What is the difference between area and perimeter?

Area is the space inside a shape, measured in square units (like cm² or ft²). Perimeter is the distance around its outer edge, measured in plain linear units (like cm or ft). Area covers a surface; perimeter traces a border.

Can a shape have the same area but a different perimeter?

Yes. A 1 × 16 rectangle and a 4 × 4 square both have an area of 16, but their perimeters are 34 and 16 respectively. Long, thin shapes have much larger perimeters than compact ones, even at equal area.

Why is area in square units and perimeter is not?

Area multiplies two lengths together, so the units multiply too — centimeters times centimeters gives centimeters squared. Perimeter only adds lengths along the edge, so the unit stays a plain length, never squared.

How do I find perimeter if I only know the area?

It depends on the shape, because area alone does not fix the perimeter. For a rectangle you also need one side: if area is 48 cm² and the length is 8 cm, the width is 6 cm, so the perimeter is 2(8 + 6) = 28 cm.

What is the formula for the area and perimeter of a rectangle?

For a rectangle with length l and width w, the area is A = l × w and the perimeter is P = 2(l + w). For example, an 8 × 5 rectangle has an area of 40 square units and a perimeter of 26 units.

How do you find the area and perimeter of a circle?

Use the radius r. The area is A = πr² and the perimeter, called the circumference, is C = 2πr, with π ≈ 3.14159. A circle of radius 7 has an area of about 153.9 square units and a circumference of about 44 units.

How do you calculate the area and perimeter of an irregular shape?

Split the figure into shapes you know — rectangles, triangles, and circle pieces. Add their areas (subtract any cut-out parts) for the total area, and trace only the outside edges, ignoring internal split lines, for the perimeter.

Is this area and perimeter calculator free to use?

Yes. Every calculator on MathInSite is completely free, with no login, no sign-up, and no ads blocking the tool. You get instant area and perimeter results, labeled diagrams, and step-by-step solutions for each shape.

Area & Perimeter Calculator

Formula

Enter your values

Enter your values and press Calculate — your answer, visual and step-by-step solution appear here.

What Are Area and Perimeter?

This area and perimeter calculator finds both measurements at once for rectangles, squares, triangles, circles, parallelograms, trapezoids, and more — each with a labeled diagram and a full step-by-step solution. But it helps to know what the two numbers actually mean, because students mix them up constantly.

Area is the amount of flat space inside a shape — the surface you would cover with paint, carpet, or tiles. It is always measured in square units (cm², m², in², ft²) because you are filling a two-dimensional region. Perimeter is the total distance around the outside edge of a shape — the length you would walk if you traced its border. It is measured in plain linear units (cm, m, in, ft) because it is just a length. For a circle, the perimeter has its own name: the circumference.

	Area	Perimeter
What it measures	Space inside the shape (a surface)	Distance around the edge (a border)
Units	Square units — cm², m², ft²	Linear units — cm, m, ft
Real-world use	Paint, flooring, carpet, turf, sod	Fencing, trim, baseboard, framing, ribbon

Quick rule: If the answer ends in "squared" (and fills a region), it is area. If it is a plain length (and follows the edge), it is perimeter.

When Do You Need Area vs. Perimeter?

The fastest way to tell which one a problem wants is to ask whether you are covering a surface or going around an edge. Painting a wall covers a surface, so you need area; fencing a yard goes around the edge, so you need perimeter.

Use area when you are covering or filling a flat region:

Painting a wall or ceiling (how much paint to buy)
Laying flooring, carpet, tile, or hardwood
Buying turf, sod, or mulch for a lawn or garden bed
Sizing a rug, tablecloth, or sheet of material

Use perimeter when you are going around the outside edge:

Building a fence around a yard or pen
Installing baseboard, crown molding, or trim around a room
Framing a picture or putting edging around a garden
Measuring ribbon, rope, or string to wrap a border

Memory hook: Paint the inside = area. Pace the outside = perimeter. Same shape, two very different numbers.

How to Find Area and Perimeter by Hand

Whatever the shape, the method is the same three steps. Once you can do it on paper, the calculator just confirms your work and shows the arithmetic.

Identify the shape. Is it a rectangle, triangle, circle, or something else? The shape decides which formulas you use, so name it first.
Pick the right formula. Look up the area formula and the perimeter formula for that shape (the reference table below has all of them). Write the formula down before you plug in numbers.
Substitute and compute. Replace each letter with its measurement, then do the arithmetic. Label area with square units and perimeter with linear units.

Worked example — a rectangle 8 cm long and 5 cm wide:

Identify the shape. It is a rectangle, so length = 8 cm and width = 5 cm.
Area: A = l × w. A = 8 × 5 = 40 cm².
Perimeter: P = 2(l + w). P = 2 × (8 + 5) = 2 × 13 = 26 cm.

