Area and Perimeter Calculator
Area and perimeter of nine plane shapes, each one drawn to scale with the measurements you gave.
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Area 12
A = w × h
A = 3 × 4
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Perimeter 14
P = 2 × (w + h)
P = 2 × (3 + 4)
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Diagonal 5
d = √(w² + h²)
d = √(3² + 4²)
Pick a shape, give it its measurements, and get the area, the perimeter and whatever else those measurements determine — a diagonal, an apothem, an arc.
Every answer is drawn. The figure is to scale with itself, so the same shape at any size gives the same picture and a rectangle that really is a hundred times longer than it is wide looks like one.
How it works
Each shape uses the formula that belongs to it, and every one of them is exact — with a single exception, which is said out loud rather than hidden. The perimeter of an ellipse cannot be written down in elementary functions at all: it is an elliptic integral, which is where that family of functions got its name. What is shown here is Ramanujan's second approximation, and the page labels it as an approximation whenever it appears. How close it is depends on how elongated the ellipse is: better than one part in a thousand million up to about two to one, around one part in a hundred thousand at ten to one, and about two parts in ten thousand at a hundred to one, and never more than four however flat it gets.
The trapezium asks for all four sides rather than the two parallel ones and a height. That is not pedantry: the two parallel sides and the height fix the area and say nothing whatever about the perimeter, because the legs can lean any way they like without changing either of those three numbers. Four sides pin the shape down, and the height falls out of them.
A ring has two edges, so its perimeter is both circles added together. A sector's perimeter is its arc plus the two straight radii, not the arc alone — the arc is given separately because that is usually the number people came for.
Measurements have no unit here. Give centimetres and the area is in square centimetres; give miles and it is in square miles. Angles are in degrees, in and out.
Examples
| Case | Input | Result |
|---|---|---|
| A hexagon of side 1 | Regular polygon, 6 sides, side 1 | area 2.598076, perimeter 6, apothem 0.866025 |
| A trapezium with a right angle | Trapezium, 8 / 4 / 3 / 5 | height 3, area 18, perimeter 20 |
| Four sides that close nothing | Trapezium, 10 / 1 / 2 / 2 | refused — the legs cannot span the difference |
Frequently asked questions
How is this different from the triangle solver?
This page takes the three sides and gives the area and the perimeter, alongside eight other shapes. The triangle solver takes any three measurements — sides, angles, or a mixture — gives every remaining one, and finds the second triangle in the case where two of them fit the same measurements. Both use the same code underneath, so they never disagree.
Why does the trapezium want four sides?
Because the two parallel sides and the height, which is what school teaches, do not describe one trapezium. Infinitely many share those three numbers and they have different perimeters. Asking for the legs too settles it, and the height is then calculated rather than asked for.
Is the ellipse perimeter exact?
No, and it is the only figure on this page that is not. No formula in elementary functions gives it. The approximation used here agrees with the true perimeter to about ten decimal places for an ellipse up to twice as long as it is wide, and loses accuracy as it is drawn out — roughly five decimal places at ten to one and four at a hundred to one. The area beside it is exact at any shape.
Why is my shape drawn as a line?
Because it is one. The figure is drawn to scale, so a rectangle of 1 by 10000 comes out as a line — nothing is widened to make it look more like a rectangle. The page says so when it happens rather than leaving it looking like a fault.
What unit are the answers in?
Whatever you brought. The calculation is unitless: lengths come back in your length unit and areas in its square. If you need to change unit afterwards, the area and length converters are linked below.
Good to know
- Every formula here is exact except the perimeter of an ellipse, which has no exact form and is marked as an approximation wherever it appears.
- The figure shows shape and proportion, never scale. Three numbers with no unit cannot honestly be drawn at a size.