Areas

Area of a Triangle #

Half the rectangle that boxes it in.

A triangle is exactly half of the rectangle that boxes it in

A rectangle 6 across and 4 up holds 24 squares.

Put the top corner anywhere along the top edge — any triangle, not a neat one.

Drop a line from that corner. Now there are two rectangles, one either side.

It cuts each one corner to corner, taking half of each: half of 24 is 12.

Area of a Parallelogram #

Slide the slanted end and it is a rectangle.

Slide the slanted end across and a parallelogram becomes a rectangle

Cut off the slanted triangle at the dashed line and slide it to the other end.

It leaves a gap at one end and fills the matching gap at the other.

Now it is a rectangle of area 7 × 4 = 28.

Area of a parallelogram = b × h.

Area of a Trapezium #

Average the parallel sides, times the height.

A trapezium behaves like a rectangle whose width is the average of its parallel sides

A trapezium has two parallel sides, and they are different lengths.

Swing the two bottom corners up into the gaps on top. Nothing is added or lost.

The 10 and the 6 have met at 8, so the area is that rectangle: 8 × 4 = 32.

Area of a trapezium $=\frac{1}{2}\times \left(a+b\right)\times h$= 1/2 × (a + b) × h.

Area of a Composite Figure #

Split it up, or subtract the gap.

Split the figure into shapes with formulas, or subtract the piece that is missing

A floor 7 m by 5 m with a 3 by 2 corner missing. Two ways to find what is left.

Subtract: the full floor is 7 × 5 = 35, the corner 3 × 2 = 6, so $35-6=29m^2$35 − 6 = 29 m².

Or split: a 7 × 3 strip below and a 4 × 2 block above — 21 + 8 = 29 again.