d4t_formulas/assets/formulas/formulas.d4rt

[

{
  "name": "Temperature converter",
  "description": "Example of simple formula, just unit conversion",
  "input": [
    {"name": "Input", "unit": "Kelvin" }
  ],
  "output": {"name": "Output", "unit": "Kelvin" },
  "d4rtCode": "Output = Input;",
  "tags": ["converter", "temperature" ]
},
  // Free fall distance (vertical)
  {
    "name": "Free Fall Distance",
    "description": r"""
Calculates vertical displacement under constant gravity

$$h = \frac{1}{2}gt^2$$

Where:
- $g$: Gravitational acceleration $9.81\ \mathrm{m/s^2}$ on Earth
- $t$: Time in free fall (seconds)

![Free Fall Diagram](https://altcalculator.com/wp-content/uploads/2023/08/Free-Fall.png)""",
    "input": [
      {"name": "t", "unit": "second"},  // Time in seconds
      {"name": "g", "unit": "meters per second"}  // Gravitational acceleration
    ],
    "output": {"name": "h", "unit": "meter"},  // Height in meters
    "d4rtCode": "h = 0.5 * g * pow(t, 2);",
    "tags": ["physics", "kinematics"]
  },
  
  // Newton's Law of Universal Gravitation
  {
    "name": "Gravitational Force",
    "description": r'''
Newton's law of universal gravitation

\(F = G\frac{m_1m_2}{r^2}\)

Where:
- $G$: Gravitational constant $6.674\times 10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
- $m_1, m_2$: Masses of two objects
- $r$: Distance between centers of masses

![Gravitation](https://upload.wikimedia.org/wikipedia/commons/thumb/3/33/NewtonsLawOfUniversalGravitation.svg/1200px-NewtonsLawOfUniversalGravitation.svg.png)''',
    "input": [
      {"name": "m1", "unit": "kilogram"},  // Mass 1
      {"name": "m2", "unit": "kilogram"},  // Mass 2
      {"name": "r", "unit": "meter"}     // Distance between masses
    ],
    "output": {"name": "F", "unit": "newton"},  // Force in newtons
    "d4rtCode": "F = (6.67430e-11 * m1 * m2) / pow(r, 2);",
    "tags": ["physics", "astronomy", "gravity"]
  },
  
  // Kinetic Energy
  {
    "name": "Kinetic Energy",
    "description": r'''
Energy possessed by a moving object

$$KE = \frac{1}{2}mv^2$$

Where:
- $m$: Mass of object
- $v$: Velocity of object

![Kinetic Energy](https://upload.wikimedia.org/wikipedia/commons/thumb/4/44/Kinetic_energy.svg/1200px-Kinetic_energy.svg.png)''',
    "input": [
      {"name": "m", "unit": "kilogram"},  // Mass
      {"name": "v", "unit": "meters per second"}  // Velocity
    ],
    "output": {"name": "KE", "unit": "joule"},  // Energy in joules
    "d4rtCode": "KE = 0.5 * m * pow(v, 2);",
    "tags": ["physics", "energy", "mechanics"]
  },
  
  // Projectile Motion Range
  {
    "name": "Projectile Range",
    "description": r"""Calculates horizontal distance of projectile motion
                 $$R = \frac{v^2 \sin(2\theta)}{g}$$
                 Where:
                 - $v$: Initial velocity
                 - $\theta$: Launch angle
                 - $g$: Gravitational acceleration
                 ![Projectile Motion](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Projectile_motion_diagram.png/800px-Projectile_motion_diagram.png)""",
    "input": [
      {"name": "v", "unit": "meters per second"},  // Initial velocity
      {"name": "a", "unit": "degree"}   // Launch angle
    ],
    "output": {"name": "R", "unit": "meter"},  // Horizontal distance
    "d4rtCode": """
       var radians = a * (pi / 180);
       R = (pow(v, 2) * sin(2 * radians)) / 9.80665;
    """,
    "tags": ["physics", "kinematics", "projectile"]
  },

  {
    "name": "Newton's Second Law",
    "description": r'''
Force equals mass times acceleration

$$F = m \cdot a$$

Where:
- $m$: Mass of object ($\mathrm{kg}$)
- $a$: Acceleration ($\mathrm{m/s^2}$)

