Polygons & Area Techniques | Math Learning Guide

📋 Prerequisites

Polygons are everywhere in competition geometry — from simple quadrilaterals to irregular shapes you have to dissect. The key strategies: know your special quadrilaterals, master the area formulas, and learn to carve weird shapes into triangles and rectangles you can actually compute.

📚 Key Concepts

Interior & Exterior Angles

For any n-sided polygon (n-gon):

Sum of interior angles: $ (n - 2) \times 180° $
Each interior angle (regular n-gon): $ \frac{(n-2) \times 180°}{n} $
Sum of exterior angles: always $ 360° $ (for any convex polygon!)
Each exterior angle (regular n-gon): $ \frac{360°}{n} $

$Triangle check: n=3 \\to (3-2) \\times 180 = 180$°. ✓

$Quadrilateral check: n=4 \\to (4-2) \\times 180 = 360$°. ✓

Special Quadrilaterals

Parallelogram: Both pairs of opposite sides parallel

Opposite sides equal, opposite angles equal
Diagonals bisect each other
$Area = base \\times height$

Rectangle: Parallelogram with right angles

Diagonals are equal
$Area = length \\times width$

Rhombus: Parallelogram with all sides equal

Diagonals are perpendicular bisectors
Area = $ \frac{d_1 \times d_2}{2} $ (half the product of diagonals)

Square: Rectangle + rhombus (all sides equal, all angles 90°)

Area = $side^{2}$
$Diagonal = side \\times \sqrt{2}$

Trapezoid: Exactly one pair of parallel sides (the bases)

Area = $ \frac{b_1 + b_2}{2} \times h $ (average of the two bases $\times$ height)

Kite: Two distinct pairs of adjacent sides equal

Diagonals are perpendicular
Area = $ \frac{d_1 \times d_2}{2} $ (same as rhombus)

Area Formula Summary

Triangle: $ \frac{1}{2}bh $
Rectangle/Square: $ lw $ or $ s^2 $
Parallelogram: $ bh $
Trapezoid: $ \frac{b_1+b_2}{2}h $
Rhombus/Kite: $ \frac{d_1 d_2}{2} $
Regular n-gon: $ \frac{1}{2} \times \text{perimeter} \times \text{apothem} $
Equilateral triangle: $ \frac{\sqrt{3}}{4}s^2 $

Dissection / Cut-and-Paste Method

For irregular shapes:

Cut it up into triangles, rectangles, and other known shapes
Calculate each piece's area separately
Add them up

Alternative: Subtraction method — surround the shape with a rectangle, subtract the areas you don't want.

Shoelace Formula

For any polygon when you know the coordinates of all vertices $(x_1,y_1), (x_2,y_2), ..., (x_n,y_n)$ in order:

\[ \text{Area} = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right| \]

Where $ (x_{n+1}, y_{n+1}) = (x_1, y_1) $ (loop back to the start).

Mnemonic: list the coordinates in order, repeat the first at the end, cross-multiply diagonally, subtract, take absolute value, halve it.

Pick's Theorem

For a simple polygon whose vertices are lattice points (grid points):

\[ \text{Area} = I + \frac{B}{2} - 1 \]

Where:

$I$ = number of interior lattice points
$B$ = number of boundary lattice points

Great for grid-based problems where counting points is easier than calculating dimensions.

⚠️ Common Mistake: For the area of a trapezoid, use the average of the two bases $times height, not just base \\times height. And for a rhombus/kite, area is half the product of the diagonals$ — don't forget the 1/2! Also: interior angle formula gives the total sum, divide by n for each angle (only for regular polygons).

💡 Key Insight: When a geometry problem asks for area and the shape is weird, don't panic. Ask: can I draw lines to split this into triangles and rectangles? Can I put a box around it and subtract the corners? Can I use the shoelace formula if I assign coordinates? Competition problems almost always have a clean dissection — look for right angles, symmetry, and hidden triangles.

✏️ Example Problems

📝 Example 1 (Trapezoid area)

A trapezoid has bases of length 8 and 14, and height 6. What is its area?

Formula: Area of trapezoid = $ \frac{b_1 + b_2}{2} \times h $

Step 1: Identify the values.

$ b_1 = 8 $, $ b_2 = 14 $, $ h = 6 $

Step 2: Plug into formula.

\[ \text{Area} = \frac{8 + 14}{2} \times 6 = \frac{22}{2} \times 6 = 11 \times 6 = 66 \]

Intuition: It's the average of the two parallel sides, times the distance between them.

Answer: 66

📝 Example 2 (Interior angles of a polygon)

What is the measure of each interior angle of a regular hexagon (6 sides)?

Step 1: Find the total sum of interior angles.

Sum = $ (n - 2) \times 180° $

For n = 6:

\[ \text{Sum} = (6 - 2) \times 180° = 4 \times 180° = 720° \]

Step 2: Divide by number of angles (regular hexagon: all angles equal).

\[ \text{Each angle} = \frac{720°}{6} = 120° \]

Quick check with exterior angles: Each exterior angle = 360°/6 = 60°, so interior = 180° - 60° = 120°. ✓

Answer: 120

📝 Example 3 (Dissection — area of composite shape)

$A rectangle of 10 \\times 8 has a right triangle with legs 3 and 4 cut out from one corner.$
What is the area of the remaining shape?

Strategy: Subtraction method — total area minus the cut-out area.

Step 1: Area of the rectangle.

$ \text{Area}_{rect} = 10 \times 8 = 80 $

Step 2: Area of the cut-out triangle.

$ \text{Area}_{tri} = \frac{1}{2} \times 3 \times 4 = 6 $

Step 3: Subtract.

\[ \text{Remaining area} = 80 - 6 = 74 \]

Answer: 74