = 360 int + ext = 180 at each vertex
Exterior angle sum $= 360^{\circ}$ (any convex polygon)
Always $360^{\circ}$
Triangle, pentagon, decagon — the exterior sum is the same.
Pair with interior
$\text{int} + \text{ext} = 180^{\circ}$ at every single vertex.
Regular shortcut
Regular $n$-gon exterior angle $= \frac{360^{\circ}}{n}$.
Quick book-notes · Exterior angles
  • $\text{int} + \text{ext} = 180^{\circ}$ at each vertex.
  • Exterior angle sum $= 360^{\circ}$ for any convex polygon.
  • Regular $n$-gon: each exterior $= \frac{360^{\circ}}{n}$.
Micro-check: What is the sum of the exterior angles of a 12-sided convex polygon?
5
Regular Polygons
+5 XP

A regular polygon has every side the same length AND every interior angle the same size. Because all $n$ interior angles are equal, you can divide the total sum by $n$ to find each individual angle: $\dfrac{(n - 2) \times 180^{\circ}}{n}$.

Two equivalent ways to find each angle of a regular $n$-gon:
• Interior route: $\dfrac{(n - 2) \times 180^{\circ}}{n}$.
• Exterior route: Each exterior $= \dfrac{360^{\circ}}{n}$, so each interior $= 180^{\circ} - \dfrac{360^{\circ}}{n}$.
Both give the same answer. Regular pentagon: $108^{\circ}$. Regular hexagon: $120^{\circ}$. Regular octagon: $135^{\circ}$.

Regular polygon - each interior angle 120 each Regular hexagon: (6-2)x180/6 = 720/6 = 120
Regular $n$-gon: each interior $= \dfrac{(n - 2) \times 180^{\circ}}{n}$
Two pathways
Interior-sum method OR exterior-angle method — pick whichever is faster.
Memorise three
Pentagon $108^{\circ}$, hexagon $120^{\circ}$, octagon $135^{\circ}$.
Equal sides AND angles
"Equilateral" alone is not regular — angles must match too.
Quick book-notes · Regular polygons
  • Regular = all sides AND all angles equal.
  • Each interior $= \frac{(n - 2) \times 180^{\circ}}{n}$.
  • Or: each interior $= 180^{\circ} - \frac{360^{\circ}}{n}$.
True or false: Each interior angle of a regular hexagon is $120^{\circ}$.

$(6 - 2) \times 180^{\circ} / 6 = 720^{\circ} / 6 = 120^{\circ}$.

6
Finding a Missing Angle
+5 XP

Given a polygon with all but one of its interior angles known, find the missing angle in two steps: (1) compute the total $(n - 2) \times 180^{\circ}$, (2) subtract the known angles. The same logic works in reverse to find $n$ if you know each angle is, say, $144^{\circ}$.

Method for missing angle:
1. Count $n$ — compute $S = (n - 2) \times 180^{\circ}$.
2. Add the known interior angles to get $K$.
3. The missing angle $= S - K$.
To find $n$ when each angle of a regular polygon equals $A$: use $(n - 2) \times 180^{\circ} = nA$ and solve for $n$. Alternatively, find the exterior angle ($180^{\circ} - A$) and divide $360^{\circ}$ by it.

Missing angle in a pentagon 110 95 130 x 100 x = 540 - 110 - 95 - 130 - 100 = 105
Missing $= S - \text{(known angles)}$
Compute $S$ first
Don't subtract until you know the total angle sum.
Watch units
Everything in degrees — don't mix with radians.
Sanity check
In a convex polygon, every interior angle is $< 180^{\circ}$.
Quick book-notes · Missing angles
  • Total $S = (n - 2) \times 180^{\circ}$.
  • Missing $= S - K$ where $K$ is the sum of known angles.
  • To find $n$: use $\frac{360^{\circ}}{\text{exterior}} = n$.
Micro-check: The sum of the interior angles of a polygon is $1260^{\circ}$. How many sides does it have?

