Calculating...
Select a shape, enter the measurements, and see the area, perimeter, and steps.
What Is an Area and Perimeter Calculator?
An area and perimeter calculator finds the space inside a two-dimensional shape and the distance around its edge. This tool covers the most common shapes taught in school and searched online: circle, rectangle, square, triangle, trapezoid, parallelogram, and ellipse. You enter the measurements, choose the shape, and the calculator returns both values with the formula and working.
Area is measured in square units because it counts how many unit squares fit inside the shape. Perimeter is measured in ordinary length units because it is a one-dimensional walk around the boundary. Keeping the two ideas separate is one of the first steps in geometry.
Shapes Covered
How to Find the Area and Perimeter of a Circle
A circle is defined by its radius, the distance from the centre to any point on the edge. The area is π times the radius squared, and the perimeter, called the circumference, is two times π times the radius. For a circle with radius 5, the area is about 78.54 square units and the perimeter is about 31.42 units.
These formulas are exact when you leave π in the answer. The calculator uses a numerical approximation of π so the result is a decimal you can use in practical problems. If you need an exact answer, write the result in terms of π instead.
How to Find the Area and Perimeter of a Rectangle and Square
A rectangle has two pairs of equal sides. Multiply its width by its height to find the area. Add all four sides, or use the shortcut two times the sum of width and height, to find the perimeter. A square is a special rectangle with all sides equal, so its area is side squared and its perimeter is four times the side.
These are the simplest area and perimeter formulas, but they are also the most widely used. Rooms, screens, paper, fields, and tiles are usually rectangles or squares. Knowing how to move between length, width, area, and perimeter helps with flooring, framing, fencing, and layout problems.
How to Find the Area and Perimeter of a Triangle
When you know all three sides of a triangle, Heron's formula gives the area. First compute the semi-perimeter s = (a + b + c) / 2. Then the area is the square root of s(s-a)(s-b)(s-c). The perimeter is simply the sum of the three sides. The calculator checks the triangle inequality so you cannot enter sides that do not form a real triangle.
The triangle inequality says that the sum of any two sides must be greater than the third side. If it fails, the three lengths cannot meet at corners. This check prevents impossible results and reminds you to verify your measurements.
How to Find the Area and Perimeter of a Trapezoid and Parallelogram
A trapezoid has one pair of parallel sides called bases. Its area is the average of the two bases multiplied by the height, the perpendicular distance between the bases. Its perimeter is the sum of all four sides, including the two legs that are not parallel.
A parallelogram has two pairs of parallel sides. Its area is base times height, where the height is measured perpendicular to the base, not along the slanted side. Its perimeter is two times the sum of a base and a side. These formulas are useful for roofs, ramps, and slanted surfaces where the vertical height differs from the slant length.
How to Find the Area and Perimeter of an Ellipse
An ellipse is a stretched circle with a semi-major axis a and a semi-minor axis b. Its area is π times a times b, which is the straightforward analogue of the circle area formula. The perimeter of an ellipse has no simple exact formula, so the calculator uses Ramanujan's widely accepted approximation.
When the two axes are equal, the ellipse becomes a circle and the approximation gives the exact circumference. As the ellipse becomes more stretched, the approximation remains accurate enough for nearly every school and engineering purpose.
Common Area and Perimeter Formulas
| Shape | Area | Perimeter |
|---|---|---|
| Circle | π r² | 2 π r |
| Rectangle | w × h | 2(w + h) |
| Square | s² | 4s |
| Triangle | √(s(s-a)(s-b)(s-c)) | a + b + c |
| Trapezoid | ((a+b)/2) × h | a + b + c + d |
| Parallelogram | base × height | 2(base + side) |
| Ellipse | π a b | ≈ π[3(a+b) - √((3a+b)(a+3b))] |
Area and Perimeter FAQ
What is the difference between area and perimeter? +
Area measures the space inside a shape in square units. Perimeter measures the distance around the outside of a shape in length units.
How do I find the area of a triangle when I only know the three sides? +
Use Heron's formula. Compute the semi-perimeter s = (a+b+c)/2, then area = √(s(s-a)(s-b)(s-c)).
Can a shape have the same area and perimeter number? +
Yes. A square with side 4 has area 16 and perimeter 16. The units are different, so the values being equal is a coincidence, not a deep rule.
What units should I use for area and perimeter? +
Use the same length unit for all inputs. The perimeter will be in that unit. The area will be in that unit squared, such as square metres or square centimetres.
How do I calculate the perimeter of an ellipse? +
There is no simple exact formula. Use Ramanujan's approximation: perimeter ≈ π[3(a+b) - √((3a+b)(a+3b))], where a and b are the semi-axes.
Why does the triangle inequality matter? +
The triangle inequality guarantees that three lengths can actually form a triangle. If the sum of any two sides is not greater than the third, the sides cannot meet at three corners.
External Resources
Authoritative references for area, perimeter, and geometry fundamentals.
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