Area Calculator for Common Shapes and Square Units
Choose a two-dimensional shape, enter its dimensions and calculate area with the correct formula. Keep exact fractions and π forms, convert physical square units, and review every substitution step before using the result.
Last Updated: July 26, 2026
Online Area Calculator
Use one shared linear unit for every dimension. Integers, decimals, fractions, proper mixed numbers and scientific notation are supported.
Your area result and formula steps will appear here.
Calculation steps
- Choose a shape and enter its dimensions.
Area unit conversions
| Area unit | Equivalent value |
|---|---|
| Calculate an area with a physical input unit to see conversions. | |
Recent calculations
- Your last six results will appear here.
How to Use This Area Calculator
- Choose the two-dimensional shape that matches your problem.
- Select one input length unit. Convert every dimension to this unit before entering values.
- Choose the result area unit and decimal precision. Generic dimensions remain in generic square units.
- Enter the requested measurements. Labels identify perpendicular heights, semi-axes, radii, diagonals and angles where those distinctions matter.
- Select Calculate Area. Read the exact result first, then review the marked decimal approximation, substitution, steps and unit table.
Use the diagram or wording in your source problem to identify each measurement. A slanted triangle side is not automatically its height. The height must meet the selected base at a right angle. Likewise, the two trapezoid values labelled as bases must be parallel sides.
What Area Measures
Area measures the size of a two-dimensional region. It answers questions such as how much floor a rectangle covers, how much surface lies inside a circle, or how much material fills a flat polygon outline. Area differs from perimeter, which measures boundary length, and from volume, which measures three-dimensional space.
The unit must be squared because the calculation combines two independent length directions. A rectangle that is 4 metres long and 3 metres wide covers 12 one-metre-by-one-metre squares, so its area is 12 m². If the same numbers represented feet, the answer would be 12 ft², a different physical area.
Area Formulas Included
| Shape | Required measurements | Area formula |
|---|---|---|
| Square | Side s | A = s² |
| Rectangle | Length l, width w | A = lw |
| Triangle | Base b, perpendicular height h | A = bh/2 |
| Parallelogram | Base b, perpendicular height h | A = bh |
| Trapezoid | Parallel sides a and b, height h | A = (a + b)h/2 |
| Rhombus or convex kite | Perpendicular diagonals d₁ and d₂ | A = d₁d₂/2 |
| Circle | Radius r | A = πr² |
| Ellipse | Semi-axes a and b | A = πab |
| Annulus | Outer radius R, inner radius r | A = π(R² − r²) |
| Circular sector | Radius r, central angle θ in degrees | A = (θ/360)πr² |
| Regular polygon | Side count n, side length s | A = ns²/[4 tan(π/n)] |
Formula references: OpenStax Contemporary Mathematics formula review and OpenStax area section.
Worked Example: Rectangle Area
Suppose a rectangular room is 12 metres long and 7.5 metres wide. The formula requires the two perpendicular side lengths.
A = 12 × 7.5
A = 90 m²
The exact result is 90 m². Because both entered dimensions are in metres, multiplying them produces square metres. Converting the final unrounded value gives approximately 968.751938 ft², not 90 ft².
Worked Example: Triangle Area
For a triangle with base 10 centimetres and perpendicular height 6 centimetres, multiply the two measurements and take one half.
A = 10 × 6 ÷ 2
A = 30 cm²
The height may lie inside or outside the drawn triangle, but it must represent the perpendicular distance from the opposite vertex to the line containing the selected base.
Exact Results with π
Curved shapes often involve π. For a circle with radius 3 metres, the exact area is 9π m². The decimal value is approximately 28.274334 m² when rounded to six places. The calculator keeps 9π separate from its decimal approximation, so a rounded display is never presented as an exact value.
The same rule applies to ellipses, annuli and sectors. An ellipse uses its semi-axes, which are half the full major and minor axis lengths. An annulus subtracts the inner circular region from the outer one. A sector uses the central angle as a fraction of 360 degrees.
Regular Polygon Area
A regular polygon has equal side lengths and equal interior angles. Its area formula uses the tangent function. For a regular hexagon with six sides of length 2 units, the calculator evaluates 6 × 2² divided by 4 tan(π/6), giving approximately 10.392305 square units.
The symbolic tangent expression remains the mathematical formula. Its decimal value is approximate because π and the tangent calculation generally do not terminate. Irregular polygons require different information, such as coordinates, diagonals or a decomposition into simpler shapes.
Square Unit Conversion
A length conversion factor must be squared when converting area. Since 1 metre equals 100 centimetres, 1 m² equals 100², or 10,000 cm². Applying the linear factor only once is a common error.
This calculator first keeps the shape calculation in the selected input unit. For physical units, it converts the unrounded area through exact metre-based factors. It uses the international inch of exactly 0.0254 metre, the international foot of exactly 0.3048 metre, the international mile of exactly 1609.344 metres, and the international acre of exactly 43,560 international square feet, equal to 4,046.8564224 m².
