Geometry

Package: GEOMETRY_PACKAGE. Registered by createEngine(); for a slimmer engine, register it explicitly (see choosing packages).

Work out the size of a shape from its measurements: the area it covers, the perimeter around its edge, the volume it holds, or its surface area. Name the measure, the shape, and the shape’s dimensions, and the formula is applied for you, no need to remember whether it was πr² or 2πr.

Flat shapes

A flat shape lies on a surface, like a floor plan or a sheet of paper, so it has an area (how much surface it covers) and a perimeter (the distance around its edge; for a circle, the circumference). Write the measure, the shape, and the dimension that fixes its size, such as a circle’s radius or a square’s side.

area of circle radius 5 // 78.54
circumference of circle radius 5 // 31.42
area of square side 4 // 16
perimeter of square side 4 // 16
area of triangle base 3, height 4 // 6

A shape with two dimensions takes them as a pair, separated by a comma:

area of rectangle width 4, height 6 // 24
perimeter of rectangle width 4, height 6 // 20

The comma matters. Without it, 4 height reads as a number times a value called height (the ordinary “4x means 4 times x” rule), so the comma is what keeps the two measurements apart. It also means the dimension words (width, height, radius, …) stay ordinary words you can still use as names elsewhere.

Solid shapes

A solid shape takes up space, like a ball or a tin, so it has a volume (how much it holds) and a surface area (how much material would wrap it). The dimensions are written the same way, with a comma between two.

volume of sphere radius 3 // 113.10
surface area of sphere radius 3 // 113.10
volume of cube side 2 // 8
volume of cylinder radius 2, height 5 // 62.83
volume of cone radius 2, height 6 // 25.13

Dimensions with units

A dimension can carry its unit, and the answer then comes out in the unit its measure has. A perimeter is a length, so it stays in metres. An area is a length times a length, so it is in square metres (m²): the size of a square one metre on each side. A volume is three lengths multiplied, so it is in cubic metres (m³). This is the answer the formula gives when you write it out by hand, pi * (5 m)^2, and it converts the way any area or volume does.

area of circle radius 5 m // 78.54 m²
area of circle radius 5 m in cm² // 785,398.16 cm²
volume of sphere radius 2 m // 33.51 m³
perimeter of circle radius 5 m in cm // 3,141.59 cm
area of rectangle width 3 m, height 4 m // 12.00 m²
volume of cube side 2 ft // 8.00 ft³

When the dimensions are in different units, they are all read in the unit of the first one written, so a width in metres and a height in centimetres give square metres. A dimension with no unit beside one with a unit is read in that unit, the way a bare number in a list of quantities is. A length with no square or cube of its own, such as the furlong, answers in square or cubic metres, as (5 furlong)^2 does.

area of rectangle width 3 m, height 400 cm // 12.00 m²
area of rectangle width 300 cm, height 4 m // 120,000.00 cm²
area of rectangle width 3 m, height 4 // 12.00 m²
area of square side 2 furlong // 161,874.26 m²

An area is not a length, so converting one to centimetres rather than square centimetres is refused. A dimension that is not a length at all, a mass, a duration or an amount of money, is refused by name rather than read as a number.

area of circle radius 5 m in cm // an area cannot be converted to a length
area of circle radius 5 kg // a radius is a length, not a mass: give it in a unit of length, such as m or ft
area of rectangle width $3, height 4 m // a width is a length, not money: give it in a unit of length, such as m or ft

The shapes and their dimensions

• circle: radius (area, perimeter / circumference)
• square: side (area, perimeter)
• rectangle: width, height (area, perimeter)
• triangle: base, height (area)
• sphere: radius (volume, surface area)
• cube: side (volume, surface area)
• cylinder: radius, height (volume, surface area)
• cone: radius, height (volume)

Dimensions written as plain numbers give a plain number, as in the first examples on this page; dimensions with units give a length, an area or a volume. A measure a shape does not define, or a missing dimension, is reported as an error naming what it needed rather than a wrong number.