api: YAML JSON TOON Database
version: 1.0.0
format: json
dataset:
  slug: geometric-formulas
  title: Geometric Formulas
  description: Common formulas for area, perimeter, surface area, and volume.
  summary: 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_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.
  caveats:
    - "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."
  related:
    - common-mathematical-constants
    - trigonometric-identities
    - spatial-data-structures
  category: Mathematics
  category_slug: mathematics
  tags: math,geometry,formulas
  view_count: 0
  entry_count: 20
  fields:
    - section: data
      field: shape
      type: string
      example: Circle
    - section: data
      field: formula
      type: string
      example: A = pi r^2
    - section: data
      field: variables
      type: string
      example: "A: area, r: radius"
data:
  - shape: Circle
    formula: A = pi r^2
    variables: "A: area, r: radius"
  - shape: Rectangle
    formula: A = l w
    variables: "A: area, l: length, w: width"
  - shape: Triangle
    formula: A = 1/2 b h
    variables: "A: area, b: base, h: height"
  - shape: Sphere Surface Area
    formula: A = 4 pi r^2
    variables: "A: surface area, r: radius"
  - shape: Sphere Volume
    formula: V = 4/3 pi r^3
    variables: "V: volume, r: radius"
  - shape: Cylinder Volume
    formula: V = pi r^2 h
    variables: "V: volume, r: radius, h: height"
  - shape: Pythagorean Theorem
    formula: a^2 + b^2 = c^2
    variables: "c: hypotenuse, a,b: legs of right triangle"
  - shape: Square
    formula: A = s^2
    variables: "A: area, s: side length"
  - shape: Parallelogram
    formula: A = b h
    variables: "A: area, b: base, h: perpendicular height"
  - shape: Trapezoid
    formula: A = (a + b) h / 2
    variables: "A: area, a and b: parallel sides, h: height between them"
  - shape: Ellipse
    formula: A = pi a b
    variables: "A: area, a: semi-major axis, b: semi-minor axis"
  - shape: Regular Polygon
    formula: A = a P / 2
    variables: "A: area, a: apothem, P: perimeter"
  - shape: Triangle (Heron's formula)
    formula: A = sqrt(s (s - a) (s - b) (s - c))
    variables: "A: area, a, b, c: side lengths, s: semi-perimeter (a + b + c) / 2"
  - shape: Cube Volume
    formula: V = s^3
    variables: "V: volume, s: edge length"
  - shape: Rectangular Prism Volume
    formula: V = l w h
    variables: "V: volume, l: length, w: width, h: height"
  - shape: Cylinder Surface Area
    formula: A = 2 pi r (r + h)
    variables: "A: total surface area, r: radius, h: height"
  - shape: Cone Volume
    formula: V = pi r^2 h / 3
    variables: "V: volume, r: base radius, h: height"
  - shape: Pyramid Volume
    formula: V = B h / 3
    variables: "V: volume, B: base area, h: height"
  - shape: Distance Formula
    formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
    variables: "d: distance, (x1, y1) and (x2, y2): points"
  - shape: Law of Cosines
    formula: c^2 = a^2 + b^2 - 2 a b cos(C)
    variables: "a, b, c: side lengths, C: angle opposite c"