![Newton's Second Law](https://upload.wikimedia.org/wikipedia/commons/thumb/7/73/Newtonslawsofmotion.jpg/800px-Newtonslawsofmotion.jpg)''',
    "input": [
      {"name": "m", "unit": "kilogram"},  // Mass
      {"name": "a", "unit": "meters per square second"} // Acceleration
    ],
    "output": {"name": "F", "unit": "newton"},  // Force in newtons
    "d4rtCode": "F = m * a;",
    "tags": ["physics", "mechanics", "newton"]
  },

  // Apgar Score
  {
    "name": "Apgar Score",
    "description": "Newborn health assessment scoring system\n\nScores 0-2 for:\n1. Heart rate\n2. Breathing\n3. Muscle tone\n4. Reflexes\n5. Skin color\nTotal score 0-10",
    "input": [
      {"name": "HeartRate", "values": ["Absent", "< 100 bpm>", "> 100 bpm"] },
      {"name": "Breathing", "values": ["Absent", "Weak, irregular", "Strong, robust cry"] },
      {"name": "MuscleTone", "values": ["None", "Some", "Flexed arms/leg, resists extension"] },
      {"name": "Reflexes", "values": ["No response", "Grimace on aggressive stimulation", "Cry on stimulation"] },
      {"name": "SkinColor", "values": ["Blue or pale", "Blue extremities, pink body", "Pink"] }
    ],
    "output": {"name": "Result", "unit": "string"},
    "d4rtCode": """
      var total = indexOf("HeartRate") + indexOf("Breathing") + indexOf("MuscleTone") + indexOf("Reflexes") + indexOf("SkinColor");
      late var interpretation;
      if( total < 4 ) {
        interpretation = 'Critical condition';
      }
      else if( total < 7 ){
        interpretation = 'Needs assistance';
      }
      else {
        interpretation = 'Normal';
      }
      Result = 'Score: \$total - \$interpretation';
    """,
    "tags": ["medical", "pediatrics", "assessment"]
  }
  ,
  {
    "name": "Compare price per mass",
    "description": "Compares two products by their price per mass and returns which is cheaper, including price per kg for each product.",
    "input": [
      {"name": "price1", "unit": "currency"},
      {"name": "mass1", "unit": "kilogram"},
      {"name": "price2", "unit": "currency"},
      {"name": "mass2", "unit": "kilogram"}
    ],
    "output": {"name": "Result", "unit": "string"},
    "d4rtCode": """
      var p1 = price1 / mass1;
      var p2 = price2 / mass2;
      if (p1 < p2) {
        Result = 'first product is cheaper at \${p1.toStringAsFixed(2)} currency/kg vs \${p2.toStringAsFixed(2)} currency/kg';
      } else if (p2 < p1) {
        Result = 'second product is cheaper at \${p2.toStringAsFixed(2)} currency/kg vs \${p1.toStringAsFixed(2)} currency/kg';
      } else {
        Result = 'both products have the same price per mass at \${p1.toStringAsFixed(2)} currency/kg';
      }
    """,
    "tags": ["comparison", "shopping", "economics"]
  }
  ,

  // Einstein's Mass-Energy Equivalence
  {
    "name": "Mass-Energy Equivalence",
    "description": r'''
Einstein's famous equation showing the relationship between mass and energy

$$E = mc^2$$

Where:
- $E$: Energy (Joules)
- $m$: Mass (kilograms)
- $c$: Speed of light $299,792,458$ $\mathrm{m/s}$

This equation shows that mass can be converted to energy and vice versa.''',
    "input": [
      {"name": "m", "unit": "kilogram"}  // Mass
    ],
    "output": {"name": "E", "unit": "joule"},  // Energy
    "d4rtCode": "E = m * pow(299792458, 2);",
    "tags": ["physics", "relativity", "energy"]
  },

  // Ohm's Law
  {
    "name": "Ohm's Law",
    "description": r'''
Relationship between voltage, current, and resistance in electrical circuits

$$V = IR$$

Where:
- $V$: Voltage (Volts)
- $I$: Current (Amperes)
- $R$: Resistance (Ohms)

This fundamental law describes how current flows through resistive materials.''',
    "input": [
      {"name": "I", "unit": "ampere"},  // Current
      {"name": "R", "unit": "ohm"}     // Resistance
    ],
    "output": {"name": "V", "unit": "volt"},  // Voltage
    "d4rtCode": "V = I * R;",
    "tags": ["physics", "electricity", "electronics"]
  },

