Area & Perimeter Calculator - Calculate Any Shape with Formulas Free

Calculate area and perimeter for any shape. Free geometry tool for rectangles, circles, triangles, and polygons with step-by-step formulas and solutions.

Units:

Unit

All inputs and outputs use the selected unit

Select Shape

Rectangle

A quadrilateral with four right angles and opposite sides equal.

Rectangles are fundamental in architecture, used in doors, windows, and floor plans.

The golden rectangle (ratio 1:1.618) is considered aesthetically pleasing in design.

Area: A = length × width

Perimeter: P = 2(length + width)

Measurements

Length

Width

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Need to find the area of a room or determine the boundary length of a garden? Whether you're working on homework, planning a home improvement project, or solving a professional design challenge, this calculator helps you determine both area and perimeter for over 30 different shapes. From simple rectangles and circles to complex irregular polygons and ovals, you'll get accurate results with step-by-step explanations. The tool handles everything from basic geometric forms to specialized shapes like ellipses, composite L and T forms, and even custom irregular polygons using coordinate points. Simply select your shape, enter the measurements, and get instant results with formulas explained clearly.

How to Use Area & Perimeter Calculator

Select a Shape

Browse shapes by category to find what you need. Basic forms include rectangles, squares, and circles. For triangles, choose from right, equilateral, isosceles, or use Heron's formula. Explore regular polygons like pentagons and hexagons, curved forms such as ellipses and sectors, composite shapes like L and T forms, or special geometries including kites and star patterns.

Enter Measurements

Provide the measurements needed for your chosen shape. You can work in metric units like millimeters, centimeters, meters, or kilometers, or use imperial units such as inches, feet, yards, or miles. The tool checks your entries and will alert you if any values need correction.

View Results

After clicking "Calculate", you'll see the area and perimeter values. When applicable, the tool also shows related measurements such as a circle's diameter or a right triangle's hypotenuse. You can copy any result with a single click. Every shape includes helpful information about its properties and where you might encounter it in practice.

Calculator Features

📐

Wide Variety of Shapes

From simple rectangles to complex composite forms

📏

Multiple Units

Support for metric (mm, cm, m, km) and imperial (in, ft, yd, mi) units

✅

Input Validation

Automatic validation with helpful error messages for invalid inputs

📋

Copy Results

One-click copy to clipboard for area and perimeter values

📚

Helpful Information

Learn about each shape's properties and where it appears in everyday life

🔢

Extra Details

See related measurements like diameter, hypotenuse, or arc length when they apply

🎯

Accurate Results

Reliable calculations based on standard mathematical formulas

📱

Responsive Design

Works perfectly on desktop, tablet, and mobile devices

Complete Function List

Over 30 different geometric forms from basic to advanced:
Eight unit options covering metric and imperial systems:
Input checking with clear error messages:
Quick copy function for calculated values:
Background information about each shape:
Related measurements shown when applicable:
Shapes grouped by type for easier browsing:
Works well on phones, tablets, and computers:
Precise mathematical calculations:
Automatic triangle validation:
Coordinate-based input for custom polygons:
Support for combined shapes like L and T forms:
Unique geometries including star patterns and curved forms:

Common Calculations & Examples

Example 1: Calculate Rectangle Area and Perimeter

Problem: Find the area and perimeter of a rectangle with length 8 cm and width 5 cm

Steps:

Select "Rectangle" from Basic Shapes
Enter length: 8
Enter width: 5
Select unit: cm
Click "Calculate"

Result: Area: 40 cm², Perimeter: 26 cm

Explanation: Multiply the length by the width to get the area: 8 × 5 = 40 square centimeters. For the perimeter, add the length and width, then double the result: 2(8 + 5) = 26 centimeters. These calculations help when planning floor layouts or estimating materials for rectangular areas.

Example 2: Calculate Circle Area and Perimeter

Problem: Find the area and circumference of a circle with radius 6 cm

Steps:

Select "Circle" from Basic Shapes
Enter radius: 6
Select unit: cm
Click "Calculate"

Result: Area: 113.10 cm², Perimeter: 37.70 cm

Explanation: Square the radius and multiply by π to find the area: π × 6² ≈ 113.10 square centimeters. The circumference equals 2π times the radius, giving approximately 37.70 centimeters. The tool also displays the diameter, which is 12 centimeters. These measurements are important when working with round objects such as wheels, pipes, or circular furniture.

