Area of plane figures
Area measures the amount of surface a flat shape covers, in square units. Most area formulas grow out of the rectangle.
Squares and rectangles
A rectangle's area is length times width, A = lw, measured in square units such as square metres. A room 4.5 m by 3.2 m has an area of 14.4 square metres. A square of side s has area s squared. Converting units squares the conversion factor: 1 square metre is 100 x 100 = 10,000 square centimetres.
Parallelograms and triangles
A parallelogram has area base times height, A = bh, where the height is perpendicular to the base. Any triangle is half of a parallelogram, so its area is one half base times height. A triangle with base 10 and height 7 has area 35.
Heron's formula
When only the three side lengths are known, Heron's formula gives the area: with semi-perimeter \(s = \frac{a + b + c}{2}\), \(A = \sqrt{s(s - a)(s - b)(s - c)}\). For sides 13, 14, and 15, s = 21 and the area is \(\sqrt{21 \times 8 \times 7 \times 6} = \sqrt{7056} = 84\). It is named after Heron of Alexandria, in the first century CE.
Trapezoids, rhombuses, and kites
A trapezoid's area is half the sum of its bases times the height. Rhombuses and kites, whose diagonals are perpendicular, have area half the product of their diagonals. These all follow by cutting the shapes into triangles and rectangles.
Regular polygons
A regular polygon's area is half its apothem times its perimeter, A = (1/2)aP, where the apothem is the distance from the centre to the midpoint of a side. A regular hexagon with side 4 has perimeter 24 and apothem 2 root 3, about 3.46, so its area is about 41.6 square units.
Circles and composite shapes
A circle's area is pi r squared. For composite shapes, split them into familiar pieces and add, or subtract cut-outs. A running track shaped like a rectangle 100 m by 60 m with semicircles on both short ends encloses 6,000 plus pi times 30 squared, about 8,827 square metres.
Scaling area
If every length is multiplied by k, area is multiplied by k squared. Doubling a garden's length and width quadruples its area and the amount of seed needed.
Worked example
A wall 5 m wide and 2.6 m high has a door 0.9 m by 2.1 m and a window 1.2 m by 1.0 m. The paintable area is 13 - 1.89 - 1.2 = 9.91 square metres. If one litre of paint covers 10 square metres and two coats are needed, about 2 litres are required.
Key takeaways
Area is measured in square units, and converting square units squares the length conversion. Rectangles have area lw, parallelograms bh, and triangles half bh, with heights perpendicular to the base. Heron's formula finds a triangle's area from its sides. Trapezoids use half the sum of the bases times the height; rhombuses and kites use half the product of the diagonals. Regular polygons use half the apothem times the perimeter, and circles pi r squared. Split composite shapes into pieces. Scaling lengths by k scales area by k squared.
Flashcard
Length x width
Check your understanding
A triangle has sides 13, 14, and 15. What is its area?