Formula and method for the Area Calculator
This calculator finds the area and perimeter of four common shapes — rectangle, square, circle, and triangle — using the standard closed-form formula for each. Pick a shape, enter its dimensions in any consistent unit, and the calculator applies the matching formula exactly; there is no approximation or iteration involved.
The formulas used
- Rectangle: Area = length × width (A = l × w). Perimeter = 2 × (length + width).
- Square: Area = side² (A = s²). Perimeter = 4 × side.
- Circle: Area = π × radius² (A = πr²). Circumference = 2 × π × radius.
- Triangle (from three side lengths): Heron's formula. First find the semi-perimeter s = (a + b + c) / 2, then Area = √[s(s − a)(s − b)(s − c)]. Perimeter is simply a + b + c.
How the calculation works
Enter your measurements in the same unit system throughout — mixing feet and inches, or centimeters and meters, produces incorrect results. Every formula above expects consistent units for all of a shape's dimensions; the calculator does not convert between unit systems, so the area comes out in whatever unit you entered, squared.
Common mistakes
- Units: area is in square units (ft², m²), never linear units. A square that is 3 feet per side has an area of 9 ft², not 9 ft.
- Diameter vs. radius: the circle formula uses radius (half the diameter). Entering the diameter where radius is expected roughly quadruples the computed area.
- Invalid triangle sides: three side lengths only form a real triangle if each one is shorter than the sum of the other two (the triangle inequality). Otherwise Heron's formula has no valid answer.
- Irregular shapes: decompose complex shapes into rectangles, triangles, and circles, calculate each separately, then add or subtract the areas as needed.
Real-world applications
- Flooring, paint, and tiling projects use area to estimate material quantities (add roughly 10-15% for waste and cuts)
- Landscaping, fencing, and framing use perimeter to plan linear material like edging or fence panels
- Land, garden, and room measurements use rectangle or triangle area for irregular plots split into simple shapes
- Circular objects — pipes, tanks, pools, rugs — use the circle formulas for material and coverage estimates
Frequently Asked Questions
What is the formula for the area of a rectangle?
Multiply length by width: A = l × w. A rectangle 14 ft long and 9 ft wide has an area of 14 × 9 = 126 square feet. The perimeter is 2 × (l + w).
How do you find the area of a triangle from three side lengths?
Use Heron's formula: compute the semi-perimeter s = (a + b + c) / 2, then area = √[s(s − a)(s − b)(s − c)]. The three sides must satisfy the triangle inequality — each side shorter than the sum of the other two — or no triangle exists.
What is the difference between area and perimeter?
Area is the amount of surface enclosed by a shape, measured in square units such as ft² or m². Perimeter (called circumference for a circle) is the total distance around the outside edge, measured in the same linear units you entered, not squared.
Which unit does the result use?
This calculator does not convert units. The area comes out in whatever unit you entered, squared. Enter feet and the area is in square feet; enter meters and it is in square meters. Use the same unit for every dimension of a shape.