What is the most area-efficient shape for a given perimeter?

The circle encloses the maximum area for a given perimeter (isoperimetric inequality). Among rectangles, the square maximizes area for a fixed perimeter. This is why bubbles are spherical and why hexagons tile efficiently in honeycombs.

How do I find the area of an irregular shape?

Divide it into regular shapes (rectangles, triangles), find each area, then sum them. For very irregular shapes, the Shoelace formula works: A = (1/2)|sum(x_i*y_{i+1} - x_{i+1}*y_i)| using vertex coordinates.

What is the difference between a parallelogram and a rectangle?

A rectangle is a special parallelogram where all angles are 90 degrees. Both have Area = base x height, but the rectangle's height equals its width. A parallelogram's height is always less than its slanted side.

How do I calculate the area of a regular hexagon?

For a regular hexagon with side length a: Area = (3*sqrt(3)/2) x a^2. With a = 10: Area = (3*sqrt(3)/2) x 100 = 259.8 sq units. Perimeter = 6a = 60 units.

What is the difference between area and perimeter?

Area measures the space inside a shape (in square units), while perimeter measures the total length of the boundary (in linear units).

How do I find the area of an irregular shape?

Divide the irregular shape into simpler shapes (rectangles, triangles, circles), calculate each area separately, then add them together.

What units should I use for area?

Use square versions of your length unit. If you measure in feet, area is in square feet (ft2). Never mix units in a single calculation.

Area & Perimeter Calculator

Shape & Dimensions

Shape

Unit

Area

—

Enter dimensions above

Area & Perimeter Formulas

Shape	Area	Perimeter / Circumference	Notes
Square	`A = s²`	`P = 4s`	s = side length
Rectangle	`A = l × w`	`P = 2(l + w)`	diagonal = √(l²+w²)
Triangle	`A = ½bh` or Heron's	`P = a + b + c`	Heron: s=(a+b+c)/2; A=√(s(s-a)(s-b)(s-c))
Circle	`A = πr²`	`C = 2πr`	d = 2r
Trapezoid	`A = ½(a+b)h`	`P = a+b+c+d`	a, b = parallel sides
Parallelogram	`A = b × h`	`P = 2(a+b)`	h = perpendicular height
Ellipse	`A = πab`	`C ≈ π[3(a+b)−√((3a+b)(a+3b))]`	Ramanujan approximation
Sector	`A = ½r²θ`	`Arc = rθ`	θ in radians; perimeter = 2r + arc

Area Converter

Value

From

Converted Area

—

Enter a value to convert

Result

—

Reverse

—

How to Use This Calculator

Select the Shape

Choose rectangle or square as the shape you want to calculate.

Enter the Dimensions

Type the width and height (for rectangles) or side length (for squares).

View Area and Perimeter

Both the area and perimeter are computed and displayed with the step-by-step solution.

Formula & Methodology

Rectangle Area

A = W × H

Multiply the width by the height to find the area of a rectangle.

Rectangle Perimeter

P = 2(W + H)

Add the width and height, then multiply by two to get the total perimeter.

Square Area

A = s²

The area of a square is the side length squared; its perimeter is 4s.

About this calculation · Accuracy & Methodology

Rectangle vs. Square Properties

Property	Rectangle	Square
Sides	Two pairs of equal sides	All four sides equal
Area formula	W × H	s²
Perimeter formula	2(W + H)	4s
Diagonals	Equal in length	Equal and perpendicular
Angles	All 90°	All 90°

Key Terms

Area: The amount of two-dimensional space enclosed within a shape, measured in square units.
Perimeter: The total length of the boundary around a two-dimensional shape.
Width (W): The horizontal measurement of a rectangle.
Height (H): The vertical measurement of a rectangle; also called length in some contexts.
Square: A special rectangle where all four sides are equal in length.

Real-World Examples

Example 1

Living Room Floor

W = 15 ft, H = 20 ft

Area = 300 ft², Perimeter = 70 ft — useful for flooring and baseboard estimates

Example 2

Garden Bed

s = 4 m

Area = 16 m², Perimeter = 16 m — a square raised garden bed

Area and Perimeter in Everyday Problem Solving

Practical Applications

Area calculations determine how much paint covers a wall, how many tiles fill a floor, or how much sod a lawn requires. Perimeter is used to calculate fencing, trim, or border material. These two measurements are the foundation of virtually every construction, landscaping, and interior-design estimate.

Maximizing Area for a Given Perimeter

Among all rectangles with the same perimeter, the square encloses the maximum area. This principle, known as the isoperimetric property for rectangles, has practical implications—if you have a fixed length of fencing, building a square pen gives your animals the most space.