Example 3: Calculate Right Triangle Area and Perimeter

Problem: Find the area and perimeter of a right triangle with legs 3 cm and 4 cm

Steps:

Select "Right Triangle" from Triangles
Enter leg a: 3
Enter leg b: 4
Select unit: cm
Click "Calculate"

Result: Area: 6 cm², Perimeter: 12 cm, Hypotenuse: 5 cm

Explanation: Multiply the two legs and divide by two for the area: (3 × 4) / 2 = 6 square centimeters. Use the Pythagorean theorem to find the hypotenuse: √(3² + 4²) = √25 = 5 centimeters. Add all three sides for the perimeter: 3 + 4 + 5 = 12 centimeters. This example shows the well-known 3-4-5 right triangle pattern.

Example 4: Calculate Regular Hexagon Area and Perimeter

Problem: Find the area and perimeter of a regular hexagon with side length 4 cm

Steps:

Select "Regular Hexagon" from Regular Polygons
Enter side length: 4
Select unit: cm
Click "Calculate"

Result: Area: 41.57 cm², Perimeter: 24 cm

Explanation: For a regular hexagon, the area formula is (3√3 × side²) / 2, which gives approximately 41.57 square centimeters when the side is 4. The perimeter is simply six times the side length, resulting in 24 centimeters. Hexagons are efficient for packing space, as seen in honeycomb structures, and are commonly used in engineering designs.

Example 5: Finding the Perimeter of a Rectangular Garden

Problem: You need to install fencing around a rectangular garden that measures 12 feet by 8 feet. How much fencing material do you need?

Steps:

Select "Rectangle" from Basic Shapes
Enter length: 12
Enter width: 8
Select unit: ft
Click "Calculate"

Result: Area: 96 ft², Perimeter: 40 ft

Explanation: The perimeter tells you the total distance around the garden, which is exactly what you need for fencing. For this rectangle, the boundary length is 2(12 + 8) = 40 feet. This means you'll need 40 feet of fencing material to go around the entire garden. The area (96 square feet) helps you know how much space you have for planting. This type of calculation is essential for any project involving boundaries, whether it's fencing, edging, or trim work.

Example 6: Calculating Both Area and Perimeter for a Room

Problem: You're planning to install new flooring and baseboards in a room that's 15 feet long and 10 feet wide. Find both the floor area and the perimeter for baseboard installation.

Steps:

Select "Rectangle" from Basic Shapes
Enter length: 15
Enter width: 10
Select unit: ft
Click "Calculate"

Result: Area: 150 ft², Perimeter: 50 ft

Explanation: The area calculation (15 × 10 = 150 square feet) tells you how much flooring material to purchase. The perimeter (2(15 + 10) = 50 feet) shows how much baseboard trim you'll need to go around the room. Having both values helps you plan your project accurately and avoid buying too much or too little material. This is a common scenario in home improvement where you need both measurements for different purposes - one for surface coverage and one for linear trim or edging.

Example 7: Determining the Boundary Length of an Oval Pool

Problem: An oval swimming pool has a length (semi-major axis) of 20 feet and a width (semi-minor axis) of 12 feet. Find the area for pool liner and the perimeter for coping stones.

Steps:

Select "Ellipse" from Curved Shapes
Enter semi-major axis (a): 20
Enter semi-minor axis (b): 12
Select unit: ft
Click "Calculate"

Result: Area: 753.98 ft², Perimeter: 101.31 ft

Explanation: The area calculation (π × 20 × 12 ≈ 753.98 square feet) helps you determine how much pool liner material you'll need to cover the bottom and sides. The perimeter (approximately 101.31 feet) tells you how many coping stones you need if each stone covers 1 foot. Ovals and ellipses use more complex formulas than circles, but this calculator handles the calculations automatically. The perimeter of an ellipse requires special formulas because it's not as simple as a circle's circumference, making this tool particularly valuable for curved shapes.

Example 8: Working with Irregular-Shaped Plots

Problem: A land plot has four corners with coordinates: (0,0), (30,0), (30,20), and (0,20) feet. Calculate both the area and perimeter of this irregular quadrilateral.

Steps:

Select "Irregular Polygon" from Special Shapes
Enter coordinates as JSON: [{"x":0,"y":0},{"x":30,"y":0},{"x":30,"y":20},{"x":0,"y":20}]
Select unit: ft
Click "Calculate"

Result: Area: 600 ft², Perimeter: 100 ft

Explanation: For irregular shapes, you provide coordinate points instead of standard measurements. The calculator uses the shoelace method to find the area and calculates the perimeter by measuring distances between consecutive points. In this case, it's actually a rectangle, but the coordinate method works for any polygon shape. This approach is invaluable for land surveys, custom designs, or any situation where you have GPS coordinates or survey data rather than simple length and width measurements. The tool handles the complex geometry automatically.

How do you calculate perimeter?

