Every flat shape has two everyday measurements: how much surface it covers (area) and how far it is around (perimeter). This page gathers the formulas for 14 common shapes in one place. Choose a shape in the tool and it shows the full formula set as a table, works each formula with your own dimensions and draws a labeled diagram, so you can check homework, a floor plan or a cutting list without hunting through a textbook.
How to use the 2D formula sheet
- Pick a Shape: square, rectangle, triangle (three sides), right triangle, equilateral triangle, parallelogram, rhombus, isosceles trapezoid, circle, circle sector, annulus, ellipse, regular polygon or stadium.
- Enter the dimensions that appear for that shape, such as the three sides of a triangle or the two diagonals of a rhombus. The sector angle is in degrees.
- Choose the Units. The area comes out in square units and lengths in the same unit you entered.
- Read the area at the bottom of the tape, the perimeter and extra measures above it, the formulas with your numbers under Formulas with your numbers, and the general formula table below the diagram.
For deeper work on one shape, such as solving backward from an area, use the dedicated tools like the trapezoid calculator or the regular polygon calculator.
Area and perimeter formulas
| Shape | Area | Perimeter |
|---|---|---|
| Square (side s) | s² | 4s |
| Rectangle (l × w) | lw | 2(l + w) |
| Triangle (sides a, b, c) | √(s(s − a)(s − b)(s − c)), s = half the perimeter | a + b + c |
| Triangle (base b, height h) | ½bh | sum of the sides |
| Right triangle (legs a, b) | ½ab | a + b + √(a² + b²) |
| Equilateral triangle (side s) | (√3 ⁄ 4)s² ≈ 0.4330s² | 3s |
| Parallelogram (base b, side a, height h) | bh | 2(a + b) |
| Rhombus (diagonals p, q) | pq ⁄ 2 | 4 × ½√(p² + q²) |
| Trapezoid (bases a, b, height h) | ½(a + b)h | a + b + both legs |
| Circle (radius r) | πr² | 2πr |
| Sector (radius r, angle θ) | (θ ⁄ 360°)πr² | 2r + (θ ⁄ 360°)2πr |
| Annulus (radii R > r) | π(R² − r²) | 2π(R + r) for both edges |
| Ellipse (semi-axes a, b) | πab | ≈ π(a + b)(1 + 3h ⁄ (10 + √(4 − 3h))) |
| Regular polygon (n sides of s) | ns² ⁄ (4 tan(180° ⁄ n)) | ns |
| Stadium (radius r, straight a) | πr² + 2ra | 2πr + 2a |
In the ellipse row, h = (a − b)² ⁄ (a + b)².
Worked example: a trapezoid garden bed
A raised bed runs 20 ft along a fence, 14 ft along the front, and is 8 ft deep, with both short sides angled equally.
Choose Trapezoid (isosceles) and enter a = 20, b = 14, h = 8 in feet.
- Area: ½ × (20 + 14) × 8 = 136 ft², the figure you need for soil or mulch.
- Each slanted side spans (20 − 14) ÷ 2 = 3 ft sideways while rising 8 ft, so its length is √(8² + 3²) ≈ 8.544 ft.
- Perimeter for edging: 20 + 14 + 2 × 8.544 ≈ 51.09 ft.
Patterns that make the formulas easier to remember
Everything starts with the rectangle
A parallelogram can be cut along its height and slid into a rectangle, so its area is base × height. A triangle is half of a parallelogram, so it gets ½bh. A trapezoid is two triangles sharing a height, giving ½(a + b)h. A regular polygon splits into n identical triangles whose height is the apothem, which is why its area is half the perimeter times the apothem. A circle behaves like a polygon with endlessly many sides: half of 2πr times r is πr².
Scaling changes area faster than length
Multiply every length by k and the perimeter grows by k, but the area grows by k². Doubling the side of a 5 cm square takes the area from 25 cm² to 100 cm², four times as much. That is why a 16-inch pizza holds four times the area of an 8-inch one.
Same perimeter, different area
With a fixed 40 cm of border, the shape you choose changes the enclosed area a lot:
| Shape with a 40 cm perimeter | Area |
|---|---|
| 15 × 5 rectangle | 75 cm² |
| Equilateral triangle | 76.98 cm² |
| Square | 100 cm² |
| Regular hexagon | 115.47 cm² |
| Regular octagon | 120.71 cm² |
| Circle | 127.32 cm² |
The circle always wins, a fact known as the isoperimetric inequality. The more sides a regular polygon has, the closer it gets to the circle.
Common mistakes
- Using a slanted side as the height in ½bh or bh.
- Mixing units, such as a length in feet and a width in inches.
- Entering a diameter where the formula expects a radius, which makes a circle’s area four times too large.
- Forgetting that a sector angle must be converted to radians before using ½r²θ. The tool does this for you.
For the three-dimensional counterparts (volumes and surface areas), see the 3D shape formulas.
Frequently asked questions
What is the difference between area and perimeter?
Perimeter is the distance around the outside of a shape, measured in plain length units such as feet. Area is the amount of flat surface inside it, measured in square units such as square feet. Fencing a yard is a perimeter question; seeding it is an area question.
Why do so many area formulas contain one-half?
Triangles, trapezoids, sectors and regular polygons can all be built from, or cut out of, a rectangle or parallelogram of the same base and height. A triangle is exactly half of such a parallelogram, which is where the ½ in ½bh comes from, and the other formulas inherit it.
Which height do I use in base-times-height formulas?
Always the perpendicular height: the straight-line distance at a right angle to the base. For a parallelogram or a slanted triangle this is shorter than the slanted side. Using the slanted side instead makes the area too large.
Is there an exact formula for the perimeter of an ellipse?
Not in terms of ordinary functions. The exact perimeter is an elliptic integral. This page uses Ramanujan's 1914 approximation, which is accurate to a few parts per million for everyday ellipses, and also shows the exact value from a fast-converging series.
How do the formulas change if I use different units?
They do not change, but the answers carry the units you enter: lengths in feet give areas in square feet. Convert every dimension to the same unit before you start, because mixing inches and feet is the most common source of wrong answers.