Area of a Triangle Using Sine

Worked example

1. 1

   $a=8,b=11,C=50^{\circ }$ — The 50° is where the 8 and the 11 meet, so it is the angle C that the formula needs.

2. 2

   $\text{Area}=\frac{1}{2}\times 8\times 11\times \sin 50^{\circ }$ — Put the numbers into the formula.

3. 3

   $\frac{1}{2}\times 8\times 11=44$ — Do the half and the two sides first.

4. 4

   $\sin 50^{\circ }=0.7660$ — Find the sine on the calculator, in degree mode.

5. 5

   $44\times 0.7660=33.7{\text{ cm}}^2$ — Multiply and give 3 significant figures. It is an area, so the units are squared.

Worked example

1. 1

   $360^{\circ }÷6=60^{\circ }$ — The six triangles fill a full turn of 360° around the centre.

2. 2

   $\text{centre to corner}=6\text{ cm}$ — Each triangle has two equal sides and a 60° angle. Its other two angles are (180° − 60°) ÷ 2 = 60° too, so all its sides are 6 cm.

3. 3

   $\frac{1}{2}\times 6\times 6\times \sin 60^{\circ }=15.59$ — The area of one triangle: two sides of 6 cm with the 60° between them.

4. 4

   $6\times 15.59=93.5{\text{ cm}}^2$ — There are six of them. Give 3 significant figures.

Worked example

1. 1

   $\frac{1}{2}\times x\times x\times \sin 120^{\circ }=50$ — Both equal sides are x, with the 120° between them. Make the formula equal to the area.

2. 2

   $\frac{1}{2}{x}^2\sin 120^{\circ }=50$ — x × x is x².

3. 3

   $x^2\sin 120^{\circ }=100$ — Double both sides to get rid of the half.

4. 4

   $x^2=\frac{100}{\sin 120^{\circ }}$ — Divide both sides by sin 120°.

5. 5

   $x^2=115.47$ — Work it out. sin 120° = 0.8660.

6. 6

   $x=\sqrt{115.47}=10.7\text{ cm}$ — Take the square root, positive because it is a length, and give 3 significant figures.

Worked example

1. 1

   $\frac{1}{2}\times 9\times 14\times \sin C=40$ — Put the area and the two sides into the formula. C is the angle between the 9 and the 14.

2. 2

   $63\sin C=40$ — Half of 9 × 14 is 63.

3. 3

   $\sin C=\frac{40}{63}=0.6349$ — Divide both sides by 63.

4. 4

   $C=39.4^{\circ }$ — Press inverse sine. This is the acute angle.

5. 5

   $C=180^{\circ }-39.4^{\circ }=140.6^{\circ }$ — The sine of 140.6° is the same, so 140.6° would also give an area of 40 cm². Use it only if the question says the angle is obtuse or the diagram shows it wide.