Formula

Area of a triangle from two sides and the included angle

Mensuration • Plane Figures • Triangle

Formula

A = \frac{1}{2} ab sin C

A triangle with sides

a

and

b

enclosing an angle

C

between them has area

\frac{1}{2} ab sin C

How to use it

1The angle must be the one between the two sides used; an angle elsewhere in the triangle will not do.
2When a line from a vertex splits a triangle into two, apply it to each piece and add: the two areas must total the whole, which turns the splitting line into the only unknown.
3At $C = 9 0^{\circ}$ it reduces to $\frac{1}{2} ab$ , the right-angled case.

Two sides of a triangle are

10

cm and

6

cm and the angle between them is

3 0^{\circ}

. Find the area.

Step-by-step

1
$A = \frac{1}{2} \times 10 \times 6 \times sin 3 0^{\circ}$
Why: The $3 0^{\circ}$ is enclosed by the two given sides, so the formula applies directly.
2
$= 30 \times \frac{1}{2} = 15$ cm $^{2}$

Answer

15

^{2}

Watch out

Since

sin C = sin (18 0^{\circ} - C)

, the area alone cannot distinguish an angle from its supplement. Using a non-included angle is the standard misapplication.

Builds on

Also known as

half ab sin C, area with included angle, sine area formula, splitting a triangle by a cevian