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Area: cheat sheet

Area, volume, circles and Pythagoras · The words and methods from the lessons, with an example of each.

perpendicular height

The distance from the base to the opposite side or corner, at a right angle (∟) to the base. A slanted side is longer, and is not the height.

h at 90∘ to bh \text{ at } 90^\circ \text{ to } b

area of a triangle

Two copies make a parallelogram with the same base and height; the triangle is half.

A=12×b×hA = \tfrac{1}{2} \times b \times h
A triangle and its turned copy make a parallelogram
A triangle and its turned copy make a parallelogram

square units

Area counts squares: cm², m². A 1 m square is 100 cm by 100 cm.

1 m2=10 000 cm21 \text{ m}^2 = 10\,000 \text{ cm}^2

composite area

Cut the shape into pieces you know and add their areas, or take the missing pieces away from a big rectangle. Each piece uses its own lengths.

A=8×6−5×3=33 m2A = 8 \times 6 - 5 \times 3 = 33\ \text{m}^2
An L-shape with its side lengths
An L-shape with its side lengths

area of a parallelogram

Cut the triangle off one end and move it: a rectangle with the same base and height.

A=b×hA = b \times h
A parallelogram cut and rearranged into a 6 by 4 rectangle
A parallelogram cut and rearranged into a 6 by 4 rectangle

composite area

Split the shape into rectangles and triangles and add the parts, or subtract a missing part from a bigger shape. Put every length in the same unit first.

24+9=33 m224 + 9 = 33 \text{ m}^2
The board

estimating an irregular area

Count the full squares, add half the part squares, multiply by the area of one square. A finer grid gives a closer estimate.

A≈(full+12×part)×one squareA \approx (\text{full} + \tfrac{1}{2} \times \text{part}) \times \text{one square}