Geometry · 2D Shapes
Area
Calculator
Calculate area and perimeter for 11 two-dimensional shapes. Supports multiple units with conversion, shape illustrations, and step-by-step formulas.
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Area
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Perimeter
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Shape
▭ Rectangle
A = l × w\nP = 2(l + w)
Area in cm², Perimeter in cm
Area
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units²
Perimeter
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units
Area
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Perimeter
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Extra 1
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Extra 2
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📋 Step-by-Step
Area & Perimeter Formulas
Area measures the 2D space a shape occupies (in square units). Perimeter measures the total length of the boundary (in linear units). These are foundational geometry concepts used in architecture, engineering, land surveying, and everyday calculations.
All Area Formulas
Rectangle: A = l×w P = 2(l+w)
Square: A = s² P = 4s
Circle: A = πr² C = 2πr
Triangle: A = ½bh or Heron's: √(s(s-a)(s-b)(s-c))
Ellipse: A = πab P ≈ π[3(a+b)−√((3a+b)(a+3b))]
Trapezoid: A = ½(b₁+b₂)h P = b₁+b₂+c+d
Parallelogram:A = b×h P = 2(b+s)
Rhombus: A = (d₁×d₂)/2 P = 4s
Reg. Polygon: A = (n×s²)/(4tan(π/n))
Sector: A = (θ/360)×πr² Arc = (θ/360)×2πr
Annulus: A = π(R²−r²) P = 2π(R+r)
What is Heron's formula and when do you use it?
Heron's formula calculates a triangle's area from its three side lengths: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. You use it when you know all three sides but not the height. It's especially useful in surveying, construction, and navigation. For example, if you know three fence posts' positions, Heron's formula gives the enclosed area without needing to measure any heights.
Why is the area of a circle πr²?
Archimedes proved this by exhaustion — inscribing polygons with more and more sides inside a circle, showing their areas approach πr². The modern proof uses calculus: integrate the thin circular rings from r=0 to r=R: ∫₀ᴿ 2πr dr = πR². Intuition: "unrolling" a circle into a triangle with base = circumference (2πr) and height = r gives area = ½ × base × height = ½ × 2πr × r = πr².
What is the most area-efficient shape (largest area for given perimeter)?
Among all shapes with the same perimeter, the circle encloses the maximum area — this is the Isoperimetric Inequality. For a perimeter P, the maximum area is P²/(4π). A square is the most efficient rectangle (for the same perimeter as a rectangle, a square has 27% more area on average). This is why cells, bubbles, and honeycombs tend toward circular or hexagonal shapes — hexagons tile perfectly AND give the most area per perimeter compared to triangles and squares.
How is the area of a trapezoid derived?
A trapezoid has two parallel sides (bases b₁ and b₂) and a height h. You can split it into a rectangle of width b₁ and a triangle, but the elegant derivation: if you make two copies of the trapezoid and flip one, you can arrange them into a parallelogram with base (b₁+b₂) and height h. The parallelogram's area = (b₁+b₂)×h. Each trapezoid is half of this: A = ½(b₁+b₂)×h. This also makes the formula intuitive — it's the average of the two bases, multiplied by the height.