Area of Triangles, Parallelograms and Trapezoids

The idea in one sentence

You already know a rectangle’s area is base × height, and every formula on this page is some shape being cut up and rearranged into a rectangle.

Parallelogram: slide the triangle across

Take a parallelogram. Cut straight down from the top-left corner to make a right triangle, slide that triangle round to the right-hand end, and you have a rectangle — same area, because nothing was added or thrown away.

The catch is what \(h\) means. It is the perpendicular height, measured straight up from the base, not the length of the slanted side. A parallelogram can lean so far it is almost flat; its slanted sides stay the same length while its area shrinks towards nothing. Only the perpendicular height tracks that.

Triangle: it’s half a parallelogram

Take any triangle and make a second copy of it, rotated half a turn. The two fit together into a parallelogram with the same base and height. So a triangle is exactly half of one:

Any side can be the base, as long as \(h\) is measured perpendicular to that side. All three choices give the same area, which is a useful check when a question hands you more measurements than you need.

For an obtuse triangle the height lands outside the shape, and you extend the base with a dotted line to meet it. The formula doesn’t care.

Trapezoid: average the two bases

A trapezoid has one pair of parallel sides, \(a\) and \(b\), a distance \(h\) apart.

The same doubling trick explains it. Two copies of the trapezoid, one turned upside down, fit into a parallelogram whose base is \(a + b\) and whose height is \(h\). Half of that is \(\frac{(a+b)h}{2}\).

Read the formula as average width × height, and it stops being something to memorise. A trapezoid is a rectangle whose top has been tilted; its average width is the width halfway up. If \(a = b\) the tilt disappears and the formula collapses back to \(bh\) — a parallelogram. If \(a = 0\) the top pinches shut and it gives \(\frac{1}{2}bh\) — a triangle. One formula covers all three.

Anything else: cut it up

An L-shaped room, an arrowhead, a hexagonal garden: there is no formula, and there doesn’t need to be. Chop the shape into rectangles and triangles, find each piece, and add. Or draw the smallest rectangle that contains the whole shape and subtract the corners you don’t want — often fewer steps.

Both methods are the same idea as every formula above, done by hand instead of in advance.

Common mix-ups

• Using the slanted side as the height. The most common error on this whole topic. The height must meet the base at a right angle.
• Forgetting the ½ on a triangle. Every triangle is half of something.
• Adding the bases without halving on a trapezoid, which doubles the answer.
• Mixing units. A base in metres and a height in centimetres gives nothing useful. Convert first.
• Square units. Area is always cm², m², in² — never plain cm.

Check yourself

Find the area of a triangle with base 10 and height 6, a parallelogram with base 8 and perpendicular height 5, and a trapezoid with parallel sides 6 and 10 that are 4 apart.

\(\tfrac{1}{2}(10)(6) = 30\). Then \(8 \times 5 = 40\). Then the average width is \(\frac{6 + 10}{2} = 8\), so \(8 \times 4 = 32\).