acalculator

What is the area of a shape?

Pick a shape and enter its measurements to get its area and the formula used.

Your numbers

Area
96

Rectangle area: 96 square units.

Formula
A = l × w

Area: 96. Rectangle area: 96 square units.

How to calculate

Finds the area of a rectangle, square, triangle, circle, ellipse, trapezoid, parallelogram, circle sector, or regular polygon from its measurements.

Example with the default inputs (Shape Rectangle, Length (l) 12, Width (w) 8): Rectangle area: 96 square units.

Method: A = the shape’s standard area formula, for example A = l × w for a rectangle and A = πr² for a circle.

  • Every length is more than 0 and all are in the same unit; the area is in that unit squared.
  • A height is measured at right angles to the base.
  • Three sides must make a triangle: each side shorter than the other two together.
  • A regular polygon has n equal sides and equal angles; a sector angle is more than 0 and at most 360°.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Shape Rectangle, Length (l) 12, Width (w) 8 gives Area 96, Formula A = l × w.Source: hand calculation in content.mdx: 12 × 8 = 96
  2. Shape Square, Side (s) 5 gives Area 25.Source: hand calculation in content.mdx: 5² = 25
  3. Shape Triangle (base and height), Base (b) 10, Height (h) 4 gives Area 20.Source: hand calculation in content.mdx: 10 × 4 ÷ 2 = 20
  4. Shape Triangle (three sides), Side a 5, Side b 6, Side c 7 gives Area 14.696938.Source: hand calculation in content.mdx: p = 9, √(9 × 4 × 3 × 2) = √216 = 6√6; Python 3: math.sqrt(216) = 14.696938456699069
  5. Shape Circle, Radius (r) 3 gives Area 28.274334.Source: hand calculation in content.mdx: π × 9; Python 3: math.pi * 9 = 28.274333882308138
  6. Shape Ellipse, Semi-axis a 5, Semi-axis b 3 gives Area 47.12389.Source: hand calculation in content.mdx: π × 15; Python 3: math.pi * 15 = 47.12388980384689
  7. Shape Trapezoid, Base (b) 10, Top (t) 6, Height (h) 4 gives Area 32.Source: hand calculation in content.mdx: (10 + 6) × 4 ÷ 2 = 32
  8. Shape Parallelogram, Base (b) 10, Height (h) 4 gives Area 40.Source: hand calculation in content.mdx: 10 × 4 = 40
  9. Shape Circle sector, Radius (r) 3, Angle in degrees (θ) 60 gives Area 4.712389.Source: hand calculation in content.mdx: (π/3) × 9 ÷ 2 = 1.5π; Python 3: (math.pi / 3) * 9 / 2 = 4.71238898038469
  10. Shape Regular polygon, Number of sides (n) 6, Side (s) 5 gives Area 64.951905.Source: hand calculation in content.mdx: 6 × 25 ÷ (4 tan 30°) = 75√3 ÷ 2; Python 3: 6 * 25 / (4 * math.tan(math.pi / 6)) = 64.95190528383291

How it works

Pick a shape, then enter the measurements it needs. The calculator uses the standard area formula for the shape (π = 3.14159265…):

  • Rectangle, length l, width w: A = l × w
  • Square, side s: A = s²
  • Triangle (base and height), base b, height h: A = b × h ÷ 2
  • Triangle (three sides), sides a, b, c: Heron's formula, A = √(p(p − a)(p − b)(p − c)) with p = (a + b + c) ÷ 2, half the perimeter. The calculator uses an equivalent form that keeps its accuracy for long, thin triangles: with the sides sorted so that a ≥ b ≥ c, A = √((a + (b + c))(c − (a − b))(c + (a − b))(a + (b − c))) ÷ 4.
  • Circle, radius r: A = π × r²
  • Ellipse, semi-axes a and b: A = π × a × b
  • Trapezoid, parallel sides b and t, height h: A = (b + t) × h ÷ 2
  • Parallelogram, base b, height h: A = b × h
  • Circle sector, radius r, angle θ in degrees: A = θ ÷ 360 × π × r², the same as θ × r² ÷ 2 with θ in radians (θ × π ÷ 180)
  • Regular polygon, n sides of length s: A = n × s² ÷ (4 × tan(π ÷ n))

The page also shows the formula it used.

Assumptions

  • Every length is more than 0, and all lengths are in the same unit. The area is in that unit squared (feet give ft²).
  • A height is measured at right angles to the base.
  • Three sides must make a triangle: each side must be shorter than the other two together. If they do not, there is no answer.
  • A sector angle is typed in degrees, more than 0 and at most 360. A regular polygon has 3 to 1,000 equal sides with equal angles.
  • A semi-axis is half of the ellipse's full width in that direction. With a = b = r the ellipse is a circle.

Worked examples by hand

Rectangle 12 × 8. A = 12 × 8 = 96.

Square with side 5. A = 5² = 25.

Triangle with base 10 and height 4. A = 10 × 4 ÷ 2 = 20.

Triangle with sides 5, 6 and 7. p = (5 + 6 + 7) ÷ 2 = 9, so A = √(9 × 4 × 3 × 2) = √216 = 6√6 = 14.696938.

Circle with radius 3. A = π × 9 = 28.274334.

Ellipse with semi-axes 5 and 3. A = π × 5 × 3 = 15π = 47.123890.

Trapezoid with parallel sides 10 and 6 and height 4. A = (10 + 6) × 4 ÷ 2 = 32.

Parallelogram with base 10 and height 4. A = 10 × 4 = 40.

Sector with radius 3 and angle 60°. 60° = π ÷ 3 radians, so A = (π ÷ 3) × 9 ÷ 2 = 1.5π = 4.712389.

Regular hexagon with side 5. A = 6 × 25 ÷ (4 × tan 30°) = 150 ÷ 2.3094011 = 64.951905 (which is 75√3 ÷ 2).

Other questions people ask

How do I calculate area?

Use the formula for the shape. For a rectangle, multiply the length by the width: a 12 × 8 room has an area of 96 square units. Break an irregular shape into rectangles and triangles, find the area of each, and add them.

What units is the area in?

The square of the unit you measured in. Measure in feet and the area is in square feet (ft²); measure in meters and it is in square meters (m²). Use the same unit for every length. 1 m² is about 10.76 ft².

How do I find the area of a triangle?

Multiply the base by the height and halve it: A = b × h ÷ 2, where the height is measured at right angles to the base. If you only know the three sides, use Heron's formula instead: with p half the perimeter, A = √(p(p − a)(p − b)(p − c)).

How do I find the area of a circle?

Multiply π by the radius squared: A = πr². A circle with radius 3 has an area of 9π, about 28.27. If you know the diameter, halve it first; a circle with diameter 6 has the same area.

How do I find the area of a trapezoid?

Add the two parallel sides, multiply by the height, and halve it: A = (b + t) × h ÷ 2. With parallel sides 10 and 6 and a height of 4, the area is 16 × 4 ÷ 2 = 32. It is the average width times the height.

How do I find the area of a sector (a slice of a circle)?

Multiply the angle in radians by the radius squared and halve it: A = θr² ÷ 2. In degrees, the sector is θ ÷ 360 of the whole circle: a 60° slice of a circle with radius 3 has an area of 60 ÷ 360 × 9π = 1.5π, about 4.71.