EQN-04-03

Quadrilaterals & Polygons

Angle sums and the defining properties of squares, rectangles, parallelograms, and regular polygons.

~6 min read•Difficulty 2/5

sum of interior angles = (n - 2) \times 180 °

Triangle $(n = 3) : 180$ °. Quadrilateral $(n = 4) : 360$ °. Pentagon $(n = 5) : 540$ °.

For a regular polygon (all angles equal), each interior angle $= (n - 2) \cdot 180$ ° / n.

The exterior angles of ANY polygon always sum to 360°. For a regular polygon, each exterior angle $= 360$ ° / n.

Parallelogram: opposite sides equal and parallel; opposite angles equal; area $=$ base $\times$ height.
Rectangle: a parallelogram with four right angles; diagonals equal.
Rhombus: all sides equal; diagonals perpendicular.
Square: rectangle $+$ rhombus (all sides equal, all angles 90°).
Trapezoid: exactly one pair of parallel sides; area $= \frac{1}{2} (b_{1} + b_{2}) \cdot h .$

Use the given marks (equal-side ticks, right-angle squares) and the angle-sum rules — never assume a shape is regular or a square from the picture.

Diagram

A hexagon: interior angles sum to $(6 - 2) \times 180 = 720$ °.

Worked examples

Example 1 (difficulty 2/5)

What is the sum of the interior angles of a hexagon?

◎ Predict the answer first:

Example 2 (difficulty 3/5)

Each interior angle of a regular polygon is 150°. How many sides does it have?

◎ Predict the answer first:

Interior-angle sum $= (n - 2) \cdot 180$ °; exterior angles always total 360°.
For "how many sides", use exterior angle $= 360$ °/n — usually quicker than the interior formula.
A square is the only quadrilateral that is both a rectangle and a rhombus.

Using

n \cdot 180

° for the interior angle sum

Why: The "180 per side" idea is over-applied.

Fix: It is

(n - 2) \cdot 180

°, because a polygon splits into

(n - 2)

triangles.

Assuming a four-sided figure is a square because it looks like one

Why: The drawing is persuasive.

Fix: A square needs equal sides AND right angles — both must be given or proven.