Formula

Area of a triangle from base and height

Mensuration • Plane Figures • Triangle

Formula

A = \frac{1}{2} bh

A triangle with base

b

and perpendicular height

h

measured to that base has area

\frac{1}{2} bh

; any of the three sides may be taken as the base, provided

h

is the perpendicular distance from the opposite vertex to it.

How to use it

1In a right-angled triangle the two legs are already perpendicular, so they serve directly as base and height.
2Where the apex lies above a point of the base, the base may be split and the two pieces added: $\frac{1}{2} b_{1} h + \frac{1}{2} b_{2} h = \frac{1}{2} (b_{1} + b_{2}) h$ .
3To maximise the area, maximise base and height separately and then check that one configuration attains both.

A triangle has base

14

cm and the opposite vertex lies

9

cm above the line of that base. Find its area.

Step-by-step

1
$A = \frac{1}{2} \times 14 \times 9$
Why: The height must be the perpendicular distance, not the length of a slanting side.
2
$= 63$ cm $^{2}$

Answer

63

^{2}

Watch out

The height is the perpendicular distance to the chosen base, not a side of the triangle; using a slant side as

h

overstates the area. For an obtuse triangle the foot of that perpendicular falls outside the base, which does not change the formula.

Also known as

half base times height, area of a triangle, 1/2 b h, triangle area formula