  // Hooke's Law
  {
    "name": "Hooke's Law",
    "description": r'''
Force exerted by a spring is proportional to its displacement

$$F = -kx$$

Where:
- $F$: Restoring force (Newtons)
- $k$: Spring constant (N/m)
- $x$: Displacement from equilibrium (meters)

The negative sign indicates the force opposes the displacement.''',
    "input": [
      {"name": "k", "unit": "newton per meter"},  // Spring constant
      {"name": "x", "unit": "meter"}             // Displacement
    ],
    "output": {"name": "F", "unit": "newton"},   // Force
    "d4rtCode": "F = -k * x;",
    "tags": ["physics", "elasticity", "oscillations"]
  },

  // Centripetal Force
  {
    "name": "Centripetal Force",
    "description": r'''
Force required to keep an object moving in circular motion

$$F = \frac{mv^2}{r}$$

Where:
- $F$: Centripetal force (Newtons)
- $m$: Mass of object (kilograms)
- $v$: Velocity (m/s)
- $r$: Radius of circular path (meters)

This force acts toward the center of the circle.''',
    "input": [
      {"name": "m", "unit": "kilogram"},         // Mass
      {"name": "v", "unit": "meters per second"}, // Velocity
      {"name": "r", "unit": "meter"}             // Radius
    ],
    "output": {"name": "F", "unit": "newton"},   // Force
    "d4rtCode": "F = (m * pow(v, 2)) / r;",
    "tags": ["physics", "circular motion", "centripetal"]
  },

  // Wave Equation
  {
    "name": "Wave Equation",
    "description": r'''
Relationship between wave speed, frequency, and wavelength

$$v = f\lambda$$

Where:
- $v$: Wave speed (m/s)
- $f$: Frequency (Hertz)
- $\lambda$: Wavelength (meters)

This applies to all types of waves including sound and light.''',
    "input": [
      {"name": "f", "unit": "hertz"},    // Frequency
      {"name": "lambda", "unit": "meter"} // Wavelength
    ],
    "output": {"name": "v", "unit": "meters per second"}, // Wave speed
    "d4rtCode": "v = f * lambda;",
    "tags": ["physics", "waves", "frequency"]
  },

  // Pythagorean Theorem
  {
    "name": "Pythagorean Theorem",
    "description": r'''
Fundamental relation in Euclidean geometry among the three sides of a right triangle

$$a^2 + b^2 = c^2$$

Where:
- $a$, $b$: Legs of the right triangle
- $c$: Hypotenuse of the right triangle

The square of the hypotenuse is equal to the sum of squares of the other two sides.''',
    "input": [
      {"name": "a", "unit": "meter"},  // First leg
      {"name": "b", "unit": "meter"}   // Second leg
    ],
    "output": {"name": "c", "unit": "meter"},  // Hypotenuse
    "d4rtCode": "c = sqrt(pow(a, 2) + pow(b, 2));",
    "tags": ["trigonometry", "geometry", "pythagorean"]
  },

  // Sine Rule
  {
    "name": "Sine Rule",
    "description": r'''
Relationship between the sides and angles of any triangle

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Where:
- $a$, $b$, $c$: Sides of the triangle
- $A$, $B$, $C$: Angles opposite to sides $a$, $b$, $c$ respectively

This rule is useful for solving triangles when certain combinations of angles and sides are known.''',
    "input": [
      {"name": "a", "unit": "meter"},      // Side a
      {"name": "A", "unit": "degree"},     // Angle A in degrees
      {"name": "B", "unit": "degree"}      // Angle B in degrees
    ],
    "output": {"name": "b", "unit": "meter"},  // Side b
    "d4rtCode": """
       var angleARad = A * (pi / 180);
       var angleBRad = B * (pi / 180);
       b = (a * sin(angleBRad)) / sin(angleARad);
    """,
    "tags": ["trigonometry", "triangle", "sine"]
  },

  // Cosine Rule
  {
    "name": "Cosine Rule",
    "description": r'''
Generalization of the Pythagorean theorem for any triangle

$$c^2 = a^2 + b^2 - 2ab\cos(C)$$

Where:
- $a$, $b$, $c$: Sides of the triangle
- $C$: Angle opposite to side $c$

This rule relates all three sides of a triangle to one of its angles.''',
    "input": [
      {"name": "a", "unit": "meter"},      // Side a
      {"name": "b", "unit": "meter"},      // Side b
      {"name": "C", "unit": "degree"}      // Angle C in degrees
    ],
    "output": {"name": "c", "unit": "meter"},  // Side c
    "d4rtCode": """
       var angleCRad = C * (pi / 180);
       c = sqrt(pow(a, 2) + pow(b, 2) - 2*a*b*cos(angleCRad));
    """,
    "tags": ["trigonometry", "triangle", "cosine"]
  },