Conversion reference: NIST revised unit conversion factors. This page labels acres as international acres to avoid confusion with obsolete survey-foot-based values.
Zero Dimensions and Degenerate Shapes
Zero is a valid mathematical boundary input. A rectangle with zero width, a circle with zero radius, or a sector with a zero-degree angle covers no two-dimensional region, so the exact area is 0. The calculator labels such an outcome as degenerate rather than pretending it is an ordinary positive-area shape.
An annulus with equal inner and outer radii has zero width and zero area. An inner radius of zero reduces the annulus formula to a full circle. A 360-degree sector also equals a full circle. Negative dimensions are rejected because ordinary geometric lengths cannot be negative.
Important Shape Assumptions
- Triangle and parallelogram height means perpendicular distance to the selected base.
- The two trapezoid base values must represent parallel sides.
- Rhombus and convex-kite diagonal mode assumes perpendicular diagonals. It is not a formula for every quadrilateral.
- Ellipse inputs are semi-axis lengths, not full axis lengths.
- Annulus circles are concentric, and the outer radius must be at least the inner radius.
- Sector angle is a central angle measured in degrees from 0 through 360.
- Regular polygon mode assumes every side and angle is equal.
Exact Fractions, Precision and Rounding
Dimension values are parsed into reduced fractions before formula arithmetic. A value such as 1/3, 2 1/2 or 1e-20 does not first pass through ordinary floating-point storage. Square, rectangle, triangle, parallelogram, trapezoid and rhombus-or-kite results therefore remain exact within the stated input limits.
π-based shapes keep an exact rational coefficient times π. Regular polygon decimals use trigonometry and are approximate. To reduce last-digit boundary errors, the calculator carries π and tangent calculations through a 120-place internal path before final display rounding. The chosen decimal places affect display rounding only. They do not change the exact coefficient, the stored calculation, or the values used for unit conversions.
Keep enough precision for the task and round once at the end. A measurement with limited real-world accuracy does not become more accurate because the calculator shows extra decimal places.
Common Area Mistakes
- Using perimeter when the problem asks for the region inside the boundary.
- Mixing centimetres and metres in one formula without converting first.
- Applying a linear unit factor once instead of squaring it for area.
- Using a slanted side as triangle or parallelogram height.
- Entering a full ellipse axis where the formula needs a semi-axis.
- Using diameter as circle radius, which makes the calculated area four times too large.
- Rounding π, dimensions or intermediate conversions too early.
- Using a regular-polygon formula for a shape with unequal sides or angles.
Method, Limits and Result Review
Length and angle fields accept up to 50 digits with scientific exponents from −100 through 100. Fraction denominators must be nonzero. Regular polygons require an integer side count from 3 through 10,000. Sector angles must stay between 0 and 360 degrees, and annulus radii must be ordered correctly.
This page performs direct area calculations only. It does not solve missing triangle sides or angles, calculate perimeter or circumference, find 3D surface area or volume, estimate material cost, process map coordinates, or decompose an arbitrary irregular outline. Those tasks require different inputs and assumptions.
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Frequently Asked Questions
How do you calculate the area of a rectangle?
Multiply the rectangle's length by its width. Both dimensions must use the same linear unit, and the answer uses the corresponding square unit.
How do you calculate the area of a triangle?
Multiply the base by the perpendicular height, then divide by 2. A slanted side is not the height unless it is perpendicular to the chosen base.
Why is area measured in square units?
Area counts the two-dimensional region covered by unit squares, so multiplying one length by another produces units such as cm², m² or ft².
Does this calculator accept fractions and scientific notation?
Yes. Dimension fields accept integers, decimals, fractions, proper mixed numbers and scientific notation within the stated input limits.
Can I enter dimensions in different units?
No. Choose one shared input unit and convert every dimension to it before calculating. This prevents hidden mixed-unit errors.
How does the calculator handle pi?
Circle, ellipse, annulus and sector results keep an exact reduced coefficient times π. A separate decimal approximation uses the selected number of places.
What happens when a dimension is zero?
Zero is accepted. If it collapses the shape, the calculator returns exact area 0 and identifies the result as a degenerate shape.
What is the area formula for a trapezoid?
For parallel sides a and b with perpendicular height h, the area is A = (a + b)h/2.
How do you find the area of a rhombus or kite?
For a rhombus or convex kite with perpendicular diagonals d₁ and d₂, use A = d₁d₂/2.
How is annulus area calculated?
Subtract the inner circle area from the outer circle area: A = π(R² − r²). The outer radius must be at least the inner radius.
How is regular polygon area calculated?
For n equal sides of length s, use A = ns²/[4 tan(π/n)]. The trigonometric result is shown as an approximation.
Is an international acre the same as a US survey acre?
This calculator uses the international acre, exactly 43,560 international square feet or 4,046.8564224 square metres. Legacy survey units differ slightly.