Geometry
Areas on the same base
Use the fact that figures on the same base and between the same parallels have equal areas.
Take a parallelogram and push its top side sideways, keeping it on the same base and the same two parallel lines. It gets longer and thinner-looking, its slanting sides stretch, its perimeter grows — and its area does not budge. That is the whole idea of this topic, and once you trust it, a great many area problems stop needing any calculation at all.
The rule, twice over
Two parallelograms on the same base and between the same parallels have equal areas. A triangle on the same base and between the same parallels as a parallelogram has half its area. Both follow from base × height, because between two parallel lines the height cannot change.
Worked example
A parallelogram and a triangle share a base of 10 units, between the same parallels 6 units apart. Find both areas.
The parallelogram's area is base × height: 10 × 6 = 60 square units.
The height is the gap between the parallels, not the length of a slanting side.
Try it together
A parallelogram of area 72 square units stands on a base of 9 units. A triangle stands on the same base, between the same parallels. Find the triangle's area.
You are not told the height, but you can get at it — and it turns out you do not even need it.
1.Divide the area by the base to get the height: what is 72 ÷ 9?
Watch out
"Between the same parallels" is the part that does the work. Two triangles on the same base with their tips at different heights have different areas, and no amount of leaning changes that.
Have a go
Have a go on your own: a triangle and a parallelogram are on the same base between the same parallels, and the triangle's area is 17 square centimetres. What is the parallelogram's area, in square centimetres?
Hint: This time you are going from the half to the whole.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.