Distance from the center to the midpoint of a side in a regular polygon.

Why do exterior angles sum to 360°?

Because they form a full rotation around the shape.

What is a regular polygon?

A polygon with equal sides and angles.

What are real-life examples of polygons?

Buildings, tiles, honeycombs, and road signs.

Polygon Calculator with Step-by-Step Solutions

What is a Polygon?

Polygon is a closed two-dimensional shape with straight sides. The word "polygon" comes from the Greek words "poly" meaning "many" and "gonia" meaning "angle". Polygons are fundamental geometric shapes that appear in mathematics, architecture, nature, and everyday life.

Polygons are classified based on the number of sides they have, with each type having specific properties and formulas for calculation.

Key Characteristics of Polygons:

Closed Shape: All sides connect to form a closed loop
Straight Sides: All sides are line segments (no curves)
Vertices: Points where sides meet (corners)
Interior Angles: Angles inside the polygon at each vertex
Exterior Angles: Angles formed by extending one side at each vertex
Diagonals: Line segments connecting non-adjacent vertices

Polygon Classification by Sides:

Triangle (3 sides): The simplest polygon
Quadrilateral (4 sides): Includes squares, rectangles, parallelograms
Pentagon (5 sides): Common in architecture and nature
Hexagon (6 sides): Efficient shape for tiling and honeycombs
Heptagon (7 sides): Less common but mathematically interesting
Octagon (8 sides): Used in stop signs and architecture
Nonagon (9 sides): Nine-sided polygon
Decagon (10 sides): Ten-sided polygon
n-gon: General term for polygon with n sides

Our Polygon Calculator helps you calculate properties of both regular and irregular polygons with step-by-step solutions for educational and professional use.

Types of Polygons

Understanding different polygon classifications is essential for accurate calculations:

Regular vs Irregular

Regular polygons have all sides equal and all angles equal. Irregular polygons have sides and/or angles of different measures.

Regular: All sides = s, All angles = θ
Irregular: Sides and angles vary

Convex vs Concave

Convex polygons have all interior angles less than 180°. Concave polygons have at least one interior angle greater than 180°.

Convex: All interior angles < 180°
Concave: At least one interior angle > 180°

Simple vs Complex

Simple polygons have sides that don't intersect. Complex polygons have sides that cross each other.

Simple: No intersecting sides
Complex: Self-intersecting sides

Triangles

Three-sided polygons classified as equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal), or right (one 90° angle).

Equilateral: a = b = c
Isosceles: a = b ≠ c
Scalene: a ≠ b ≠ c

Quadrilaterals

Four-sided polygons including squares, rectangles, parallelograms, trapezoids, rhombuses, and kites, each with specific properties.

Square: All sides equal, all angles 90°
Rectangle: Opposite sides equal, all angles 90°

Special Polygons

Polygons with specific mathematical properties like regular polygons, cyclic polygons (all vertices on a circle), and tangential polygons (all sides tangent to a circle).

Cyclic: All vertices on circle
Tangential: All sides tangent to circle

Polygon Formulas

Complete reference of polygon formulas for various calculations:

Area of Regular Polygon

A = (1/4) × n × s² × cot(π/n)

Calculates the area of a regular polygon with n sides of length s. Alternative formula: A = (1/2) × P × a where P is perimeter and a is apothem.

Perimeter of Polygon

P = Σ sᵢ (sum of all sides)

Calculates the perimeter by summing all side lengths. For regular polygons: P = n × s.

Interior Angles

Sum = (n-2) × 180°
Each (regular) = (n-2) × 180° / n

Calculates the sum of interior angles in an n-sided polygon. For regular polygons, calculates each interior angle.

Exterior Angles

Sum = 360°
Each (regular) = 360° / n

The sum of exterior angles is always 360° for any convex polygon. For regular polygons, calculates each exterior angle.

Shoelace Formula (Area)

A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|

Calculates the area of any polygon given its vertex coordinates (x₁,y₁), (x₂,y₂), ..., (xₙ,yₙ).

Heron's Formula (Triangle)

A = √[s(s-a)(s-b)(s-c)]
s = (a+b+c)/2

Calculates the area of a triangle when all three side lengths are known, without needing the height.