The method depends on the shape you're working with. For rectangles and squares, add all four sides together, or use the formula 2(length + width) for rectangles. For circles, multiply the diameter by π (approximately 3.14159), or use 2π × radius. Triangles require adding all three side lengths. Regular polygons like hexagons or octagons simply multiply the side length by the number of sides. The key is identifying which measurements you have available - some shapes need side lengths, others need radius or diameter, and some require multiple measurements. This calculator automatically selects the right formula based on your chosen shape and entered values, so you don't need to memorize formulas.

Can you convert perimeter to area?

Generally, you cannot directly convert perimeter to area because they measure different things - perimeter is a linear measurement (distance around) while area is a two-dimensional measurement (space inside). However, for specific shapes with known relationships, you can work backwards. For example, if you know a square's perimeter, you can divide by 4 to get the side length, then square it to find the area. For circles, if you know the circumference (perimeter), you can find the radius by dividing by 2π, then calculate area using π × radius². The relationship depends entirely on the shape's geometry. This calculator helps by allowing you to calculate both values when you have the necessary measurements.

How do I find the perimeter of an irregular shape?

For irregular shapes, you'll use the irregular polygon option in this calculator. You'll need to provide the coordinates of each corner point (vertex) of your shape. Enter them as a JSON array format like [{"x":0,"y":0},{"x":4,"y":0},{"x":4,"y":4},{"x":0,"y":4}]. The tool uses the shoelace method to accurately determine the area, and it calculates the boundary length by measuring the distance between each consecutive point and summing them up. This approach works for any polygon, no matter how complex or irregular, as long as you can provide the coordinate points. It's particularly useful for land surveys, custom designs, or any situation where you have coordinate data rather than standard measurements.

What's the formula for calculating the perimeter of an oval?

Ovals and ellipses use a more complex formula than circles. The exact perimeter of an ellipse requires an infinite series calculation, but there are good approximations. The most commonly used approximation is Ramanujan's formula: P ≈ π[3(a + b) - √((3a + b)(a + 3b))], where 'a' is the semi-major axis (half the longest diameter) and 'b' is the semi-minor axis (half the shortest diameter). For practical purposes, a simpler approximation is P ≈ π × √[2(a² + b²)]. This calculator uses precise methods to give you accurate results for ellipses and ovals. Simply enter the semi-major and semi-minor axis lengths, and you'll get both the area and perimeter (circumference) calculated accurately.

How do I calculate both area and perimeter together?

This calculator makes it easy to find both values simultaneously. After selecting your shape, enter all the required measurements. For most shapes, you'll need the same inputs for both calculations - like side lengths for a rectangle or radius for a circle. Some shapes, like triangles, might need different measurements for area versus perimeter (base and height for area, all three sides for perimeter). The calculator has a "Calculate Both" mode that handles this automatically. Once you click calculate, you'll see both the area (in square units) and perimeter (in linear units) displayed clearly. The results show the formulas used and allow you to copy either value. This is especially helpful when planning projects where you need both measurements, such as determining how much material to buy (area) and how much edging or fencing you'll need (perimeter).

What units are supported?

You can work with eight different units. The metric options are millimeters, centimeters, meters, and kilometers. Imperial choices include inches, feet, yards, and miles. Pick your unit before entering any measurements, and everything will use that same unit.

What is the difference between area and perimeter?

Area measures the space contained within a shape, expressed in square units such as square centimeters or square meters. Perimeter is the total distance around the shape's outer edge, measured in linear units like centimeters or meters. For instance, a rectangle measuring 5 by 3 units has an area of 15 square units and a perimeter of 16 units.

How accurate are the calculations?

The tool uses standard mathematical formulas with precise number handling. Results are shown with up to six decimal places, while the internal calculations maintain even greater precision. Very large or very small numbers may appear in scientific notation.

Can I calculate composite shapes?

Yes, you can work with composite forms. The tool includes options for L and T shapes made from rectangles. Enter the measurements for each rectangular section, and it will calculate both the area and perimeter correctly, making sure shared edges are handled properly.

What shapes are included?

The tool supports over 30 different forms organized into categories. You'll find basic shapes like rectangles and circles, various triangle types, regular polygons with different numbers of sides, curved forms including ellipses and sectors, composite shapes such as L and T forms, and special geometries like star patterns and irregular polygons.

How do I copy results?

Click the copy button next to any calculated value to save it to your clipboard. A brief confirmation appears to let you know it worked. You can then paste the number wherever you need it.

A Step-by-Step Guide to Finding Perimeter

Finding the boundary length of any shape follows a logical process. First, identify what type of shape you're working with - is it a regular polygon, a circle, or something more complex? For rectangles and squares, add all four sides together. For circles, you'll need either the radius or diameter to use the formula 2πr or πd. Triangles require adding all three side lengths. Regular polygons like pentagons or hexagons simply multiply the side length by the number of sides. The key is knowing which measurements you have available. Some shapes need just one measurement (like a square's side), while others need multiple values (like a triangle's three sides). This calculator handles all these variations automatically, so you can focus on entering your measurements rather than memorizing formulas.

