Area of a Triangle Flashcards

All 6 cards in this deck

What is the formula for the area of any triangle using two sides and an angle?

Area $=\tfrac12 ab\sin C$

$C$ is the angle included between sides $a$ and $b$.

True or false? $\tfrac12 ab\sin C$ works with any two sides and any angle of the triangle.

False.

The angle must be the one included between the two sides used.

What are the steps to find the area of a triangle from two sides and the included angle?

e.g. sides $8$ cm and $11$ cm with an included angle of $72^\circ$

1. Write the formula: Area $=\tfrac12 ab\sin C$
2. Substitute: $0.5\times 8\times 11\times\sin 72^\circ$
3. Evaluate: $41.8$ cm$^2$ (3 s.f.)

What are the steps to find an included angle when the area is given?

e.g. a triangle with sides $8$ cm and $10$ cm has area $20$ cm$^2$

1. Substitute into $\tfrac12 ab\sin C=$ Area: $0.5\times 8\times 10\times\sin C=20$
2. Rearrange: $\sin C=\dfrac{20}{40}=0.5$
3. Inverse sine: $C=30^\circ$ (or $150^\circ$ if obtuse)

What are the steps to find $x$ when the area formula gives an equation in $x$?

e.g. sides $(x+3)$ m and $(2x-1)$ m, included angle $45^\circ$, area $6\sqrt2$ m$^2$

1. Set up the equation: $\tfrac12(x+3)(2x-1)\sin 45^\circ=6\sqrt2$
2. Expand and rearrange: $2x^2+5x-27=0$
3. Solve with the quadratic formula: $x=2.63$ (3 s.f.)

True or false? $\tfrac12 ab\sin C$ cannot be used when the included angle is obtuse.

False.

The sine of an obtuse angle is positive, so the formula still gives a positive area.