The 2D Shape Calculator calculates all the main properties of a flat shape at once — not just the area, but also the perimeter, diagonals, heights and angles.
For a triangle it uses Heron’s formula for area and the law of cosines for the three angles; for a parallelogram it finds the height and both diagonals from two sides and the angle between them. It covers squares, rectangles, triangles, circles, ellipses, parallelograms, trapezoids, rhombuses and regular polygons.
2D Shape Calculator
All the main properties of 2D shapes — area, perimeter, diagonals and angles — for squares, rectangles, triangles, circles, ellipses, parallelograms, trapezoids, rhombuses and regular polygons.
Formula
| Shape | Area | Other properties |
|---|---|---|
| Square | a² | P = 4a, d = a√2 |
| Rectangle | lw | P = 2(l + w), d = √(l² + w²) |
| Triangle | √(p(p − a)(p − b)(p − c)) | angles by the law of cosines |
| Circle | πr² | C = 2πr, d = 2r |
| Ellipse | πab | P ≈ π[3(a + b) − √((3a + b)(a + 3b))] |
| Parallelogram | ab × sin θ | h = b sin θ, two diagonals |
| Trapezoid | ½(a + b)h | perimeter, midline (a + b) ÷ 2 |
| Rhombus | a² × sin θ | P = 4a, two diagonals |
| Regular polygon | ns² ÷ (4 tan(π/n)) | apothem, circumradius, interior angle (n − 2) × 180° ÷ n |
Where: a, b, c = sides, l = length, w = width, r = radius, h = height, θ = angle between two sides, n = number of sides, s = side length.
How the Calculation Works
Step 1: The calculator uses your inputs to find the area with the shape’s formula.
Step 2: It uses geometry rules to find the remaining properties. For example, the law of cosines, cos A = (b² + c² − a²) ÷ 2bc, gives each triangle angle.
Step 3: All results are listed together in your chosen units.
Variables Explained
| Property | Meaning |
|---|---|
| Perimeter | Distance around the shape |
| Diagonal | Straight line between opposite corners |
| Apothem | Distance from a regular polygon’s center to the middle of a side |
| Circumradius | Distance from the center to a corner |
| Midline | Average of a trapezoid’s two bases |
Example Calculation
Triangle with sides 5, 6 and 7 cm
Area: p = 9, A = √(9 × 4 × 3 × 2) ≈ 14.70 cm²
Angles: A ≈ 44.42°, B ≈ 57.12°, C ≈ 78.46° (they add up to 180°)
Parallelogram
Given: base 8 cm, side 5 cm, angle 60°
Area = 8 × 5 × sin 60° ≈ 34.64 cm²; height ≈ 4.33 cm; diagonals ≈ 11.36 cm and 7 cm
Regular hexagon
Given: side 4 cm
Area ≈ 41.57 cm²; perimeter 24 cm; apothem ≈ 3.46 cm; interior angle 120°
Common Area Conversion Table
| Unit | Equivalent |
|---|---|
| 1 m² | 10,000 cm² |
| 1 m² | 10.7639 ft² |
| 1 ft² | 144 in² |
Shape Properties
Quadrilateral family
Squares, rectangles, rhombuses and parallelograms all have two pairs of parallel sides. A square is both a rectangle (right angles) and a rhombus (equal sides). A trapezoid has only one pair of parallel sides.
Ellipse perimeter
An ellipse has no exact simple perimeter formula, so the calculator uses Ramanujan’s approximation, which is very accurate for everyday shapes.
Dimensions Explained
For trapezoids, enter the perpendicular height and both slanted legs; the legs affect only the perimeter. For rhombuses and parallelograms, the angle is the interior angle between two adjacent sides.
Frequently Asked Questions
What are the properties of a 2D shape?
The main ones are area, perimeter, side lengths, angles and, for many shapes, diagonals.
How do you find the angles of a triangle from its sides?
Use the law of cosines: cos A = (b² + c² − a²) ÷ (2bc), and similarly for the others.
How do you find the diagonal of a rectangle?
d = √(l² + w²). An 8 × 6 rectangle has a 10-unit diagonal.
What is the interior angle of a regular polygon?
(n − 2) × 180° ÷ n. A regular hexagon’s angles are 120°.
What is the perimeter of an ellipse?
Approximately π[3(a + b) − √((3a + b)(a + 3b))] using Ramanujan’s formula.
