🟦Perimeter & Area

Perimeter is the distance once around the outside of a shape. Walk the boundary and count the metres: that is the perimeter. It is a length, so it is measured in cm, m, km.

Area is how much flat space the shape covers — how many unit squares fit inside it. It is measured in square units: cm², m².

Mixing them up is the single most common error in geometry, and the units are the giveaway. If your answer to an area question is in plain cm, something has gone wrong.

For a rectangle:

• ◆Perimeter = 2 x (length + width)
• ◆Area = length x width

A rectangle 7 cm by 3 cm has perimeter 2(7 + 3) = 20 cm and area 7 x 3 = 21 cm². Same shape, completely different questions.

Take any triangle, make a second identical copy, rotate it 180° and slot the two together. You get a parallelogram with the same base and the same height. So:

Area of a triangle = (1/2) x base x height

The height must be perpendicular to the base — measured at a right angle to it, not along a slanted side. In an obtuse triangle the perpendicular height can even fall outside the triangle, and that is fine.

Any of the three sides may be chosen as the base, as long as the height used is the one perpendicular to that side. All three choices give the same area.

A parallelogram can be cut along a vertical line and the triangle slid from one end to the other, turning it into a rectangle of the same base and the same perpendicular height. So:

Area of a parallelogram = base x perpendicular height

Problems often supply the slanted side as a decoy. It is longer than the true height and must not be used.

A trapezoid has one pair of parallel sides, of lengths a and b. Two copies fit together into a parallelogram of base (a + b) and height h, so one trapezoid is half of that:

Area of a trapezoid = ((a + b) / 2) x h

Read that as: the average of the two parallel sides, times the height. The bit sticking out at one end exactly fills the gap at the other.

Real floors and gardens are rarely simple rectangles. For an L-shape, either

• ◆cut it into two rectangles and add the areas, or
• ◆start with the full enclosing rectangle and subtract the missing corner.

Both routes give the same answer, and doing it both ways is a free check.

A surprise about the perimeter of an L-shape cut from a rectangle: it is exactly the same as the perimeter of the original rectangle. The steps that go inwards are matched by steps that come back out.

Formulas also run backwards. If a rectangle has area 96 cm² and one side is 8 cm, then 8 x ? = 96, so the other side is 96 / 8 = 12 cm. If instead the perimeter is 96 with one side 8, halve first: 96 / 2 = 48 is one length plus one width, so the other side is 48 - 8 = 40 cm.