Specialized Polygon Formulas

Number of Diagonals: D = n(n-3)/2
Apothem of Regular Polygon: a = s / [2 × tan(π/n)]
Radius of Circumscribed Circle: R = s / [2 × sin(π/n)]
Radius of Inscribed Circle: r = s / [2 × tan(π/n)]
Central Angle: θ = 360° / n
Interior Angle from Exterior: Interior = 180° - Exterior

Specific Polygon Formulas

Triangle Area: A = ½ × base × height
Square Area: A = s²
Rectangle Area: A = length × width
Parallelogram Area: A = base × height
Trapezoid Area: A = ½(a+b) × h
Rhombus Area: A = ½ × d₁ × d₂
Regular Pentagon Area: A = (1/4)√(5(5+2√5)) × s²
Regular Hexagon Area: A = (3√3/2) × s²
Regular Octagon Area: A = 2(1+√2) × s²

Real-World Applications of Polygons

Polygon calculations are essential in numerous fields and everyday situations:

Architecture & Construction

Calculating floor areas for rooms and buildings
Determining material quantities (flooring, tiles, paint)
Designing roof structures and trusses
Planning land plots and property boundaries
Creating blueprints and architectural plans

Computer Graphics & Gaming

3D modeling and polygon meshes
Collision detection in video games
Terrain generation and mapping
UI element design and layout
Image processing and computer vision

Engineering & Manufacturing

CAD (Computer-Aided Design) systems
CNC machining and tool paths
Structural analysis and design
Material optimization and cutting plans
Quality control and measurement

Surveying & Geography

Calculating land areas and property boundaries
Topographic mapping and contour lines
GPS and GIS (Geographic Information Systems)
Urban planning and zoning
Environmental monitoring and analysis

Art & Design

Geometric art and patterns
Graphic design and layout
Textile and fabric design
Packaging design and optimization
Logo and icon design

Everyday Life

Planning garden layouts and landscaping
Home renovation and DIY projects
Cooking and baking (cutting shapes)
Sewing and pattern making
Sports field dimensions and layouts

Solved Polygon Examples

Step-by-step solutions to common polygon problems:

Example 1: Regular Hexagon

Calculate the area and perimeter of a regular hexagon with side length 10 cm.

1. Perimeter: P = n × s = 6 × 10 = 60 cm

2. Area formula: A = (3√3/2) × s²

3. Substitute: A = (3√3/2) × 10²

4. Calculate: A = (3√3/2) × 100

5. Result: A = 150√3 ≈ 259.81 cm²

Perimeter = 60 cm, Area ≈ 259.81 cm²

Example 2: Interior Angles

Find the sum of interior angles and each interior angle of a regular octagon.

1. Sum formula: S = (n-2) × 180°

2. Substitute: S = (8-2) × 180° = 6 × 180°

3. Calculate: S = 1080°

4. Each angle: θ = S / n = 1080° / 8

5. Result: Each interior angle = 135°

Sum = 1080°, Each angle = 135°

Example 3: Triangle Area

Calculate the area of a triangle with sides 13 cm, 14 cm, and 15 cm.

1. Heron's formula: s = (a+b+c)/2

2. Calculate s: s = (13+14+15)/2 = 21

3. Area: A = √[s(s-a)(s-b)(s-c)]

4. Substitute: A = √[21(21-13)(21-14)(21-15)]

5. Calculate: A = √[21×8×7×6] = √7056 = 84 cm²

Area = 84 cm²

Example 4: Quadrilateral

Find the area of a rectangle with length 12 m and width 8 m.

1. Rectangle area formula: A = length × width

2. Substitute: A = 12 × 8

3. Calculate: A = 96 m²

4. Perimeter: P = 2(length + width)

5. Calculate: P = 2(12+8) = 40 m

Area = 96 m², Perimeter = 40 m

Example 5: Regular Pentagon

Calculate the area of a regular pentagon with side length 7 cm.

1. Area formula: A = (1/4)√(5(5+2√5)) × s²

2. Calculate constant: √(5(5+2√5)) ≈ √(5×9.472)) ≈ √47.36 ≈ 6.882

3. Substitute: A = (1/4) × 6.882 × 7²

4. Calculate: A = (1/4) × 6.882 × 49

5. Result: A ≈ 84.30 cm²

Area ≈ 84.30 cm²

Example 6: Irregular Polygon

Find the perimeter of an irregular pentagon with sides 5, 7, 6, 8, and 9 units.

1. Perimeter formula: P = sum of all sides

2. Add sides: 5 + 7 + 6 + 8 + 9

3. Calculate: 5+7=12, 12+6=18, 18+8=26, 26+9=35

4. Result: P = 35 units

5. Note: For area, use coordinates or divide into triangles

Perimeter = 35 units

Polygon Practice Problems

Test your understanding with these practice problems:

Problem 1: Find the area of a regular hexagon with side length 8 cm.

Solution:

Area = (3√3/2) × s² = (3√3/2) × 8² = (3√3/2) × 64 = 96√3 ≈ 166.28 cm²

Problem 2: What is the sum of interior angles in a 12-sided polygon?