  // Trigonometric Identity
  {
    "name": "Trigonometric Identity",
    "description": r'''
Fundamental Pythagorean identity in trigonometry

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

Where:
- $\theta$: Any angle in radians or degrees

This identity is derived from the Pythagorean theorem applied to the unit circle.''',
    "input": [
      {"name": "theta", "unit": "degree"}  // Angle in degrees
    ],
    "output": {"name": "result", "unit": "scalar"},  // Result (should be 1)
    "d4rtCode": """
       var thetaRad = theta * (pi / 180);
       result = pow(sin(thetaRad), 2) + pow(cos(thetaRad), 2);
    """,
    "tags": ["trigonometry", "identity", "sine", "cosine"]
  },

  // Gravitational Potential Energy
  {
    "name": "Gravitational Potential Energy",
    "description": r'''
Energy possessed by an object due to its position in a gravitational field

$$PE = mgh$$

Where:
- $m$: Mass of object (kilograms)
- $g$: Gravitational acceleration ($9.81\ \mathrm{m/s^2}$ on Earth)
- $h$: Height above reference point (meters)

This energy increases with height and can be converted to kinetic energy.''',
    "input": [
      {"name": "m", "unit": "kilogram"},  // Mass
      {"name": "h", "unit": "meter"},     // Height
      {"name": "g", "unit": "meters per square second"}  // Gravitational acceleration
    ],
    "output": {"name": "PE", "unit": "joule"},  // Potential energy
    "d4rtCode": "PE = m * g * h;",
    "tags": ["physics", "energy", "mechanics", "gravity"]
  },

  // Linear Momentum
  {
    "name": "Linear Momentum",
    "description": r'''
Quantity of motion possessed by a moving object

$$p = mv$$

Where:
- $p$: Momentum ($\mathrm{kg \cdot m/s}$)
- $m$: Mass (kilograms)
- $v$: Velocity (m/s)

Momentum is conserved in isolated systems.''',
    "input": [
      {"name": "m", "unit": "kilogram"},  // Mass
      {"name": "v", "unit": "meters per second"}  // Velocity
    ],
    "output": {"name": "p", "unit": "kilogram meter per second"},  // Momentum
    "d4rtCode": "p = m * v;",
    "tags": ["physics", "mechanics", "momentum"]
  },

  // Density
  {
    "name": "Density",
    "description": r'''
Mass per unit volume of a substance

$$\rho = \frac{m}{V}$$

Where:
- $\rho$: Density ($\mathrm{kg/m^3}$)
- $m$: Mass (kilograms)
- $V$: Volume (cubic meters)

Density is an intrinsic property of materials.''',
    "input": [
      {"name": "m", "unit": "kilogram"},  // Mass
      {"name": "V", "unit": "cubic meter"}  // Volume
    ],
    "output": {"name": "rho", "unit": "kilogram per cubic meter"},  // Density
    "d4rtCode": "rho = m / V;",
    "tags": ["physics", "mechanics", "material"]
  },

  // Pressure
  {
    "name": "Pressure",
    "description": r'''
Force applied perpendicular to a surface per unit area

$$P = \frac{F}{A}$$

Where:
- $P$: Pressure (Pascals)
- $F$: Force (Newtons)
- $A$: Area (square meters)

Pressure is transmitted equally in all directions in fluids.''',
    "input": [
      {"name": "F", "unit": "newton"},  // Force
      {"name": "A", "unit": "square meter"}  // Area
    ],
    "output": {"name": "P", "unit": "pascal"},  // Pressure
    "d4rtCode": "P = F / A;",
    "tags": ["physics", "mechanics", "fluid"]
  },

  // Work
  {
    "name": "Work",
    "description": r'''
Energy transferred when a force moves an object

$$W = Fd\cos(\theta)$$

Where:
- $W$: Work (Joules)
- $F$: Force (Newtons)
- $d$: Displacement (meters)
- $\theta$: Angle between force and displacement