Notice the same two numbers, 8 and 5, produce a surface of 40 square centimeters and a border of 26 centimeters. How to find area always means multiplying lengths together (so the units square); how to find perimeter always means adding lengths along the edge (so the units stay linear).

Area and Perimeter Formulas by Shape

This is the reference core. Start with the master table below — it lists the area formula and perimeter formula for every shape this area and perimeter calculator solves — then read the per-shape cards underneath for diagrams and variable definitions.

Shape	Area formula	Perimeter formula
Square	A = s²	P = 4s
Rectangle	A = l × w	P = 2(l + w)
Triangle	A = ½ × b × h	P = a + b + c
Circle	A = πr²	C = 2πr
Parallelogram	A = b × h	P = 2(a + b)
Trapezoid	A = ½(a + b) × h	P = a + b + c + d
Rhombus	A = ½ × d₁ × d₂	P = 4s
Regular polygon	A = ½ × P × a	P = n × s

Square

A square has four equal sides of length s. Because every side is the same, both formulas are short.

A = s² P = 4s

For a square with s = 6 m: area = 6² = 36 m², perimeter = 4 × 6 = 24 m.

Rectangle

A rectangle has a length l and a width w. The rectangle area is length times width, and the perimeter adds up all four sides — two lengths and two widths.

A = l × w P = 2(l + w)

For l = 10 ft and w = 4 ft: area = 40 ft², perimeter = 2(10 + 4) = 28 ft.

Triangle

The area of a triangle is half its base b times its perpendicular height h. The perimeter is just the three side lengths a, b, and c added together.

A = ½ × b × h P = a + b + c

If you only know two sides of a right triangle and need the third before adding the perimeter, our Pythagorean theorem calculator finds the missing side in one step. Note that h is the straight-up height, not a slanted side.

Circle

A circle is defined by its radius r (center to edge). Its area uses πr², and its perimeter — called the circumference — is 2πr. Use π ≈ 3.14159.

A = πr² C = 2πr

For r = 7 cm: area = π × 49 ≈ 153.94 cm², circumference = 2π × 7 ≈ 43.98 cm. For more options — diameter, sector, and arc length — see our dedicated circle area and circumference calculator.

Parallelogram

A parallelogram slants, but its area is still base times perpendicular height, the same as a rectangle. The perimeter adds the two pairs of equal sides, a and b.

A = b × h P = 2(a + b)

For base = 9 cm, height = 5 cm, slant side = 6 cm: area = 45 cm², perimeter = 2(9 + 6) = 30 cm.

Trapezoid

A trapezoid has two parallel sides, a and b, separated by height h. Its area averages the two parallel sides and multiplies by the height; its perimeter sums all four sides.

A = ½(a + b) × h P = a + b + c + d

For parallel sides 8 and 12 cm, height 5 cm: area = ½(8 + 12) × 5 = 50 cm². Add the two slanted sides for the perimeter.

Rhombus

A rhombus is a "pushed-over" square — four equal sides of length s. The easiest area uses its two diagonals, d₁ and d₂, and the perimeter is simply four times one side.

A = ½ × d₁ × d₂ P = 4s

For diagonals 6 cm and 8 cm with side 5 cm: area = ½ × 6 × 8 = 24 cm², perimeter = 4 × 5 = 20 cm.

Regular Polygon

A regular polygon has n equal sides of length s. Its area uses the apothem (a, the distance from the center to the middle of a side): area equals half the perimeter times the apothem. The perimeter is the number of sides times the side length.

A = ½ × P × a P = n × s

For a regular hexagon (n = 6) with s = 4 cm and apothem ≈ 3.46 cm: perimeter = 24 cm, area = ½ × 24 × 3.46 ≈ 41.6 cm².

Composite and Irregular Shapes

Real homework and DIY problems are rarely a single clean shape — they are L-shaped rooms, a rectangle with a triangular gable, or a rectangle capped by a semicircle. You do not need a special formula. You break the figure into shapes you already know.

Split it into known shapes. Draw lines to divide the figure into rectangles, triangles, and circle pieces — for example, an L-shape becomes two rectangles.
Find each area, then add (or subtract). Compute the area of every piece and total them. If part is cut out (a hole or notch), subtract that piece instead.
Trace the outer edge for perimeter. Walk the entire outside boundary and add only the edges on the outside — ignore the internal split lines you drew, since they are not part of the border.

Watch out: When you split a shape, the dividing line adds to the areas you sum, but it is not part of the perimeter. Only outside edges count toward the distance around.

Working Backwards: Find Length, Width, or Max Area

Word problems often hand you the area or perimeter and ask for a missing dimension. You can run any formula in reverse with a little algebra.

