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Angles & Polygons

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Form 1 Pathway: N/A

First Principles

Objective: Master the geometry of interior and exterior angles in both regular and irregular polygons from fundamental principles.

The Walker's Analogy: Picture yourself walking along the boundary of a fenced shamba in Eldoret. At every corner post, you turn by an exterior angle. When you return to your starting corner facing your original direction, your body has completed exactly one full rotation — \(360^\circ\). This is why the exterior angles of any convex polygon always add up to \(360^\circ\), regardless of how many sides it has!


Interior Angle Sum
540°
Each Interior Angle
108°
Each Exterior Angle
72°
Exterior Sum
360°

Move the slider or click the button to see the angles calculate dynamically.

1. Triangulation: Where \((n-2)\times180^\circ\) Comes From

Pick any single corner (vertex) of an \(n\)-sided polygon and draw straight lines (diagonals) to all other non-adjacent corners. You will divide the polygon into exactly \((n-2)\) non-overlapping triangles.

Since the interior angles of every triangle add up to \(180^\circ\), the sum of all interior angles of an \(n\)-sided polygon is:

\[\text{Sum of Interior Angles} = (n - 2) \times 180^\circ\]

2. Straight-Line Relationship at Every Vertex

At any single corner of a polygon, extending one edge creates a straight line. The interior angle and the exterior angle sit together on this straight line, meaning they are supplementary:

\[\text{Interior Angle} + \text{Exterior Angle} = 180^\circ\]

Key Formulas

1. Sum of Interior Angles

\[S_n = (n - 2) \times 180^\circ \quad \text{or} \quad S_n = (2n - 4) \times 90^\circ\]

Applies to all polygons (both regular and irregular) with \(n\) sides.

2. Interior Angle of a Regular Polygon

\[I = \frac{(n - 2) \times 180^\circ}{n} = 180^\circ - E\]

Where all \(n\) interior angles and sides are equal.

3. Sum of Exterior Angles

\[\sum E = 360^\circ\]

True for any convex polygon, regardless of the number of sides.

4. Exterior Angle of a Regular Polygon

\[E = \frac{360^\circ}{n}\]

5. Finding the Number of Sides (\(n\))

\[n = \frac{360^\circ}{\text{Exterior Angle}} = \frac{360^\circ}{180^\circ - I}\]

Worked Examples

EASY (Level 1)

Problem: Calculate the size of each interior angle of a regular hexagon (6 sides).

Solution:

  1. Method 1 (Using the Interior Sum formula): \[\text{Sum} = (n - 2) \times 180^\circ = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ\] Divide equally among the 6 vertices: \[\text{Each interior angle} = \frac{720^\circ}{6} = 120^\circ\]
  2. Method 2 (Using Exterior Angles — Faster!): \[\text{Exterior angle } E = \frac{360^\circ}{6} = 60^\circ\] \[\text{Interior angle } I = 180^\circ - 60^\circ = 120^\circ\]

Answer: \(120^\circ\)

MEDIUM (Level 2)

Problem: Each interior angle of a regular polygon is \(144^\circ\). Determine the number of sides the polygon has.

Solution:

  1. Find the size of each exterior angle: Since interior and exterior angles sum to \(180^\circ\): \[E = 180^\circ - 144^\circ = 36^\circ\]
  2. Apply the exterior sum property: The sum of all exterior angles is always \(360^\circ\): \[n = \frac{360^\circ}{E} = \frac{360^\circ}{36^\circ} = 10\]

Answer: The polygon has 10 sides (a regular decagon).

HARD (Level 3)

Problem: The interior angles of an irregular hexagon are \(x^\circ\), \((x + 10)^\circ\), \((x + 20)^\circ\), \((x + 30)^\circ\), \(2x^\circ\), and \((2x - 20)^\circ\). Find the value of the smallest interior angle.

