An area formula matches a particular geometric shape and its measurements. Multiplying length by width works directly for rectangles, but a triangle needs halving and a circle uses its radius. Identify the shape before substituting values; a familiar-looking formula is not enough.
Flat shapes and three-dimensional solids
Plane figures have two dimensions and occupy regions of a plane. Solids also extend in a third dimension and may have volume. The area of a flat circular base is different from a cylinder’s total surface area and from the volume inside it.
Six useful formulas and the measurements behind them
In order below: a square uses side a; a rectangle uses base b and perpendicular height h; a triangle uses the same base-height pair but halves the product. A trapezoid uses parallel bases B and b with height h. A rhombus uses perpendicular diagonals D and d. A circle uses radius r.
A square is a special rectangle and a special rhombus, so its area can be found in more than one valid way. For the triangle formula, the height must correspond to the selected base. Only a right triangle has a hypotenuse; the term does not apply to every triangle’s longest side.
Worked example: a rectangular wall with a triangular top
Split the outline into an 8 m wide, 5 m high rectangle and a triangle above it. The triangle has the same 8 m base and a perpendicular height of 3 m. Their interiors do not overlap, so calculate their areas separately and add them.
The total is 52 square metres. The 3 m triangle height is measured above the rectangular part, not from ground level. If the total height had been given instead, you would first subtract the rectangle’s height.
When should an area be subtracted?
Subtract the area of a hole, window or cutout from the enclosing region. A floor with a rectangular opening is the larger floor area minus the opening area. Do not add an internal border or subtract an edge: area concerns two-dimensional regions, not boundary lines.
Convert all lengths to the same unit before calculating, retain exact values involving π until the last step, and report square units. For the distinction between surface and outline, use area and perimeter; for more detail on perpendicular heights, see triangle area.
