Understanding Shapes
ICSE Class 8 · Mathematics · Chapter 16
What this chapter covers
- Curves and polygons
- Parts of a polygon
- Convex and concave polygons
- Regular polygons
- Angle sum property of a quadrilateral
- Interior angles of a polygon
- Exterior angles of a polygon
- Angles of a regular polygon
- Common mistakes
Key points
- A polygon is a simple closed figure made up only of straight line segments, named by its number of sides.
- A diagonal joins two vertices that are not adjacent; a triangle has no diagonals.
- In a convex polygon every interior angle is less than 180°; a concave polygon has at least one reflex interior angle.
- A regular polygon has all sides equal and all angles equal; it is always convex.
- The four angles of any quadrilateral add up to 360°.
- The interior angles of an n-sided polygon add up to (n - 2) × 180°.
- At each vertex, interior angle + exterior angle = 180°.
- The exterior angles of any convex polygon add up to 360°.
- In a regular n-gon, each exterior angle is 360° ÷ n and each interior angle is 180° minus that.
- The number of sides of a regular polygon is 360° divided by one exterior angle, and it must be a whole number.
Questions answered
Is a circle a polygon? Give a reason.
No. A polygon is made up only of straight line segments, and a circle has no straight parts.
How many diagonals does a quadrilateral have, and why does a triangle have none?
A quadrilateral has 2 diagonals. A triangle has none because each of its vertices is adjacent to both of the others.
A quadrilateral has one interior angle of 200°. Is it convex or concave?
Concave, because one of its interior angles is greater than 180°.
Why is a rectangle with sides 6 cm and 4 cm not a regular polygon?
Its angles are all equal, but its sides are not all equal. A regular polygon needs both.
Three angles of a quadrilateral are each 85°. Find the fourth angle.
360° - 3 × 85° = 360° - 255° = 105°.
Find the sum of the interior angles of a heptagon.
(7 - 2) × 180° = 900°.
The angle sum of a polygon is 1800°. How many sides does it have?
(n - 2) × 180 = 1800, so n - 2 = 10 and n = 12.
Each exterior angle of a regular polygon is 24°. Find the number of sides and each interior angle.
n = 360 ÷ 24 = 15. Each interior angle is 180° - 24° = 156°.