Perimeter Calculator
Perimeter is what you would measure with a tape or buy in fencing. Area is the space inside it, and the two behave very differently.
Perimeter
40.0000
- Perimeter
- 40.0000
- Area
- 96.0000
- Diagonal
- 14.4222
- Perimeter is the distance round the outside — what you would measure with a tape or buy in fencing. Area is the space inside it.
- Two shapes with the same perimeter can have very different areas. Of all shapes with a given perimeter, the circle encloses the most, which is why bubbles are round.
About perimeter and area
Perimeter is the distance all the way round: twice the length plus twice the width for a rectangle, four times the side for a square, 2πr for a circle. It is what you need for fencing, skirting, edging, guttering, framing and trim — anything sold by the length that goes around the outside.
The two are almost independent, which is the interesting part. A 12 by 8 rectangle and a 19 by 1 rectangle both have a perimeter of 40, and their areas are 96 and 19. The same amount of fencing encloses five times as much land depending on the shape you make with it.
Pushed to its conclusion: of all shapes with a given perimeter, the circle encloses the most area. A circle with a circumference of 40 has an area of 127, well above any rectangle with the same perimeter. That is why bubbles are round, why round tanks hold more per unit of steel, and why a square is the best rectangle for enclosing a field.
The practical version: if you are buying fencing to enclose an area, a squarer shape gets more land per metre of fence. If you are buying skirting for a room you already have, the shape is fixed and only the measurement matters.
What it works out
- Perimeter of a rectangle, square, circle or triangle
- Area alongside it
- The diagonal of a rectangle
The formula
Rectangle: 2(l + w) Square: 4s Circle: 2πr Triangle: a + b + c
A 12 by 8 rectangle has a perimeter of 2 × (12 + 8), which is 40, and an area of 96.
Its diagonal is √(12² + 8²) — Pythagoras again — which is 14.42. Useful when you need to know whether something fits through a doorway or across a room.
Now compare shapes with the same perimeter of 40. A 19 by 1 rectangle encloses 19. A 12 by 8 encloses 96. A 10 by 10 square encloses 100. A circle with a circumference of 40 encloses 127.
That progression is the whole point. The nearer a shape gets to a circle, the more it encloses for the same distance round — which is the isoperimetric inequality, and it is why soap bubbles, planets and pressure vessels all end up round.
For fencing a field, it means a square costs less per unit of land than a long thin plot. For skirting a room, the shape is already decided and only the measuring matters.
- l, w
- Length and width of a rectangle.
- s
- The side of a square.
- r
- The radius of a circle.
- Diagonal
- Corner to corner, by Pythagoras on the two sides.
A worked example
A rectangle 12 by 8.
That works out to 40.0000 .
- Perimeter
- 40.0000
- Area
- 96.0000
- Diagonal
- 14.4222
Questions
How do I calculate perimeter?
Add up every side. For a rectangle that is 2 × (length + width); for a circle it is 2πr, which is called the circumference.
What is the difference between perimeter and area?
Perimeter is the distance round the outside — fencing, skirting, trim. Area is the space inside — flooring, paint, turf. They are measured in different units and are almost independent of each other.
Can two shapes have the same perimeter and different areas?
Very different. A 19 by 1 rectangle and a 12 by 8 both have a perimeter of 40, and enclose 19 and 96 respectively — five times the difference from the same amount of fencing.
Which shape encloses the most for a given perimeter?
A circle, always. A circumference of 40 encloses 127, against 100 for the best possible square. Among rectangles, the squarest one wins.
How do I find the diagonal of a rectangle?
Pythagoras on the two sides: √(length² + width²). A 12 by 8 has a diagonal of 14.42.
How do I find the perimeter of an irregular shape?
Measure each side and add them. There is no formula for an irregular outline — which is why a wheel-style measure exists for walking round a boundary.