# Area and perimeter calculator for 2D shapes

> Area and perimeter of 15 plane shapes, from rectangles, triangles and trapezoids to circles, ellipses, regular polygons and rings, drawn to scale.

Interactive version: https://www.calcopenly.com/geometry/shape-area-perimeter-calculator
Subject: Geometry calculators

Choose one of 15 plane shapes and enter the dimensions that define it. The area and perimeter come back with the other lengths that shape has: diagonals, heights, arc and chord lengths, a regular polygon's apothem and circumradius, or a ring's width. A triangle from three sides uses Heron's formula, A = √(s(s − a)(s − b)(s − c)) with s half the perimeter, and a regular polygon uses A = ns²/(4 tan(180°/n)).

The default rectangle, 8 cm by 5 cm, has an area of 40 cm², a perimeter of 26 cm and a diagonal of 9.434 cm. Area sizes flooring, turf and paint; perimeter sizes fencing, edging and skirting board. The drawing is to scale, so a mistyped dimension shows up as the wrong shape.

Base and height do not fix a triangle's slanted sides, and a trapezoid needs its legs, so those two inputs give area only. An ellipse's perimeter has no closed form; Ramanujan's second approximation is used, accurate to 0.04% or better.

## Inputs

- **Shape** (options: Square, Rectangle, Triangle (base and height), Triangle (three sides), Parallelogram, Trapezoid (trapezium), Rhombus, Kite, Circle, Semicircle, Circular sector, Circular segment, Ellipse, Regular polygon, Annulus (ring))
- **Side length**
- **Number of sides**
- **Length**
- **Width**
- **Base**
- **Slanted side**
- **Trapezoid defined by** (options: Bases and height, Bases and legs)
- **Bottom base (a)**
- **Top base (b)**
- **Height**
- **Left leg (c)**
- **Right leg (d)**
- **Side a**
- **Side b**
- **Side c**
- **Diagonal p**
- **Diagonal q**
- **Shorter pair of sides (a)**
- **Longer pair of sides (b)**
- **Angle between a and b**
- **Radius**
- **Central angle**: Between 0° and 360°
- **Semi-axis a**
- **Semi-axis b**
- **Outer radius (R)**
- **Inner radius (r)**
- **Show results in** (options: Millimetres (mm), Centimetres (cm), Metres (m), Kilometres (km), Inches (in), Feet (ft), Yards (yd), Miles (mi))

## Results

- Area — main result
- Perimeter
- Diagonal
- Second diagonal
- Side length
- Height
- Arc length
- Chord length
- Apothem (inradius)
- Circumradius
- Ring width

## Formula

$$
\begin{gathered} A_\triangle = \sqrt{s(s-a)(s-b)(s-c)},\quad A_{n\text{-gon}} = \frac{n s^2}{4\tan(\pi/n)} \\ P_\text{ellipse} \approx \pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right) \end{gathered}
$$

## Worked examples

### Rectangle 8 × 5 cm

- Shape: Rectangle
- Length: 8 cm
- Width: 5 cm
- Show results in: Centimetres (cm)
- **Area: 40 cm²**
- **Perimeter: 26 cm**
- **Diagonal: 9.433981 cm**
- Checked against: 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89)

### 3-4-5 triangle by Heron's formula

- Shape: Triangle (three sides)
- Side a: 3 cm
- Side b: 4 cm
- Side c: 5 cm
- Show results in: Centimetres (cm)
- **Area: 6 cm²**
- **Perimeter: 12 cm**
- **Height: 4 cm**
- Checked against: Right triangle with legs 3 and 4: ½·3·4 = 6; height onto side 3 is the other leg, 4