Understanding Area and Perimeter

Area tells you how much space a shape contains, measured in square units. Perimeter is the total distance around the shape's edge, measured in linear units. For rectangles, multiply length by width for area, and add length and width then double it for perimeter. With circles, square the radius and multiply by π for area, and multiply the radius by 2π for the circumference. These ideas appear everywhere from building design to everyday measurements.

The Relationship Between Area and Perimeter

While area and perimeter measure different aspects of a shape, they're related in important ways. For a given perimeter, a circle encloses the maximum possible area - this is why circular shapes are efficient in nature and design. Conversely, for a given area, a circle has the minimum perimeter. This relationship explains why many containers and structures use circular or rounded forms. However, you generally cannot directly convert between them without additional information about the shape's geometry. For example, knowing only a rectangle's perimeter doesn't tell you its area - a 4×4 square and a 1×7 rectangle both have perimeters of 16 units, but very different areas (16 vs. 7 square units). Understanding this relationship helps in design optimization and material planning.

Perimeter Formulas for Common Shapes

Each shape type has its own approach to finding the boundary length. Rectangles use 2(length + width), while squares simply multiply the side by 4. Circles use 2π × radius or π × diameter. Triangles add all three sides together. Regular polygons multiply the side length by the number of sides - so a hexagon with 6-inch sides has a perimeter of 36 inches. For parallelograms, it's 2(base + side). Trapezoids require adding all four sides. Ellipses and ovals use more complex approximations since there's no simple exact formula. The key is matching the formula to your shape and the measurements you have available. This calculator includes all these formulas and automatically selects the right one based on your inputs.

Triangle Formulas

There are several ways to find a triangle's area. The most common method multiplies the base by the height and divides by two. Heron's formula works when you know all three sides, using the semi-perimeter in its calculation. For right triangles, multiply the two legs and divide by two. Equilateral triangles have their own formula involving the square of the side length and the square root of three.

Working with Ellipses and Ovals

Ellipses and ovals are more complex than circles because they have two different radii - a longer semi-major axis and a shorter semi-minor axis. The area formula is straightforward: π × a × b, where a and b are the semi-major and semi-minor axes. However, finding the exact perimeter (circumference) of an ellipse requires advanced mathematics involving elliptic integrals. Practical calculations use approximations like Ramanujan's formula, which provides excellent accuracy for most purposes. This is why specialized tools are valuable - they handle the complex mathematics behind the scenes. Ellipses appear frequently in architecture, engineering, and design, from racetracks to pool designs to planetary orbits. Understanding how to work with these curved shapes opens up many practical applications.

Regular Polygons

Regular polygons have sides and angles that are all equal. To find the area, you use a formula that involves the number of sides, the side length, and a trigonometric function. The perimeter is straightforward: just multiply the number of sides by the side length. As you add more sides, these shapes start to look more like circles. Common examples include five-sided pentagons, six-sided hexagons, eight-sided octagons, and twelve-sided dodecagons.

Composite Shapes

Composite forms combine multiple basic shapes. When working with L or T shapes, find the area of each rectangular section and add them up. Calculating the perimeter needs extra care to make sure you don't count shared edges twice. For rounded rectangles, the calculation accounts for the curved corners by adjusting the standard rectangle formula.

Area & Perimeter Calculator - Calculate Any Shape with Formulas Free

Basic Shapes (7)

Triangles (4)

Triangles (4)

Regular Polygons (6)

Regular Polygons (6)

Curved Shapes (6)

Curved Shapes (6)

Composite Shapes (4)

Composite Shapes (4)

Special Shapes (6)

Special Shapes (6)

Rectangle

Measurements

How to Use Area & Perimeter Calculator

Select a Shape

Enter Measurements

View Results

Calculator Features

Wide Variety of Shapes

Multiple Units

Input Validation

Copy Results

Helpful Information

Extra Details

Accurate Results

Responsive Design

Complete Function List

Common Calculations & Examples

Example 1: Calculate Rectangle Area and Perimeter

Example 2: Calculate Circle Area and Perimeter

Example 3: Calculate Right Triangle Area and Perimeter

Example 4: Calculate Regular Hexagon Area and Perimeter

Example 5: Finding the Perimeter of a Rectangular Garden

Example 6: Calculating Both Area and Perimeter for a Room

Example 7: Determining the Boundary Length of an Oval Pool

Example 8: Working with Irregular-Shaped Plots

Tips for Using Area & Perimeter Calculator

Frequently Asked Questions

Understanding Area & Perimeter Calculator

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