Area of Triangles

Draw a rectangle around a triangle so the triangle's base and height are the rectangle's sides. The two leftover corners fit together to fill exactly the same space as the triangle, so the triangle is precisely half the rectangle. The rectangle's area is base times height, so the triangle's is A = ½ × b × h. Everything in this topic then comes down to one thing: the height has to be perpendicular to the base — a straight-up distance measured at right angles. It is not a slanted side, no matter how conveniently that number is printed in the diagram.

The height is whichever distance is perpendicular to the base you've chosen, and it can look quite different from triangle to triangle. In a right-angled triangle the two shorter sides already meet at 90 degrees, so they are your base and height with no extra work. In an obtuse triangle the perpendicular can fall outside the shape entirely — you extend the base with a dashed line and measure to that, which is perfectly valid. And any of the three sides can serve as the base, so long as you pair it with the height that matches. All three triangles in the picture share a base of 120 and a height of 100, so all three have identical areas even though they look nothing alike.

Questions often give you the area and ask for a missing base or height. Substitute what you know into A = ½bh and solve. If the area is 48 and the height is 8, then 48 = ½ × b × 8. Tidy the right side first — half of 8 is 4, so it becomes 48 = 4b — and then it's a one-step equation giving b = 12 cm. Simplifying before you solve saves most of the arithmetic trouble here. And substitute back at the end: half of 12 times 8 is 48, so the answer checks out.

Real shapes are usually made of simpler ones. A house-shape is a rectangle with a triangle on top: the rectangle gives 18 times 9, which is 162, and the triangle gives half of 18 times 6, which is 54, for a total of 216 square centimetres. Add the pieces when the shape sticks out, and subtract when there's a hole cut in it. Units matter here in a way that surprises people: area units are always squared, and converting them isn't what you'd guess. One square metre is 10000 square centimetres, not 100, because a one-metre square is 100 cm by 100 cm. The safest habit is to convert all your lengths to a common unit before you calculate anything.

Five ways to lose a mark on triangle area. Using a slanted side as the height — the height is perpendicular to the base, and a slant is not. Forgetting the half, which gives you the rectangle instead of the triangle. Using the hypotenuse in a right-angled triangle, when you want the two sides that meet at the right angle. Mixing units, like a base in centimetres with a height in metres. And writing cm when the answer needs cm squared, which drops a mark even though the number is right. The formula is short — A = ½bh — so the marks are all in the care around it.