048i - Interior Angle Sums of Polygons

Lesson 48 — Extending to the Interior Angle Sum of Other Polygons

Strand: Measurement | Descriptor: AC9M7M05 | Duration: 45 minutes

Learning Intentions

• To apply the triangle angle sum to determine the interior angle sum of any polygon.
• To find the size of each interior angle of a regular polygon.

Success Criteria

I can:

1. Split a polygon into triangles from a single vertex.
2. Use to find the interior angle sum.
3. Find each interior angle of a regular polygon.
4. Find a missing angle in an irregular polygon.

Warmup

(6 minutes — retrieval, mini whiteboards)

1. What is the interior angle sum of a triangle?
2. A quadrilateral is split by one diagonal into two triangles. What is its angle sum?
3. Find : a quadrilateral has angles , , and .
4. How many sides has a hexagon?

Answers: 1. ; 2. ; 3. ; 4. Six.

Bridging question: Q2 shows that a quadrilateral’s angle sum comes from splitting it into triangles. Could that method work for any polygon?

Activities

Activity 1 — The Triangulation Investigation (14 min)

Pairs, ruler, polygon templates.

For each polygon, pick one vertex and draw every diagonal from it. Count the triangles formed. Record in a table.

Polygon	Sides	Triangles	Angle sum
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon

Socratic scaffolding:

Prompt	Purpose
How many triangles does a pentagon split into?	Three.
So what is its angle sum?	.
Compare triangles to sides across your table.	The triangle count is always two fewer than the number of sides.
Why two fewer?	From your chosen vertex you cannot draw a diagonal to itself or to its two neighbours — so diagonals, which cut the shape into triangles.
Write the general rule.	.
Test it on a decagon.	.
Does the triangle fit?	✓ The rule includes the case we started from.

Completed table:

Polygon		Triangles	Angle sum
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon

Emphasise: this is not a new fact to memorise — it is the triangle angle sum, applied repeatedly. Every polygon result traces back to .

Activity 2 — Regular Polygons (10 min)

For a regular polygon, all interior angles are equal, so divide the total by :

I do — regular hexagon.

We do: Find each interior angle.

Regular polygon		Angle sum	Each angle
Triangle
Square
Pentagon
Hexagon
Octagon
Decagon

Discussion — why hexagons tile. A regular hexagon’s interior angle is , and , so three hexagons meet exactly at a point with no gap. Squares work too (), and equilateral triangles (). But a regular pentagon’s does not divide exactly, which is why pentagons cannot tile the plane on their own. This is why honeycombs are hexagonal.

You do: Find the interior angle sum, and each angle if regular.

1. A nonagon ( sides)
2. A dodecagon ( sides)
3. A regular polygon with sides
4. A regular polygon with sides

(Answers: , each ; , each ; , each ; , each .)

Activity 3 — Missing Angles in Irregular Polygons (10 min)

I do. A pentagon has angles , , , and .

You do:

1. A quadrilateral has angles , , and .
2. A hexagon has five angles of , , , , and one unknown.
3. A pentagon has three angles of and two equal unknown angles. Find each.
4. A regular polygon has interior angles of . How many sides has it?

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what is the unknown?	The number of sides.
What do you know about each angle?	Each is , and there are of them.
Write the total two ways.	Total , and total .
Set them equal.	.
Solve.	, so .
Check.	, and ✓

(Answers: 1. ; 2. ; 3. , so each is ; 4. sides.)

Checks for Understanding

(5 minutes — exit ticket)

1. Find the interior angle sum of a heptagon.
2. Find each interior angle of a regular octagon.
3. A quadrilateral has angles , , and . Find .
4. A regular polygon has interior angles of . How many sides?
5. Reasoning. Explain why the formula works, referring to triangles.

Answers: 1. ; 2. ; 3. ; 4. , so ; 5. Drawing every diagonal from one vertex splits an -sided polygon into triangles, and each contributes .

Common Misconceptions

Misconception	How to pre-empt it
Using instead of .	Always count the triangles on a diagram before substituting.
Dividing by for an irregular polygon.	Division only applies when all angles are equal. Check for regularity first.
Confusing interior with exterior angles.	This lesson concerns interior angles only. Label them explicitly on diagrams.
Believing the formula fails for concave polygons.	It still holds, provided reflex interior angles are counted properly. Mention briefly.
Drawing diagonals from several vertices, over-counting triangles.	The method uses one vertex only. Model it carefully.
Assuming any regular polygon tiles the plane.	Only those whose interior angle divides exactly — a good discussion.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is the interior angle sum of a polygon with sides?

Answer

.

E2 (AMC Junior style). A regular polygon has interior angles of . How many sides has it?

Answer

A decagon.

E3 (Challenge). A polygon’s interior angle sum is . How many sides has it?

Answer

A hendecagon.

E4 (Challenge). Explain why a regular pentagon cannot tile the plane, but a regular hexagon can.

Answer

Tiles meeting at a point must have angles summing to exactly . A regular hexagon’s angle is , and ✓. A regular pentagon’s is , and — not a whole number, so pentagons always leave a gap or overlap.

E5 (Challenge). In a pentagon, the angles are in the ratio . Find all five.

Answer

The parts total , and the angle sum is , so one part is . The angles are , , , and . (Check: they total ✓)

Homework

1. Find the interior angle sum of: (a) a pentagon (b) an octagon (c) a decagon (d) a polygon with sides.
2. Find each interior angle of a regular: (a) pentagon (b) hexagon (c) nonagon (d) dodecagon.
3. Find the missing angle: (a) a quadrilateral with , , (b) a pentagon with , , , (c) a hexagon with , , , , .
4. A regular polygon has interior angles of (a) (b) (c) . Find the number of sides in each case.
5. A polygon’s interior angle sum is (a) (b) . Find the number of sides.
6. A pentagon has two angles of and three equal unknown angles. Find each unknown angle.
7. Reasoning. Explain why the interior angle of a regular polygon gets closer to as the number of sides increases.
8. Reasoning. Which regular polygons tile the plane on their own? Justify using interior angles.
9. Challenge. A hexagon has angles in the ratio . Find all six.
10. Challenge. A regular polygon has an interior angle that is exactly times its exterior angle. Find the number of sides. (Hint: interior + exterior .)

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — (a) (b) (c) . Q4 — (a) (b) (c) . Q5 — (a) (b) . Q6 — , so each is . Q7 — the total grows by per extra side while the number of angles grows by one, so each angle approaches but never reaches . Q8 — equilateral triangle (), square () and regular hexagon () — these are the only interior angles dividing exactly. Q9 — parts total , angle sum , so one part is ; angles . Q10 — interior exterior and they sum to , so exterior ; since exterior angles of a regular polygon total , sides.