Any triangle is half base times height
The right-triangle formula, extended to every triangle — including the one case a right triangle never raises.
The last lesson found a right triangle’s area by cutting a rectangle in half. Most triangles are not right triangles. This lesson is about all of them.
A triangle, A, B and C, all three points draggable. The perpendicular height from C to line AB is drawn, meeting it at F — free to fall anywhere on that line, not just between A and B.
A triangle, A, B and C, all three points draggable, with the perpendicular height dropped from C to the line through A and B always following. A live reading gives the base, the height, and base times height divided by two, and says whether the foot of the height sits between A and B or beyond one end of the base. This needs JavaScript switched on.
The height of a triangle, measured from a chosen side, is the perpendicular distance from the opposite vertex to the line through that side — not to the segment, to the whole line, which is exactly why F is allowed to land outside AB.
Splitting the triangle in two
When F sits between A and B, the height CF splits triangle ABC into two right triangles, AFC and FBC, sharing the leg CF. Each is half its own rectangle, from the last lesson: AFC is half of AF × CF, and FBC is half of FB × CF. Add them:
since AF and FB together make up the whole of AB.
When F lands beyond B instead, the same two right triangles are there, but the smaller one now has to be subtracted rather than added: triangle ABC is triangle AFC with triangle BFC cut back out of it.
since AF minus BF is exactly AB, this time. Different arrangement, same formula.
This is the last time area needs a case split at all — every shape after this one (parallelogram, trapezium) reduces to triangles without ever needing to ask where a foot lands.
F lands outside AB precisely when the triangle is obtuse at A or at B — the one shape where a right-triangle argument on its own would have nowhere to put the foot.
A triangle with base 9 units and height 4 units.
A triangle with base 5 units and height 5 units, and the foot of the height lands outside the base.
Where the foot lands changes nothing about the calculation — only which two right triangles the proof adds or subtracts to get there.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.