Area formulas by shape
The table below summarizes the standard formula and required inputs for each supported shape.
| Shape | Area formula | Inputs required |
|---|---|---|
| Circle | A = πr² | radius r |
| Triangle | A = ½ × base × height | base, height |
| Rectangle | A = base × width | base, width |
| Trapezoid | A = ½ × (a + b) × height | parallel sides a, b, height |
| Parallelogram | A = base × height | base, height |
| Ellipse | A = π × a × b | semi-axes a, b |
| Regular polygon | A = ns² ÷ (4 tan(π/n)) | sides n, side length s |
| Circular sector | A = ½ r²θ | radius r, angle θ |
- A triangle's area from base and height (½ × base × height) is only exact when the height is the perpendicular distance from the base to the opposite vertex, not a slanted side length.
- For an irregular polygon or a triangle where only the three side lengths are known, use Heron's formula instead — see the triangle calculator.
- The regular-polygon formula assumes all n sides and interior angles are equal; it does not apply to irregular polygons.
What is area?
Area is the amount of two-dimensional space enclosed by a shape's boundary, measured in square units (for example square meters or square feet). Every regular shape has a formula that derives its area from a small number of measured dimensions, such as a circle's radius or a rectangle's base and width.
The eight shapes covered here — circle, triangle, rectangle, trapezoid, parallelogram, ellipse, regular polygon and circular sector — cover the shapes most commonly encountered in school geometry, construction take-offs and everyday measurement problems. Each uses a distinct, standard formula rather than an approximation.
How to use this area calculator
- Select the shape you want to measure from the shape dropdown.
- Enter the dimensions requested for that shape — for example radius for a circle, or base and height for a triangle or parallelogram.
- For a trapezoid, enter both parallel sides (a and b) and the perpendicular height between them.
- For a regular polygon, enter the number of sides (n) and the length of one side (a); for a sector, enter the radius and the central angle in degrees.
- Read the area and any secondary result, such as perimeter or arc length, which update instantly as you change values.
The formula behind each shape's area
Each shape uses its own closed-form area formula. Angles entered for a circular sector are converted from degrees to radians before use, since the sector formula is defined in radians.
Common mistakes
- Entering a triangle's slanted side length instead of the perpendicular height, which inflates the computed area.
- Confusing radius and diameter for the circle or sector shape — the formula requires the radius, not the diameter.
- For a sector, entering the angle in radians when the calculator expects degrees (0–360°).
- For a trapezoid, swapping a non-parallel side for one of the two parallel sides a and b.
- Assuming the regular-polygon formula works for irregular shapes with unequal sides or angles.
Häufig gestellte Fragen
How do you find the area of a circle?
The area of a circle equals π times the radius squared (A = πr²). A circle with radius 5 has an area of π × 5² = π × 25 ≈ 78.5398 square units.
What is the area of a triangle?
A triangle's area equals half of its base times its perpendicular height (A = ½ × base × height). A triangle with a base of 4 and a height of 3 has an area of ½ × 4 × 3 = 6 square units.
How do you find the area of a trapezoid?
A trapezoid's area equals half the sum of its two parallel sides, multiplied by the perpendicular height between them: A = ½ × (a + b) × height. With parallel sides 3 and 5 and a height of 4, the area is ½ × (3 + 5) × 4 = 16 square units.
What is the area of a regular polygon?
A regular polygon's area equals n × s² ÷ (4 × tan(π/n)), where n is the number of sides and s is the side length. A regular hexagon (n = 6) with side length 4 has an area of (6 × 4²) ÷ (4 × tan(30°)) ≈ 41.5692 square units.
How is the area of an ellipse calculated?
An ellipse's area equals π times its two semi-axes: A = π × a × b, where a and b are the semi-major and semi-minor axis lengths. An ellipse with semi-axes 3 and 5 has an area of π × 3 × 5 ≈ 47.1239 square units.
What is the area of a circular sector?
A sector's area equals half the radius squared times the central angle in radians: A = ½r²θ. A sector with radius 5 and a 60° central angle (θ = π/3 radians) has an area of ½ × 25 × (π/3) ≈ 13.0900 square units.
Does the calculator give perimeter as well as area?
Yes, for shapes where perimeter is well defined from the given inputs — rectangle, ellipse, regular polygon and sector — the calculator also returns the perimeter (or circumference for a circle, and arc length for a sector).
Quellenangaben
- NIST Handbook 44 / NIST Guide to the SI — standard geometric unit conventions.
- Weisstein, Eric W. "Regular Polygon." MathWorld — A Wolfram Web Resource.
- Standard geometry textbook conventions (e.g. Larson, Geometry, Cengage Learning).