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Area Calculator

Select a shape, enter its dimensions, and get the area with a step-by-step calculation. Covers 8 common geometric shapes.

Formula: A = l × w

Result

Area = 40
Step-by-step:
A = l × w
A = 8 × 5
A = 40

Area Formulas for All Shapes

Shape Formula Variables
Square A = s² s = side length
Rectangle A = l × w l = length, w = width
Triangle A = ½ × b × h b = base, h = perpendicular height
Circle A = πr² r = radius
Trapezoid A = ½(a + b) × h a, b = parallel sides, h = height
Parallelogram A = b × h b = base, h = perpendicular height
Ellipse A = πab a = semi-major axis, b = semi-minor axis
Rhombus A = (d1 × d2) / 2 d1, d2 = diagonals

Triangle Area in Depth

Three common methods to find triangle area:

  1. Base × Height: A = ½ × b × h. Height must be perpendicular to the chosen base.
  2. Heron's Formula (all 3 sides known): compute semi-perimeter s = (a+b+c)/2, then A = √(s(s−a)(s−b)(s−c)). Example: sides 6, 8, 10 → s=12, A=√(12×6×4×2)=√576=24.
  3. SAS method: A = ½ab sin(C), where C is the angle between sides a and b.

Why Parallelogram and Rectangle Share a Formula

A parallelogram can be rearranged into a rectangle by cutting a triangle from one end and attaching it to the other. Both result in A = base × height. The key: height is always the perpendicular height, not the slant side.

Similarly, a trapezoid averages its two parallel sides and multiplies by the perpendicular height. When both parallel sides are equal, it becomes a rectangle: A = s × h.

Understanding π in Circle and Ellipse Formulas

The circle formula A = πr² comes from integrating thin rings from the center outward. Each ring at radius r has circumference 2πr and width dr, giving dA = 2πr dr. Integrating from 0 to R: A = πR².

The ellipse formula A = πab extends this: an ellipse with semi-axes a and b is a stretched circle. Stretching by factor k multiplies area by k, so A = π × a × b.

Area Unit Conversions

From To Multiply by
m²cm²10,000
m²ft²10.764
ft²in²144
acresft²43,560
acresm²4,047
hectaresm²10,000
km²hectares100

Real-World Area Examples

Frequently Asked Questions

What is area and how is it measured?

Area is the amount of 2D space inside a shape, measured in square units (cm², m², ft², etc.). To find area, you typically multiply two dimensions (length and width for a rectangle) or apply a shape-specific formula. Area is additive: you can find the total area of a complex shape by breaking it into simple shapes and summing their areas.

How do you find the area of a triangle?

The standard formula is A = ½ × base × height, where height is the perpendicular distance from the base to the opposite vertex. If you know all three sides, use Heron's formula: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. If you know two sides and the angle between them, use A = ½ab·sin(C).

What is the area formula for a trapezoid?

A trapezoid (trapezium in UK) has two parallel sides called bases (a and b) and a height h (perpendicular distance between the bases). Area = ½(a + b) × h. This averages the two parallel sides and multiplies by height. When a = b, this simplifies to a × h, which is the parallelogram formula.

What is the area of an ellipse?

Area of an ellipse = π × a × b, where a is the semi-major axis (half the longer diameter) and b is the semi-minor axis (half the shorter diameter). When a = b, this becomes πr², the circle formula. For example, an ellipse with semi-axes 5 and 3 has area π × 5 × 3 ≈ 47.12 square units.

How do I convert between area units?

Common conversions: 1 m² = 10,000 cm² = 1,000,000 mm². 1 ft² = 144 in². 1 yd² = 9 ft². 1 acre = 43,560 ft² = 4,047 m². 1 hectare = 10,000 m² = 2.471 acres. 1 km² = 100 hectares = 1,000,000 m². When converting, note that area units scale by the square of the length conversion: 1 m = 100 cm, so 1 m² = 100² = 10,000 cm².

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