Watch Me Solve It · 3 examples

Watch Me Solve It · Sum of interior angles
+15 XP per step
Q1
PROBLEM
Find the sum of the interior angles of an octagon ($n = 8$).
  1. 1
    Write the formula
    $S = (n - 2) \times 180^{\circ}$
  2. 2
    Substitute $n = 8$
    $S = (8 - 2) \times 180^{\circ} = 6 \times 180^{\circ}$
  3. 3
    Compute
    $S = 1080^{\circ}$
    An octagon splits into 6 triangles, each $180^{\circ}$.
Answer$S = 1080^{\circ}$.
Watch Me Solve It · Regular polygon angle
+15 XP per step
Q2
PROBLEM
Find the size of each interior angle of a regular pentagon.
  1. 1
    Find the total
    $S = (5 - 2) \times 180^{\circ} = 540^{\circ}$
  2. 2
    Divide by 5
    Each angle $= \dfrac{540^{\circ}}{5}$
  3. 3
    Compute
    Each angle $= 108^{\circ}$
    Check: $5 \times 108 = 540^{\circ}$ ✓
AnswerEach interior angle $= 108^{\circ}$.
Watch Me Solve It · Find the number of sides
+15 XP per step
Q3
PROBLEM
A regular polygon has each interior angle equal to $150^{\circ}$. How many sides does it have?
  1. 1
    Find the exterior angle
    $\text{ext} = 180^{\circ} - 150^{\circ} = 30^{\circ}$
  2. 2
    Use the exterior sum
    $n = \dfrac{360^{\circ}}{\text{ext}} = \dfrac{360^{\circ}}{30^{\circ}}$
  3. 3
    Compute
    $n = 12$
    A regular dodecagon — 12 sides.
Answer$n = 12$ sides.

Common Pitfalls

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Common Pitfalls
heads-up
Forgetting the $-2$ in the formula
Writing $n \times 180^{\circ}$ instead of $(n - 2) \times 180^{\circ}$ overshoots by exactly $360^{\circ}$.
Fix: A polygon splits into $n - 2$ triangles. Always subtract 2 first.
Confusing interior and exterior sums
Interior sum changes with $n$. Exterior sum is ALWAYS $360^{\circ}$. Mixing them gives wrong $n$.
Fix: Sum $= 360^{\circ}$ $\Rightarrow$ exterior. Sum depends on $n$ $\Rightarrow$ interior.
Calling "equilateral" the same as "regular"
A rhombus is equilateral (4 equal sides) but not regular (angles need not be equal).
Fix: Regular needs equal SIDES AND equal ANGLES.
Copy Into Your Books

Interior sum

  • $S = (n - 2) \times 180^{\circ}$
  • Triangle: $180^{\circ}$
  • Pentagon: $540^{\circ}$
  • Hexagon: $720^{\circ}$

Exterior sum

  • Always $360^{\circ}$
  • $\text{int} + \text{ext} = 180^{\circ}$
  • Regular ext $= \frac{360^{\circ}}{n}$

Regular polygons

  • Equal sides AND equal angles
  • Each int $= \frac{(n-2) \times 180^{\circ}}{n}$
  • Pentagon $108^{\circ}$, hex $120^{\circ}$, oct $135^{\circ}$

Find $n$

  • From ext: $n = \frac{360^{\circ}}{\text{ext}}$
  • From sum $S$: $n = \frac{S}{180^{\circ}} + 2$

How are you completing this lesson?

Brain Trainer · 4 problems

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Brain Trainer · Polygon Angles
4 problems

Four drills on polygon angle sums. Solve, then reveal.

  1. 1 Find the interior angle sum of a heptagon ($n = 7$).

    $(7 - 2) \times 180^{\circ}$.$900^{\circ}$

  2. 2 Each interior angle of a regular polygon is $144^{\circ}$. Find $n$.

    $\text{ext} = 36^{\circ}$; $n = 360 / 36$.$n = 10$

  3. 3 Four interior angles of a pentagon are $100^{\circ}, 115^{\circ}, 95^{\circ}, 120^{\circ}$. Find the fifth.

    $540 - 100 - 115 - 95 - 120$.$110^{\circ}$

  4. 4 Find each exterior angle of a regular nonagon ($n = 9$).

    $\frac{360^{\circ}}{9}$.$40^{\circ}$
Complete in your workbook.

Quick Check · 5 questions

1
The sum of the interior angles of a pentagon is:
+10 XP
2
Each interior angle of a regular hexagon is:
+10 XP
3
The exterior angles of a convex 20-sided polygon sum to:
+10 XP
4
A regular polygon has each interior angle equal to $156^{\circ}$. How many sides has it?
+10 XP
5
Five angles of a hexagon are $110^{\circ}, 130^{\circ}, 100^{\circ}, 145^{\circ}$ and $115^{\circ}$. Find the sixth.
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Easy 3 MARKS

Q6. A regular octagon is constructed.
(a) Find the sum of all interior angles.
(b) Find the size of EACH interior angle.
(c) Find the size of EACH exterior angle.