### Regular hexagon, side 2 m

- Shape: Regular polygon
- Side length: 2 m
- Number of sides: 6
- Show results in: Metres (m)
- **Area: 10.392305 m²**
- **Perimeter: 12 m**
- **Apothem (inradius): 1.732051 m**
- **Circumradius: 2 m**
- Checked against: Python 3.8 math: 6*4/(4*tan(pi/6)) = 6√3, 2/(2*tan(pi/6)) = √3; a hexagon's circumradius equals its side

### Square as a regular 4-gon (exact tan 45°)

- Shape: Regular polygon
- Side length: 3 cm
- Number of sides: 4
- Show results in: Centimetres (cm)
- **Area: 9 cm²**
- **Apothem (inradius): 1.5 cm**
- **Circumradius: 2.12132 cm**
- Checked against: Square of side 3: area 9, apothem 3/2; Python 3.8 math: 3/sqrt(2)

### Ellipse with a = b is a circle (edge case)

- Shape: Ellipse
- Semi-axis a: 5 cm
- Semi-axis b: 5 cm
- Show results in: Centimetres (cm)
- **Area: 78.539816 cm²**
- **Perimeter: 31.415927 cm**
- Checked against: h = 0 so Ramanujan II reduces to 2πr; Python 3.8 math: 25*pi, 10*pi

### Ellipse 10 × 6 (semi-axes)

- Shape: Ellipse
- Semi-axis a: 10 cm
- Semi-axis b: 6 cm
- Show results in: Centimetres (cm)
- **Area: 188.495559 cm²**
- **Perimeter: 51.053998 cm**
- Checked against: Python 3.8 math: 60*pi and Ramanujan II with h = 1/16; the exact perimeter by the AGM series (Python decimal, 40 digits) is 51.0539977268, 1.2e-9 higher

## Questions

### How do you find the area of a trapezoid?

Average the two parallel sides and multiply by the perpendicular height: A = (a + b)/2 × h. Bases of 10 cm and 4 cm with a height of 4 cm give 7 × 4 = 28 cm². British English calls this shape a trapezium; the formula is the same. The perimeter also needs the two slanted legs: legs of 5 cm each make it 24 cm.

### How do you find the area of a triangle from three sides?

Use Heron's formula. Take the semi-perimeter s = (a + b + c)/2, then A = √(s(s − a)(s − b)(s − c)). Sides of 5, 6 and 7 cm give s = 9 and A = √(9 × 4 × 3 × 2) = √216 ≈ 14.697 cm². Three lengths only make a triangle if each is shorter than the other two combined, so 3, 4 and 8 have no area.

### What is the area of a regular hexagon?

A = (3√3/2)s², about 2.598 times the side squared. A hexagon with 5 cm sides covers 64.952 cm², and one with 2 m sides covers 10.392 m². Any regular polygon with n sides of length s has A = ns²/(4 tan(180°/n)). A hexagon's circumradius equals its side, and its apothem is s√3/2 ≈ 0.866s.

### Is there a formula for the perimeter of an ellipse?

Not an exact one in elementary functions: the exact perimeter is a complete elliptic integral. Ramanujan's 1914 approximation, P ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))) with h = (a − b)²/(a + b)², is off by at most 0.04%, reached only as the ellipse flattens into a line. For semi-axes of 10 and 6 it gives 51.0539977, within 1.2 × 10⁻⁹ of the exact value.

### How do you convert square feet to square metres?

Multiply by 0.09290304, the square of the international foot (0.3048 m exactly, fixed in 1959). One square metre is about 10.764 ft². A 12 ft × 10 ft room is 120 ft², or 11.148 m². Area scales with the square of the length unit, so converting each side first and then multiplying gives the same result.

### How accurate is the area and perimeter calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 16 worked examples whose answers come from independent sources; for example, “Rectangle 8 × 5 cm” is checked against 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89).

### Where does the method come from?

Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld; Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II); NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm).

## Sources

- [Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld](https://mathworld.wolfram.com/Ellipse.html)
- Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II)
- [NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm)](https://www.nist.gov/pml/owm/nist-handbook-44-current-edition)