Maximum work occurs when force and displacement are parallel.''',
    "input": [
      {"name": "F", "unit": "newton"},  // Force
      {"name": "d", "unit": "meter"},   // Displacement
      {"name": "theta", "unit": "degree"}  // Angle
    ],
    "output": {"name": "W", "unit": "joule"},  // Work
    "d4rtCode": """
       var thetaRad = theta * (pi / 180);
       W = F * d * cos(thetaRad);
    """,
    "tags": ["physics", "energy", "mechanics"]
  },

  // Power
  {
    "name": "Power",
    "description": r'''
Rate at which work is done or energy is transferred

$$P = \frac{W}{t}$$

Where:
- $P$: Power (Watts)
- $W$: Work or Energy (Joules)
- $t$: Time (seconds)

Power measures how quickly energy is used or transferred.''',
    "input": [
      {"name": "W", "unit": "joule"},  // Work or Energy
      {"name": "t", "unit": "second"}  // Time
    ],
    "output": {"name": "P", "unit": "watt"},  // Power
    "d4rtCode": "P = W / t;",
    "tags": ["physics", "energy", "mechanics"]
  },

  // Coulomb's Law
  {
    "name": "Coulomb's Law",
    "description": r'''
Force between two electrically charged particles

$$F = k_e\frac{q_1q_2}{r^2}$$

Where:
- $F$: Electrostatic force (Newtons)
- $k_e$: Coulomb's constant $8.988\times 10^9\ \mathrm{N\cdot m^2/C^2}$
- $q_1, q_2$: Electric charges (Coulombs)
- $r$: Distance between charges (meters)

Like charges repel, opposite charges attract.''',
    "input": [
      {"name": "q1", "unit": "coulomb"},  // Charge 1
      {"name": "q2", "unit": "coulomb"},  // Charge 2
      {"name": "r", "unit": "meter"}      // Distance
    ],
    "output": {"name": "F", "unit": "newton"},  // Force
    "d4rtCode": "F = (8.9875517923e9 * q1 * q2) / pow(r, 2);",
    "tags": ["physics", "electricity", "electrostatics"]
  },

  // Electric Power
  {
    "name": "Electric Power",
    "description": r'''
Rate at which electrical energy is transferred

$$P = VI$$

Where:
- $P$: Power (Watts)
- $V$: Voltage (Volts)
- $I$: Current (Amperes)

This formula is fundamental in electrical circuit analysis.''',
    "input": [
      {"name": "V", "unit": "volt"},  // Voltage
      {"name": "I", "unit": "ampere"}  // Current
    ],
    "output": {"name": "P", "unit": "watt"},  // Power
    "d4rtCode": "P = V * I;",
    "tags": ["physics", "electricity", "electronics"]
  },

  // Ideal Gas Law
  {
    "name": "Ideal Gas Law",
    "description": r'''
Equation of state for an ideal gas

$$PV = nRT$$

Where:
- $P$: Pressure (Pascals)
- $V$: Volume (cubic meters)
- $n$: Amount of substance (moles)
- $R$: Universal gas constant $8.314\ \mathrm{J/(mol\cdot K)}$
- $T$: Temperature (Kelvin)

This law combines Boyle's, Charles's, and Avogadro's laws.''',
    "input": [
      {"name": "n", "unit": "mole"},     // Amount of substance
      {"name": "T", "unit": "kelvin"},   // Temperature
      {"name": "V", "unit": "cubic meter"}  // Volume
    ],
    "output": {"name": "P", "unit": "pascal"},  // Pressure
    "d4rtCode": "P = (n * 8.314462618 * T) / V;",
    "tags": ["physics", "thermodynamics", "gas"]
  },

  // Snell's Law
  {
    "name": "Snell's Law",
    "description": r'''
Law describing refraction of light at interface between media

$$n_1\sin(\theta_1) = n_2\sin(\theta_2)$$

Where:
- $n_1, n_2$: Refractive indices of the two media
- $\theta_1$: Angle of incidence
- $\theta_2$: Angle of refraction

This law explains how light bends when passing between materials.''',
    "input": [
      {"name": "n1", "unit": "scalar"},    // Refractive index 1
      {"name": "n2", "unit": "scalar"},    // Refractive index 2
      {"name": "theta1", "unit": "degree"} // Angle of incidence
    ],
    "output": {"name": "theta2", "unit": "degree"},  // Angle of refraction
    "d4rtCode": """
       var theta1Rad = theta1 * (pi / 180);
       var sinTheta2 = (n1 * sin(theta1Rad)) / n2;
       theta2 = asin(sinTheta2) * (180 / pi);
    """,
    "tags": ["physics", "optics", "light"]
  },