Answer in your workbook.
Apply Medium 3 MARKS

Q7. Four interior angles of a pentagon are $108^{\circ}, x^{\circ}, (x + 20)^{\circ}, 95^{\circ}$ and the last is $(2x - 5)^{\circ}$.
(a) Set up an equation using the angle sum.
(b) Solve for $x$.
(c) State all five angles.

Answer in your workbook.
Reason Hard 3 MARKS

Q8. The interior angle sum of a regular polygon is $1620^{\circ}$.
(a) Find $n$.
(b) Find the size of each interior angle.
(c) Find the size of each exterior angle.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. B — $540^{\circ}$ (pentagon).

2. A — $120^{\circ}$ (regular hexagon).

3. D — $360^{\circ}$ (exterior sum is constant).

4. C — 15 sides.

5. B — $120^{\circ}$.

Show Your Working Model Answers

Q6 (3 marks): (a) $S = (8 - 2) \times 180^{\circ} = 1080^{\circ}$ [1]. (b) Each interior $= 1080^{\circ} / 8 = 135^{\circ}$ [1]. (c) Each exterior $= 180^{\circ} - 135^{\circ} = 45^{\circ}$ (or $360^{\circ}/8 = 45^{\circ}$) [1].

Q7 (3 marks): (a) $108 + x + (x + 20) + 95 + (2x - 5) = 540$ ($\angle$ sum of pentagon) [1]. (b) $4x + 218 = 540 \Rightarrow 4x = 322 \Rightarrow x = 80.5$ [1]. (c) Angles: $108^{\circ}, 80.5^{\circ}, 100.5^{\circ}, 95^{\circ}, 156^{\circ}$. Check: $108 + 80.5 + 100.5 + 95 + 156 = 540^{\circ}$ ✓ [1].

Q8 (3 marks): (a) $(n - 2) \times 180 = 1620 \Rightarrow n - 2 = 9 \Rightarrow n = 11$ [1]. (b) Each interior $= 1620^{\circ} / 11 \approx 147.27^{\circ}$ [1]. (c) Each exterior $= 360^{\circ} / 11 \approx 32.73^{\circ}$ [1].

Stretch Challenge · +25 XP, +10 coins

Tiling the Plane

A regular polygon is said to tile the plane if copies of it can be placed edge-to-edge with no gaps or overlaps, fitting perfectly around every shared vertex. (a) The interior angle of a regular polygon must divide $360^{\circ}$ exactly for it to tile alone. Show that only three regular polygons can tile the plane by themselves: equilateral triangle, square, and regular hexagon. (b) For each, state how many copies meet at a shared vertex. (c) Explain why a regular pentagon cannot tile the plane alone.

Reveal solution

(a) For a regular polygon to tile alone, its interior angle $A$ must satisfy $360 / A$ is a whole number. Triangle: $A = 60^{\circ}$, $360 / 60 = 6$ ✓. Square: $A = 90^{\circ}$, $360 / 90 = 4$ ✓. Hexagon: $A = 120^{\circ}$, $360 / 120 = 3$ ✓. Heptagon: $A \approx 128.57^{\circ}$, $360 / 128.57$ is not an integer. Octagon: $A = 135^{\circ}$, $360 / 135 \approx 2.67$. No larger $n$ works because the interior angle keeps growing toward $180^{\circ}$. (b) 6 triangles, 4 squares, or 3 hexagons meet at each vertex. (c) Pentagon: $A = 108^{\circ}$, and $360 / 108 \approx 3.33$, not a whole number — three pentagons leave a gap of $360 - 3 \times 108 = 36^{\circ}$, but four overlap. So no exact fit.

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Quick Review

Interior sum

$S = (n - 2) \times 180^{\circ}$.

Exterior sum

Always $360^{\circ}$ (convex).

Each pair

$\text{int} + \text{ext} = 180^{\circ}$.

Regular angle

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

Find $n$

$n = \frac{360^{\circ}}{\text{ext}}$.

Memorise

Pent $108^{\circ}$, hex $120^{\circ}$, oct $135^{\circ}$.

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