Solution:

  1. Calculate the theoretical interior angle sum for \(n = 6\): \[S_6 = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ\]
  2. Set up the algebraic equation: \[x + (x + 10) + (x + 20) + (x + 30) + 2x + (2x - 20) = 720\]
  3. Combine like terms: \[(x + x + x + x + 2x + 2x) + (10 + 20 + 30 - 20) = 720\] \[8x + 40 = 720\]
  4. Solve for \(x\): \[8x = 720 - 40 = 680\] \[x = \frac{680}{8} = 85\]
  5. Identify the smallest angle: Compare all angle expressions: \(x = 85^\circ\), \(x+10 = 95^\circ\), \(x+20 = 105^\circ\), \(x+30 = 115^\circ\), \(2x = 170^\circ\), and \(2x - 20 = 150^\circ\). The smallest angle is \(x = 85^\circ\).

Answer: Smallest interior angle = \(85^\circ\)

Common Mistakes

Common Mistake Believing that the interior angles of all polygons add up to \(360^\circ\).

Correction Only 4-sided quadrilaterals have an interior angle sum of \(360^\circ\). For any polygon, the interior angle sum grows with the number of sides: \((n - 2) \times 180^\circ\). The number \(360^\circ\) is constant only for the exterior angle sum.

Why it feels right Because \(360^\circ\) represents a complete circle, learners frequently memorize \(360^\circ\) as the default angle sum for shapes.

Common Mistake Confusing the exterior angle with the reflex angle \((360^\circ - \text{interior})\).

Correction The exterior angle is formed by extending one edge in a straight line. Therefore, \(\text{Interior} + \text{Exterior} = 180^\circ\), not \(360^\circ\).

Why it feels right The word "exterior" sounds like "everything on the outside of the vertex", causing students to measure the entire outside arc around the vertex.

Common Mistake Dividing \((n - 2) \times 180^\circ\) by \(n\) for irregular polygons.

Correction You can only divide by \(n\) to find an individual angle if the polygon is regular (equiangular and equilateral). For irregular polygons, individual angles must be solved using algebraic equations or specific given data.

Why it feels right It is easy to assume formulas from regular shapes apply uniformly to all polygons.

Real World

Surveying and Land Subdivisions in Kenya: Land surveyors dividing shambas in Kiambu or Nakuru into polygonal plots verify boundary traverses using \((n-2)\times180^\circ\). If the sum of measured bearings differs from the theoretical sum, there is a survey error on the ground.
Gazebo and Bandas Architecture: Octagonal and hexagonal bandas found in eco-lodges across the Maasai Mara rely on exact corner roof joist angles (e.g., \(135^\circ\) for regular octagons) so that timber joints meet flush without structural weakness.
Road Roundabout and Pavement Tiling: Hexagonal interlocking pavers fit together seamlessly without gaps because each interior angle is \(120^\circ\). Three tiles meeting at a vertex create \(3 \times 120^\circ = 360^\circ\), creating a tessellation that withstands heavy lorry traffic.
Satellite Antenna Dishes: High-precision satellite tracking stations use polygonal ring mounts to distribute tension symmetrically across all pivot points.

Practice

Find the sum of the interior angles of a hexagon (a 6-sided polygon) in degrees. (Type only the number, e.g., 540)
Review the concepts above.
Calculate the size of each exterior angle of a regular pentagon (5 sides) in degrees. (Type only the number, e.g., 60)
Review the concepts above.
Find the size of each interior angle of a regular octagon (8 sides) in degrees. (Type only the number, e.g., 120)
Review the concepts above.
The sum of the interior angles of a polygon is 1440 degrees. How many sides does the polygon have? (Type only the number, e.g., 10)
Review the concepts above.
Each interior angle of a regular polygon is 156 degrees. Calculate the number of sides of this polygon. (Type only the number, e.g., 15)
Review the concepts above.
In a nonagon (9-sided polygon), three of the interior angles are 120 degrees each, and the remaining six interior angles are all equal to x degrees. Find the value of x. (Type only the number, e.g., 150)
Review the concepts above.