  // Buoyant Force (Archimedes' Principle)
  {
    "name": "Buoyant Force",
    "description": r'''
Upward force exerted on an object immersed in a fluid

$$F_b = \rho g V$$

Where:
- $F_b$: Buoyant force (Newtons)
- $\rho$: Fluid density ($\mathrm{kg/m^3}$)
- $g$: Gravitational acceleration ($\mathrm{m/s^2}$)
- $V$: Displaced volume (cubic meters)

An object floats when buoyant force equals its weight.''',
    "input": [
      {"name": "rho", "unit": "kilogram per cubic meter"},  // Fluid density
      {"name": "g", "unit": "meters per square second"},    // Gravitational acceleration
      {"name": "V", "unit": "cubic meter"}                  // Displaced volume
    ],
    "output": {"name": "Fb", "unit": "newton"},  // Buoyant force
    "d4rtCode": "Fb = rho * g * V;",
    "tags": ["physics", "fluid", "mechanics"]
  },

  // Area of Circle
  {
    "name": "Area of Circle",
    "description": r'''
Area enclosed by a circle

$$A = \pi r^2$$

Where:
- $A$: Area (square meters)
- $r$: Radius (meters)
- $\pi$: Pi ($\approx 3.14159$)

The area is proportional to the square of the radius.''',
    "input": [
      {"name": "r", "unit": "meter"}  // Radius
    ],
    "output": {"name": "A", "unit": "square meter"},  // Area
    "d4rtCode": "A = pi * pow(r, 2);",
    "tags": ["geometry", "circle", "area"]
  },

  // Circumference of Circle
  {
    "name": "Circumference of Circle",
    "description": r'''
Perimeter (distance around) a circle

$$C = 2\pi r$$

Where:
- $C$: Circumference (meters)
- $r$: Radius (meters)
- $\pi$: Pi ($\approx 3.14159$)

The circumference is proportional to the radius.''',
    "input": [
      {"name": "r", "unit": "meter"}  // Radius
    ],
    "output": {"name": "C", "unit": "meter"},  // Circumference
    "d4rtCode": "C = 2 * pi * r;",
    "tags": ["geometry", "circle", "perimeter"]
  },

  // Area of Triangle
  {
    "name": "Area of Triangle",
    "description": r'''
Area enclosed by a triangle

$$A = \frac{1}{2}bh$$

Where:
- $A$: Area (square meters)
- $b$: Base length (meters)
- $h$: Height perpendicular to base (meters)

This formula works for any triangle.''',
    "input": [
      {"name": "b", "unit": "meter"},  // Base
      {"name": "h", "unit": "meter"}   // Height
    ],
    "output": {"name": "A", "unit": "square meter"},  // Area
    "d4rtCode": "A = 0.5 * b * h;",
    "tags": ["geometry", "triangle", "area"]
  },

  // Area of Rectangle
  {
    "name": "Area of Rectangle",
    "description": r'''
Area enclosed by a rectangle

$$A = lw$$

Where:
- $A$: Area (square meters)
- $l$: Length (meters)
- $w$: Width (meters)

The area is the product of length and width.''',
    "input": [
      {"name": "l", "unit": "meter"},  // Length
      {"name": "w", "unit": "meter"}   // Width
    ],
    "output": {"name": "A", "unit": "square meter"},  // Area
    "d4rtCode": "A = l * w;",
    "tags": ["geometry", "rectangle", "area"]
  },

  // Area of Trapezoid
  {
    "name": "Area of Trapezoid",
    "description": r'''
Area enclosed by a trapezoid

$$A = \frac{1}{2}(a+b)h$$

Where:
- $A$: Area (square meters)
- $a, b$: Lengths of parallel sides (meters)
- $h$: Height (perpendicular distance between parallel sides, meters)

The area is the average of parallel sides times height.''',
    "input": [
      {"name": "a", "unit": "meter"},  // Parallel side 1
      {"name": "b", "unit": "meter"},  // Parallel side 2
      {"name": "h", "unit": "meter"}   // Height
    ],
    "output": {"name": "A", "unit": "square meter"},  // Area
    "d4rtCode": "A = 0.5 * (a + b) * h;",
    "tags": ["geometry", "trapezoid", "area"]
  },

