Grade 9 Angles of Polygons: Short Notes

9 ශ්‍රේණිය · 3 වාරය · 25 පරිච්ඡේදය · බහු අස්‍රවල කෝණ

48%

of Grade 9 term tests include it (19 of 40)

169

marks in past papers (30 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Tell interior and exterior angles apart, and use interior + exterior = 180°
Find the sum of the interior angles of any polygon
Use the fact that the exterior angles of a polygon add up to 360°
Find the angles of a regular polygon
Find the number of sides from an angle or an angle sum

The Vesak lantern frame

Nimal is making an eight-sided Vesak lantern from bamboo strips. For the frame to close neatly, every corner has to be the same angle. Which angle? With one formula you can work it out before you cut a single strip.

Interior and exterior angles

A polygon is a closed shape with straight sides. A polygon with n sides has n vertices.• Interior angle: the angle inside the polygon at a vertex.• Exterior angle: produce one side beyond the vertex; the angle between that extension and the next side.They sit on a straight line, so interior + exterior = 180°.

Cut it into triangles

From one vertex A, the diagonals cut a pentagon into 3 triangles. Each has 180°, so the interior angles add up to 3 × 180° = 540°. An n-sided polygon always gives n − 2 triangles.

🧮 Sum of interior angles

n is the number of sides. Quadrilateral: 360°, pentagon: 540°, hexagon: 720°. Use it to find a missing angle, or work backwards to find n.

S = (n − 2) × 180°

✏️ A missing angle

A pentagon has interior angles 100°, 110°, 95°, 120° and x. Find x.

1Sum for n = 5: (5 − 2) × 180° = 540°.
2Known angles: 100 + 110 + 95 + 120 = 425°.
3x = 540° − 425° = 115°.

✅ x = 115°

Walk around the edge

Walk around any polygon. At each corner you turn through the exterior angle. After one lap you face the way you started: one full turn. So the exterior angles of any polygon add up to 360°.

⚠️ Exterior angles always make 360°

Examiners found students who did not know that the exterior angles of a polygon sum to 360°, or who thought the total grows with more sides. It doesn't: triangle, hexagon or 20-gon, the exterior angles always add up to 360°. Only the interior sum grows.

🧮 Regular polygons

In a regular polygon all sides and all angles are equal, so the 360° is shared equally among the n exterior angles. Find the exterior angle first: it is the quickest route.

exterior = 360° ÷ n interior = 180° − exterior

✏️ Nimal's lantern

Nimal's lantern frame is a regular octagon. Find (i) each exterior angle (ii) each interior angle.

1(i) Exterior = 360° ÷ 8 = 45°.
2(ii) Interior = 180° − 45° = 135°.
3Check: 8 × 135° = 1080° = (8 − 2) × 180° ✓

✅ (i) 45° (ii) 135°

⚠️ Can that polygon exist?

Can a regular polygon have interior angles of 100°? Exterior = 80°, and 360 ÷ 80 = 4.5. A polygon can't have 4.5 sides, so no. The number of sides must be a whole number of at least 3. Students who skip this check write answers like 'n = 4.5'.

📌 Remember

• Interior + exterior = 180° at every vertex.• Sum of interior angles = (n − 2) × 180°.• Sum of exterior angles = 360° for every polygon.• Regular polygon: exterior = 360° ÷ n, then interior = 180° − exterior.• Number of sides: n = 360° ÷ exterior angle; it must be a whole number.

🧾 Quick revision sheet

Sum of interior angles of an n-sided polygon	(n − 2) × 180°
Sum of exterior angles of any polygon	360°
Interior + exterior at a vertex	180° (angles on a straight line)
Interior angle of a regular pentagon	108° (exterior 72°)
Interior angle of a regular hexagon	120° (exterior 60°)
Number of sides from the exterior angle	n = 360° ÷ exterior angle
Sum of interior angles of a hexagon	720°