Geometric Formulas
Common formulas for area, perimeter, surface area, and volume.
Area, surface area and volume formulas for the common plane and solid shapes, with the Pythagorean theorem, plus trapezoids, ellipses, cones, pyramids, regular polygons, Heron's formula, the distance formula and the law of cosines. Each row names the variables so the formula can be applied directly.
Use it when
- You need the formula for the area or volume of a shape and the meaning of each symbol.
- You are checking a geometry calculation.
Watch out for
- Units multiply: areas are in square units and volumes in cubic units.
- Pick the right measurement: height is perpendicular to the base, not the slanted side.
The data
20 rows
| Shape | Formula | Variables |
|---|---|---|
| Circle | A = pi r^2 | A: area, r: radius |
| Rectangle | A = l w | A: area, l: length, w: width |
| Triangle | A = 1/2 b h | A: area, b: base, h: height |
| Sphere Surface Area | A = 4 pi r^2 | A: surface area, r: radius |
| Sphere Volume | V = 4/3 pi r^3 | V: volume, r: radius |
| Cylinder Volume | V = pi r^2 h | V: volume, r: radius, h: height |
| Pythagorean Theorem | a^2 + b^2 = c^2 | c: hypotenuse, a,b: legs of right triangle |
| Square | A = s^2 | A: area, s: side length |
| Parallelogram | A = b h | A: area, b: base, h: perpendicular height |
| Trapezoid | A = (a + b) h / 2 | A: area, a and b: parallel sides, h: height between them |
| Ellipse | A = pi a b | A: area, a: semi-major axis, b: semi-minor axis |
| Regular Polygon | A = a P / 2 | A: area, a: apothem, P: perimeter |
| Triangle (Heron's formula) | A = sqrt(s (s - a) (s - b) (s - c)) | A: area, a, b, c: side lengths, s: semi-perimeter (a + b + c) / 2 |
| Cube Volume | V = s^3 | V: volume, s: edge length |
| Rectangular Prism Volume | V = l w h | V: volume, l: length, w: width, h: height |
| Cylinder Surface Area | A = 2 pi r (r + h) | A: total surface area, r: radius, h: height |
| Cone Volume | V = pi r^2 h / 3 | V: volume, r: base radius, h: height |
| Pyramid Volume | V = B h / 3 | V: volume, B: base area, h: height |
| Distance Formula | d = sqrt((x2 - x1)^2 + (y2 - y1)^2) | d: distance, (x1, y1) and (x2, y2): points |
| Law of Cosines | c^2 = a^2 + b^2 - 2 a b cos(C) | a, b, c: side lengths, C: angle opposite c |
Fields
The shape of this dataset, read from the data itself: every column, its type and a real value. Agents can use this to plan a query without fetching the whole document.
| Section | Field | Type | Example |
|---|---|---|---|
| data | shape | string | Circle |
| data | formula | string | A = pi r^2 |
| data | variables | string | A: area, r: radius |
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