  // Volume of Sphere
  {
    "name": "Volume of Sphere",
    "description": r'''
Volume enclosed by a sphere

$$V = \frac{4}{3}\pi r^3$$

Where:
- $V$: Volume (cubic meters)
- $r$: Radius (meters)
- $\pi$: Pi ($\approx 3.14159$)

The volume is proportional to the cube of the radius.''',
    "input": [
      {"name": "r", "unit": "meter"}  // Radius
    ],
    "output": {"name": "V", "unit": "cubic meter"},  // Volume
    "d4rtCode": "V = (4.0/3.0) * pi * pow(r, 3);",
    "tags": ["geometry", "sphere", "volume"]
  },

  // Surface Area of Sphere
  {
    "name": "Surface Area of Sphere",
    "description": r'''
Total surface area of a sphere

$$A = 4\pi r^2$$

Where:
- $A$: Surface area (square meters)
- $r$: Radius (meters)
- $\pi$: Pi ($\approx 3.14159$)

The surface area is four times the area of a circle with the same radius.''',
    "input": [
      {"name": "r", "unit": "meter"}  // Radius
    ],
    "output": {"name": "A", "unit": "square meter"},  // Surface area
    "d4rtCode": "A = 4 * pi * pow(r, 2);",
    "tags": ["geometry", "sphere", "surface area"]
  },

  // Volume of Cylinder
  {
    "name": "Volume of Cylinder",
    "description": r'''
Volume enclosed by a cylinder

$$V = \pi r^2 h$$

Where:
- $V$: Volume (cubic meters)
- $r$: Radius of base (meters)
- $h$: Height (meters)
- $\pi$: Pi ($\approx 3.14159$)

The volume is the area of the base times the height.''',
    "input": [
      {"name": "r", "unit": "meter"},  // Radius
      {"name": "h", "unit": "meter"}   // Height
    ],
    "output": {"name": "V", "unit": "cubic meter"},  // Volume
    "d4rtCode": "V = pi * pow(r, 2) * h;",
    "tags": ["geometry", "cylinder", "volume"]
  },

  // Surface Area of Cylinder
  {
    "name": "Surface Area of Cylinder",
    "description": r'''
Total surface area of a cylinder (including top and bottom)

$$A = 2\pi r(r + h)$$

Where:
- $A$: Total surface area (square meters)
- $r$: Radius of base (meters)
- $h$: Height (meters)
- $\pi$: Pi ($\approx 3.14159$)

This includes the lateral surface plus the two circular ends.''',
    "input": [
      {"name": "r", "unit": "meter"},  // Radius
      {"name": "h", "unit": "meter"}   // Height
    ],
    "output": {"name": "A", "unit": "square meter"},  // Surface area
    "d4rtCode": "A = 2 * pi * r * (r + h);",
    "tags": ["geometry", "cylinder", "surface area"]
  },

  // Volume of Cone
  {
    "name": "Volume of Cone",
    "description": r'''
Volume enclosed by a cone

$$V = \frac{1}{3}\pi r^2 h$$

Where:
- $V$: Volume (cubic meters)
- $r$: Radius of base (meters)
- $h$: Height (meters)
- $\pi$: Pi ($\approx 3.14159$)

The volume is one-third the volume of a cylinder with the same base and height.''',
    "input": [
      {"name": "r", "unit": "meter"},  // Radius
      {"name": "h", "unit": "meter"}   // Height
    ],
    "output": {"name": "V", "unit": "cubic meter"},  // Volume
    "d4rtCode": "V = (1.0/3.0) * pi * pow(r, 2) * h;",
    "tags": ["geometry", "cone", "volume"]
  },

  // Volume of Cube
  {
    "name": "Volume of Cube",
    "description": r'''
Volume enclosed by a cube

$$V = s^3$$

Where:
- $V$: Volume (cubic meters)
- $s$: Side length (meters)

The volume is the cube of the side length.''',
    "input": [
      {"name": "s", "unit": "meter"}  // Side length
    ],
    "output": {"name": "V", "unit": "cubic meter"},  // Volume
    "d4rtCode": "V = pow(s, 3);",
    "tags": ["geometry", "cube", "volume"]
  },

  // Surface Area of Cube
  {
    "name": "Surface Area of Cube",
    "description": r'''
Total surface area of a cube

$$A = 6s^2$$

Where:
- $A$: Total surface area (square meters)
- $s$: Side length (meters)

