Explanation
Measuring area
Area measures the two-dimensional region inside a boundary. Enter each linear dimension in metres; every answer is returned in square metres (m²). The formula depends on the shape, and a triangle's height must be perpendicular to its base.
A = w × h
| A | area in square metres (square metres (m²)) |
|---|---|
| w | rectangle width in metres (metres (m)) |
| h | perpendicular height in metres (metres (m)) |
A = b × h ÷ 2, where h is perpendicular to b
| A | area in square metres (square metres (m²)) |
|---|---|
| b | triangle base in metres (metres (m)) |
| h | perpendicular height in metres (metres (m)) |
A = πr²
| A | area in square metres (square metres (m²)) |
|---|---|
| r | circle radius in metres, from centre to circumference (metres (m)) |
A 3 m by 4 m rectangle
| Width | 3 |
|---|---|
| Height | 4 |
| Result | 12 m² |
3 m × 4 m = 12 m²
A triangle with base 3 m and perpendicular height 4 m
| Base | 3 |
|---|---|
| Height | 4 |
| Result | 6 m² |
3 m × 4 m ÷ 2 = 6 m²
A circle with radius 1 m
| Radius | 1 |
|---|---|
| Result | 3.14 m² |
π × (1 m)² = 3.1415926536 m²
Diagram of this result
The selected shape is drawn in the proportions you entered, with each measurement on a dimension line. The shaded region is the area.
The entered measurements and the area are stated in the answer and the steps.
Common mistakes
Do not use a triangle's sloping side as its height: height is measured at a right angle to the base, even if that perpendicular falls outside the triangle. A circle uses its radius, not its diameter; if you have a diameter, halve it first. Area uses squared units, so a result in metres (m) is not an area.
Assumptions and limits
All entered dimensions must be positive finite canonical decimals, with the shared limit of 30 significant digits and absolute adjusted exponent at most 100. Inputs represent metres, not a guessed or selectable unit. Decimal arithmetic uses 50 significant digits and rounds only for presentation; answers beyond the shared output range are rejected. The circle calculation uses π truncated after 50 decimal places, so π-derived areas are approximations. Comparison fixtures use relative tolerance 1e-12 plus absolute tolerance 1e-12 × the reference scale.
Why does the triangle need a perpendicular height?
The formula is half the area of a parallelogram with the same base and perpendicular altitude. A sloping edge only equals the height for a right triangle; otherwise using it overstates the triangle's area.
Does the circle input ask for radius or diameter?
It asks for radius: the distance from the centre to the edge. The area formula squares the radius. A diameter is twice the radius, so convert it before entering.
References
For geometric measurement and area formulas, see OpenStax Prealgebra 2e, measurement and geometry topics: https://openstax.org/details/books/prealgebra-2e. Formula examples in this lesson are independently checked against direct arithmetic and standard geometric definitions.