Question 1

How do you find the area of a regular polygon?

Accepted Answer

For a regular polygon with n sides of length s, use: Area = (n × s²) / (4 × tan(π/n)). For example, a regular hexagon with side 5 has area = (6 × 25) / (4 × tan(30°)) ≈ 64.95 square units.

Question 2

What is the Shoelace formula for irregular polygons?

Accepted Answer

The Shoelace formula computes the area of any polygon from its vertex coordinates: Area = 0.5 × |Σ(xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|. List the vertices in order (clockwise or counter-clockwise), then apply the summation over each consecutive pair of vertices, wrapping the last back to the first.

Question 3

What is the interior angle of a regular polygon?

Accepted Answer

The interior angle of a regular n-gon is (n − 2) × 180 / n degrees. A triangle has interior angles of 60°, a square 90°, a pentagon 108°, a hexagon 120°, and so on.

Question 4

How is area of a trapezoid calculated?

Accepted Answer

Trapezoid area = 0.5 × (base1 + base2) × height. The two bases are the parallel sides; height is the perpendicular distance between them.

Name	Sides	Interior Angle	Area Formula
Triangle	3	60°	`A = (√3/4)s²`
Square	4	90°	`A = s²`
Pentagon	5	108°	`A = (s²/4)√(25+10√5)`
Hexagon	6	120°	`A = (3√3/2)s²`
Heptagon	7	128.57°	`A = (7s²/4)cot(π/7)`
Octagon	8	135°	`A = 2(1+√2)s²`
Nonagon	9	140°	`A = (9s²/4)cot(π/9)`
Decagon	10	144°	`A = (5s²/2)√(5+2√5)`
Dodecagon	12	150°	`A = 3(2+√3)s²`
n-gon (general)	n	(n−2)×180/n°	`A = (ns²)/(4tan(π/n))`

Shape	Formula	Notes
Rectangle	`A = w × h`	Width × Height
Parallelogram	`A = b × h`	Base × perpendicular height
Trapezoid	`A = ½(b₁+b₂)×h`	Average of parallel bases × height
Rhombus	`A = (d₁×d₂)/2`	Half the product of diagonals
Irregular polygon	Shoelace formula	Requires ordered vertex coordinates

Property	Formula
Interior angle (regular)	`(n−2) × 180 / n`
Sum of interior angles	`(n−2) × 180°`
Apothem (regular)	`a = s / (2 × tan(π/n))`
Circumradius (regular)	`R = s / (2 × sin(π/n))`
Number of diagonals	`n(n−3) / 2`

Separator	Comma between x and y: `(x,y)`
Vertices	One per line or separated by commas
Order	Any consistent winding (CW or CCW)
Minimum	At least 3 vertices required
Decimals	Supported: `(1.5, 3.25)`

Multi-line	`(0,0) (4,0) (4,3) (0,3)`
Single line	`(0,0),(4,0),(4,3),(0,3)`
No parentheses	`0,0 4,0 4,3 0,3`
Spaces around comma	`(0, 0), (4, 0)`

Polygon Area Calculator

Regular Polygon Reference

Special Shape Formulas

Key Polygon Properties

How to Enter Coordinates for Irregular Polygons

Format

Accepted Input Examples

Shoelace Formula Explained

The Formula

Example: Rectangle (0,0)→(4,0)→(4,3)→(0,3)

Step 1	List vertices: (x₁,y₁), (x₂,y₂), … (xₙ,yₙ)
Step 2	Sum: Σ(xᵢ × yᵢ₊₁)
Step 3	Sum: Σ(yᵢ × xᵢ₊₁)
Step 4	Area = ½ × \|Step2 − Step3\|
Note	Wrap last vertex back to first

Forward sum	0×0 + 4×3 + 4×3 + 0×0 = 24
Backward sum	0×4 + 0×4 + 3×0 + 3×0 = 0
Area	½ × \|24 − 0\| = 12 ✓
Perimeter	4 + 3 + 4 + 3 = 14
Matches	w × h = 4 × 3 = 12 ✓