The surface area is six times the area of one face.''',
    "input": [
      {"name": "s", "unit": "meter"}  // Side length
    ],
    "output": {"name": "A", "unit": "square meter"},  // Surface area
    "d4rtCode": "A = 6 * pow(s, 2);",
    "tags": ["geometry", "cube", "surface area"]
  },

  // Perimeter of Rectangle
  {
    "name": "Perimeter of Rectangle",
    "description": r'''
Total distance around a rectangle

$$P = 2(l + w)$$

Where:
- $P$: Perimeter (meters)
- $l$: Length (meters)
- $w$: Width (meters)

The perimeter is twice the sum of length and width.''',
    "input": [
      {"name": "l", "unit": "meter"},  // Length
      {"name": "w", "unit": "meter"}   // Width
    ],
    "output": {"name": "P", "unit": "meter"},  // Perimeter
    "d4rtCode": "P = 2 * (l + w);",
    "tags": ["geometry", "rectangle", "perimeter"]
  },

  // Perimeter of Triangle
  {
    "name": "Perimeter of Triangle",
    "description": r'''
Total distance around a triangle

$$P = a + b + c$$

Where:
- $P$: Perimeter (meters)
- $a, b, c$: Side lengths (meters)

The perimeter is the sum of all three sides.''',
    "input": [
      {"name": "a", "unit": "meter"},  // Side 1
      {"name": "b", "unit": "meter"},  // Side 2
      {"name": "c", "unit": "meter"}   // Side 3
    ],
    "output": {"name": "P", "unit": "meter"},  // Perimeter
    "d4rtCode": "P = a + b + c;",
    "tags": ["geometry", "triangle", "perimeter"]
  },

  // Area of Regular Polygon
  {
    "name": "Area of Regular Polygon",
    "description": r'''
Area of a regular polygon with n sides

$$A = \frac{1}{4}ns^2\cot(\frac{\pi}{n})$$

Where:
- $A$: Area (square meters)
- $n$: Number of sides
- $s$: Side length (meters)
- $\pi$: Pi ($\approx 3.14159$)

This formula works for any regular polygon (equal sides and angles).''',
    "input": [
      {"name": "n", "unit": "scalar"},  // Number of sides
      {"name": "s", "unit": "meter"}    // Side length
    ],
    "output": {"name": "A", "unit": "square meter"},  // Area
    "d4rtCode": "A = 0.25 * n * pow(s, 2) * (cos(pi/n) / sin(pi/n));",
    "tags": ["geometry", "polygon", "area"]
  },

  // Sum of Interior Angles of Polygon
  {
    "name": "Sum of Interior Angles",
    "description": r'''
Sum of interior angles of a polygon

$$S = (n - 2) \times 180°$$

Where:
- $S$: Sum of interior angles (degrees)
- $n$: Number of sides

This formula works for any simple polygon.''',
    "input": [
      {"name": "n", "unit": "scalar"}  // Number of sides
    ],
    "output": {"name": "S", "unit": "degree"},  // Sum of angles
    "d4rtCode": "S = (n - 2) * 180;",
    "tags": ["geometry", "polygon", "angles"]
  },

  // Heron's Formula (Area of Triangle)
  {
    "name": "Heron's Formula",
    "description": r'''
Area of a triangle using only side lengths

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

Where:
- $A$: Area (square meters)
- $a, b, c$: Side lengths (meters)
- $s$: Semi-perimeter $= \frac{a+b+c}{2}$

This formula is useful when height is unknown.

**Note:** The side lengths must satisfy the triangle inequality: the sum of any two sides must be greater than the third side (a+b>c, a+c>b, b+c>a). If this condition is not met, the formula returns NaN.''',
    "input": [
      {"name": "a", "unit": "meter"},  // Side 1
      {"name": "b", "unit": "meter"},  // Side 2
      {"name": "c", "unit": "meter"}   // Side 3
    ],
    "output": {"name": "A", "unit": "square meter"},  // Area
    "d4rtCode": """
       if( a + b < c || a+c < b || b+c < a ){
          signal( "There is not a valid triangle with those longitudes" );  
       }
       var s = (a + b + c) / 2;
       A = sqrt(s * (s - a) * (s - b) * (s - c));
    """,
    "tags": ["geometry", "triangle", "area"]
  }
]