Area & Perimeter Calculator
Area & Perimeter Calculator for All Shapes
This geometry calculator computes the area and perimeter (or circumference) for seven common 2D shapes: circle, rectangle, square, triangle (two methods), trapezoid, ellipse, and regular polygon. Select the shape, enter the dimensions, and the formula used is shown alongside the results.
Shape Formulas Reference
- Circle: Area = πr², Circumference = 2πr
- Rectangle: Area = w×h, Perimeter = 2(w+h)
- Square: Area = s², Perimeter = 4s, Diagonal = s√2
- Triangle (base+height): Area = ½×b×h
- Triangle (Heron's): Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
- Trapezoid: Area = ½(a+b)×h
- Ellipse: Area = πab, Circumference ≈ π[3(a+b)−√((3a+b)(a+3b))]
- Regular Polygon: Area = ns²/(4 tan(π/n)), Perimeter = n×s
Heron's Formula Explained
Heron's formula calculates triangle area from three side lengths, without needing to know the height. First compute the semi-perimeter s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). This is especially useful in surveying, navigation, and coordinate geometry where three vertex positions are known but the height is not.
Real-World Applications
- Construction: Calculate floor area for flooring materials, paint area for walls.
- Landscaping: Compute lawn or garden area for seed and fertiliser quantities.
- Engineering: Cross-sectional areas for stress and flow calculations.
- Cartography: Estimate land areas from survey measurements.
Frequently Asked Questions
What units does this calculator use?
The calculator is unit-agnostic. If you enter dimensions in metres, areas are in square metres and perimeters in metres. If you enter in feet, results are in square feet and feet. Always use consistent units for all inputs.
How do I find the area of an irregular shape?
For irregular shapes, divide the shape into standard shapes (rectangles, triangles, circles), calculate each area separately, then add (or subtract for cutouts) them together. This decomposition method works for any polygon or compound shape.
What is the difference between perimeter and circumference?
Perimeter refers to the total boundary length of any polygon (straight-sided shape). Circumference specifically refers to the perimeter of a circle or ellipse — the continuous curved boundary. Both measure the total distance around the outside of a 2D shape.
What is a regular polygon?
A regular polygon has all sides equal in length and all interior angles equal. Examples: equilateral triangle (3 sides), square (4), pentagon (5), hexagon (6), octagon (8). As the number of sides increases, a regular polygon approaches a circle.
Why does the triangle area formula need ½?
Any triangle can be inscribed in a rectangle of width equal to the base and height equal to the triangle's height. The triangle always covers exactly half of that rectangle's area, hence the ½ factor. This is visually obvious if you draw the rectangle around any triangle.