Find a missing side from the area. A rectangle has area 48 cm² and a length of 8 cm. Since A = l × w, rearrange to w = A ÷ l = 48 ÷ 8 = 6 cm. The same idea works for a triangle's height: h = 2A ÷ b.

Find both sides from area and perimeter together. A rectangle has perimeter 26 cm and area 40 cm². From P = 2(l + w), the two sides add to 13; from A = l × w, they multiply to 40. The pair that fits is 8 and 5, so the rectangle is 8 cm × 5 cm.

Optimization fact: For a fixed perimeter, the shape with the largest possible area is always a square. A 40 m fence makes a 10 × 10 square enclosing 100 m² — more than any other rectangle with that perimeter (a 15 × 5 rectangle holds only 75 m²).

Worked Example — A Real Room

Here is a practical job: you are renovating a rectangular room that measures 12 ft by 9 ft. You need flooring (an area in square feet) and new baseboard (a perimeter in linear feet), and the baseboard is sold by the inch.

Flooring = area. A = l × w = 12 × 9 = 108 ft² of flooring. Buy about 10% extra for cuts and waste, so order roughly 119 ft².
Baseboard = perimeter. P = 2(l + w) = 2(12 + 9) = 2 × 21 = 42 ft of baseboard around the room.
Convert the perimeter to inches. 42 ft × 12 in/ft = 504 inches — the length to enter when the trim is priced per inch.

One room, two answers: flooring is an area in square feet, while baseboard is a perimeter in linear feet (here converted to inches). Mixing the units up is the single most common real-world mistake.

Common Area-vs-Perimeter Mistakes

Mistake — Using the wrong unitsArea must end in square units (m², ft²) and perimeter in plain linear units (m, ft). Writing area as "40 cm" or perimeter as "26 cm²" loses marks instantly. Square units for inside, linear units for around.

Mistake — Adding sides to get area (or multiplying to get perimeter)Area comes from multiplying lengths; perimeter comes from adding them. For an 8 × 5 rectangle, area = 8 × 5 = 40, not 8 + 5 = 13. Reach for the right operation before you start.

Mistake — Using a slanted side as the heightFor triangles, parallelograms, and trapezoids, the height h is the perpendicular distance — straight up, at a right angle to the base — not the slanted edge. Using the slant side overstates the area.

Mistake — Forgetting to halve a triangleA triangle is exactly half of the rectangle around it, so its area is ½ × b × h. Dropping the ½ doubles your answer. The same ½ appears in the trapezoid and rhombus formulas.

Mistake — Mixing up radius and diameter for a circleThe formulas A = πr² and C = 2πr use the radius (center to edge). If you are given the diameter, halve it first: r = d ÷ 2. Plugging the diameter straight in inflates area fourfold.

Frequently Asked Questions

What is the difference between area and perimeter?

Area is the space inside a shape, measured in square units (like cm² or ft²). Perimeter is the distance around its outer edge, measured in plain linear units (like cm or ft). Area covers a surface; perimeter traces a border.
Can a shape have the same area but a different perimeter?

Yes. A 1 × 16 rectangle and a 4 × 4 square both have an area of 16, but their perimeters are 34 and 16 respectively. Long, thin shapes have much larger perimeters than compact ones, even at equal area.
Why is area in square units and perimeter is not?

Area multiplies two lengths together, so the units multiply too — centimeters times centimeters gives centimeters squared. Perimeter only adds lengths along the edge, so the unit stays a plain length, never squared.
How do I find perimeter if I only know the area?

It depends on the shape, because area alone does not fix the perimeter. For a rectangle you also need one side: if area is 48 cm² and the length is 8 cm, the width is 6 cm, so the perimeter is 2(8 + 6) = 28 cm.
What is the formula for the area and perimeter of a rectangle?

For a rectangle with length l and width w, the area is A = l × w and the perimeter is P = 2(l + w). For example, an 8 × 5 rectangle has an area of 40 square units and a perimeter of 26 units.
How do you find the area and perimeter of a circle?

Use the radius r. The area is A = πr² and the perimeter, called the circumference, is C = 2πr, with π ≈ 3.14159. A circle of radius 7 has an area of about 153.9 square units and a circumference of about 44 units.
How do you calculate the area and perimeter of an irregular shape?

Split the figure into shapes you know — rectangles, triangles, and circle pieces. Add their areas (subtract any cut-out parts) for the total area, and trace only the outside edges, ignoring internal split lines, for the perimeter.
Is this area and perimeter calculator free to use?

Yes. Every calculator on MathInSite is completely free, with no login, no sign-up, and no ads blocking the tool. You get instant area and perimeter results, labeled diagrams, and step-by-step solutions for each shape. Explore more free math calculators on the homepage.