Solution:

Sum = (n-2) × 180° = (12-2) × 180° = 10 × 180° = 1800°

Problem 3: Calculate the perimeter of an irregular quadrilateral with sides 15 m, 20 m, 18 m, and 22 m.

Solution:

Perimeter = 15 + 20 + 18 + 22 = 75 m

Problem 4: Find the area of a triangle with base 24 cm and height 10 cm.

Solution:

Area = ½ × base × height = ½ × 24 × 10 = 120 cm²

Problem 5: How many diagonals does a decagon (10-sided polygon) have?

Solution:

Diagonals = n(n-3)/2 = 10(10-3)/2 = 10×7/2 = 35 diagonals

How to Solve Polygon Problems Step-by-Step

Follow this systematic approach to solve polygon problems:

1

Identify Polygon Type

Determine if the polygon is regular or irregular, convex or concave, and count the number of sides.

Example: Regular hexagon (6 equal sides, 6 equal angles) Type: Regular polygon, n = 6

2

Extract Given Information

Write down all given values: side lengths, angles, apothem, radius, or coordinates.

Given: Side length s = 10 cm
Number of sides n = 6
Need to find: Area and perimeter

3

Select Correct Formulas

Choose appropriate formulas based on polygon type and what you need to calculate.

Regular hexagon formulas:
Perimeter P = n × s
Area A = (3√3/2) × s²

4

Substitute Values

Replace variables in the formulas with your given values.

P = 6 × 10 = 60 cm
A = (3√3/2) × 10² = (3√3/2) × 100

5

Perform Calculations

Calculate step by step, following order of operations (PEMDAS).

A = (3√3/2) × 100
= 150√3
≈ 150 × 1.732
≈ 259.8 cm²

6

State Final Answer

Present your answer clearly with appropriate units and context.

The regular hexagon has:
Perimeter = 60 cm
Area = 150√3 cm² ≈ 259.8 cm²

Pro Tips for Polygon Calculations

Draw diagrams: Visualizing the polygon helps identify properties and relationships
Check units: Ensure all measurements use the same units before calculating
Use exact values: Keep values in radical form (√) when possible for precision
Verify formulas: Double-check you're using the correct formula for the polygon type
Estimate first: Do a rough calculation to check if your final answer is reasonable

Polygon Calculator FAQs – Area, Angles, Formulas & Properties

Learn everything about polygon calculations including area, perimeter, angles, diagonals, and real-world applications.

What is a polygon in geometry?

A polygon is a closed 2D shape made up of straight line segments. Examples include triangles, squares, pentagons, and hexagons. Polygons are classified based on the number of sides they have.

What is the difference between a polygon and a polyhedron?

A polygon is a two-dimensional shape with straight sides, while a polyhedron is a three-dimensional solid made of polygonal faces. Polygons are 2D, whereas polyhedra are 3D objects.

How do I calculate the area of a regular polygon?

The area of a regular polygon can be calculated using the formula A = (1/2) × perimeter × apothem. You can also use A = (1/4) × n × s² × cot(π/n), where n is number of sides and s is side length.

How do I calculate the area of an irregular polygon?

For irregular polygons, divide the shape into smaller triangles and sum their areas, or use the shoelace formula if coordinates are available.

What is the perimeter of a polygon?

The perimeter of a polygon is the total length of all its sides. For regular polygons, it is calculated as number of sides multiplied by side length.

What is an apothem and how do I find it?

The apothem is the distance from the center of a regular polygon to the midpoint of one of its sides. It can be calculated using trigonometric formulas such as a = s / [2 × tan(π/n)].

What is the sum of interior angles of a polygon?

The sum of interior angles of an n-sided polygon is (n − 2) × 180°. This formula works for all polygons.

Why is the sum of exterior angles always 360°?

When moving around a polygon, the total turning angle completes a full circle, which is 360°. Therefore, the sum of exterior angles is always 360°.

How many diagonals does a polygon have?

A polygon with n sides has n(n − 3)/2 diagonals. This formula counts all possible diagonals connecting non-adjacent vertices.

What is a regular vs irregular polygon?

A regular polygon has all sides and angles equal, while an irregular polygon has unequal sides or angles.

What units are used in polygon calculations?

Length units such as meters, centimeters, feet, and inches are used for sides and perimeter, while square units are used for area.

What are common real-life examples of polygons?

Polygons appear in buildings (rectangles), tiles (squares), honeycombs (hexagons), road signs (octagons), and engineering designs.

Is this polygon calculator accurate?

Yes, the calculator uses precise geometry formulas and algorithms to provide accurate results for area, perimeter, and angles.

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