Area Calculator
Calculate the area and perimeter of 8 shapes. Select a shape, enter its dimensions, and instantly get area, perimeter, and additional measurements like diagonal or circumradius.
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How It Works
Multiply length by width. This is the most basic area formula. Area = l × w. The diagonal uses the Pythagorean theorem: d = √(l²+w²).
Area = length × width5m × 8m = 40 m²Area = base × height / 2. The height must be perpendicular to the base. For SSS (three sides), use Heron's formula with the semiperimeter s.
Area = (b × h) / 2 or √(s(s-a)(s-b)(s-c))base=6, h=4 → Area = 12Area = π × r². Use radius (half diameter). The circumference is 2πr. Both derived from the constant relationship between radius and circular area.
Area = π × r²r=7 → Area = 153.94Area = (a + b) / 2 × h, where a and b are the two parallel sides and h is the perpendicular height between them. Average the parallel sides, multiply by height.
Area = (a + b) / 2 × ha=4, b=8, h=5 → Area = 30Area = π × a × b, where a and b are the semi-major and semi-minor axes. The perimeter (circumference) uses Ramanujan's approximation.
Area = π × a × ba=5, b=3 → Area = 47.12Area = (n × s²) / (4 × tan(π/n)). The inradius (apothem) is the distance from center to midpoint of a side. Circumradius is center to vertex.
Area = n × s² / (4tan(π/n))hexagon s=4 → Area = 41.57Quick Reference
Common examples — verify instantly above.
Rectangle
5 × 8
40 sq units
Square
6 × 6
36 sq units
Triangle
base=6, h=4
12 sq units
Circle
r=7
153.94 sq units
Trapezoid
a=4, b=8, h=5
30 sq units
Ellipse
a=5, b=3
47.12 sq units
Sector
r=5, 90°
19.63 sq units
Hexagon
side=4
41.57 sq units
Tips & Shortcuts
Always use the perpendicular height — not the slant side — when computing triangle or parallelogram area.
To find area of an irregular shape: divide into standard shapes, find each area separately, add them together.
Area scales with the square of dimensions. Doubling all lengths quadruples the area.
For circular areas, remember π ≈ 3.14159. A quick estimate: area ≈ 3 × r².
The area of a sector is a fraction of the full circle: (angle/360) × πr².
Heron's formula for triangle area works for any triangle when only the three sides are known — no height needed.
Common Mistakes
Using slant height instead of perpendicular height
Height must always be perpendicular (90°) to the base, not the length of a slanting side.
Forgetting to square the radius for circle area
Circle area = πr², not πr. The radius is squared.
Mixing diameter and radius in the circle formula
The formula uses radius (r = diameter/2). Using diameter directly gives 4× the correct area.
Using base length instead of height in triangle area
Area = base × HEIGHT / 2. The height is the perpendicular distance to the base, not a side length.
Not converting units before calculating
If length is in meters and width in centimeters, convert both to the same unit before multiplying.
Forgetting /2 in triangle and trapezoid formulas
Triangle area = bh/2 (not bh). Trapezoid area = (a+b)h/2 (not (a+b)h).
Frequently Asked Questions
Area is the amount of two-dimensional space inside a shape, measured in square units (cm², m², ft², etc.).
Area = base × height / 2. If you know all three sides, use Heron's formula: Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.
π ≈ 3.14159 is the ratio of a circle's circumference to its diameter. Circle area = πr² because the circle's spread scales with the square of its radius.
A sector is a pie slice of a circle — the region bounded by two radii and the arc between them. Area = (angle/360) × πr².
Area = (a + b) / 2 × h, where a and b are the parallel sides (bases) and h is the height (perpendicular distance between bases).
A regular polygon has all sides equal and all angles equal. Examples: equilateral triangle, square, pentagon, hexagon. Area = (n × s²) / (4 × tan(π/n)) where n=sides, s=side length.
Break the irregular shape into standard shapes (rectangles, triangles, circles) whose formulas you know, calculate each area separately, then add them together. For complex curves, numerical integration or the Shoelace formula (for polygons with known vertices) gives exact results. Alternatively, overlay a grid and count squares — this approximation improves as the grid gets finer.
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