Geometry (Classes VI–VIII) • Topic 4 of 6

Quadrilaterals & Polygons

A quadrilateral's four interior angles always add to 360°, because any quadrilateral splits into two triangles (2 × 180°). The general formula every Paper II candidate needs is the interior-angle sum of an n-sided polygon = (n − 2) × 180°; for a regular polygon each interior angle is that sum divided by n, and — the fact people forget — the exterior angles of ANY polygon always total 360°, so each exterior angle of a regular n-gon is 360° ÷ n. CTET also loves the family relationships among special quadrilaterals: every square is a rectangle and a rhombus; every rectangle and rhombus is a parallelogram; a square is the most special of all. The pedagogy hinges on this hierarchy, because van Hiele Level 1 (Analysis) is exactly where a child can list a shape's properties but a Level 2 child can reason 'a square must be a rectangle because it has all the rectangle's properties'. The stubborn misconception is denying inclusion — children insist 'a square is not a rectangle' because the words look different. CTET tests the (n − 2) × 180° formula, the 360° quadrilateral sum, exterior angles of regular polygons, and the square-rectangle inclusion question as both content and pedagogy.

✅ Solved examples

1. Three angles of a quadrilateral are 90°, 85° and 95°. Find the fourth angle.
The four interior angles of a quadrilateral add to 360°. Fourth angle = 360° − (90° + 85° + 95°) = 360° − 270° = 90°.
2. What is the sum of the interior angles of a regular hexagon, and what is each interior angle?
For n = 6, interior-angle sum = (n − 2) × 180° = 4 × 180° = 720°. Each interior angle of the regular hexagon = 720° ÷ 6 = 120°.
3. Each exterior angle of a regular polygon is 40°. How many sides does it have?
The exterior angles of any polygon sum to 360°, so the number of sides = 360° ÷ 40° = 9. It is a regular nonagon.
4. A pupil argues 'a square cannot be a rectangle because they have different names'. How should the teacher correct the underlying idea?
By definition, a rectangle is a quadrilateral with four right angles; a square has four right angles (and equal sides), so it satisfies every rectangle property — hence every square IS a rectangle. The teacher should build the shape hierarchy with overlapping property lists, not rely on the names.

✏️ Practice — try these, take hints as needed

1. The sum of the interior angles of a pentagon (5 sides) is:
Use (n − 2) × 180°.
(5 − 2) × 180.
540°
2. Three angles of a quadrilateral are 100°, 80° and 100°. The fourth angle is:
All four add to 360°.
360 − 280.
80°
3. Each interior angle of a regular octagon (8 sides) measures:
Sum = (8 − 2) × 180° = 1080°.
Divide by 8.
135°
4. The sum of the exterior angles of any convex polygon, taken one per vertex, is:
It does not depend on the number of sides.
A full turn.
360°

📝 Topic test — 8 questions

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