Area formula A = ½ ab sin C
Area of a Triangle
Area of a Triangle is a topic in Triangle Trigonometry in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.4(E), which requires students to determine the area of a triangle using trigonometric ratios.
The area of a triangle is \(\dfrac12 ab\sin C\) from two sides and the included angle, or Heron's formula from all three sides.
Theory
For a right triangle, area is \(\dfrac12(\text{base})(\text{height})\), but for a general triangle the height may be unknown. Two trig-based formulas fix that:
- SAS area: \(\text{Area}=\dfrac12 ab\sin C\), from two sides and the included angle.
- Heron's formula: from all three sides, using the semi-perimeter \(s=\dfrac{a+b+c}{2}\).
The two formulas:
How to find a triangle's area
- Two sides + included angle: apply \(\dfrac12 ab\sin C\).
- Three sides: compute \(s\), then Heron's formula.
- Solving for a side: substitute the known area and solve.
- Include units squared in the answer.
Use \(\text{Area}=\dfrac12 ab\sin C\).
| \(\text{Area}\) | \(=\) | \(\dfrac12(8)(5)\sin 30^\circ\) |
| \(=\) | \(20\cdot\dfrac12=10\) |
First the semi-perimeter, then Heron's formula.
| \(s\) | \(=\) | \(\dfrac{6+8+10}{2}=12\) |
| \(\text{Area}\) | \(=\) | \(\sqrt{12(6)(4)(2)}\) |
| \(=\) | \(\sqrt{576}=24\) |
Set up \(\dfrac12 ab\sin C=30\) with \(\sin 90^\circ=1\).
| \(\dfrac12\cdot a\cdot 12\cdot 1\) | \(=\) | \(30\) |
| \(6a\) | \(=\) | \(30\) |
| \(a\) | \(=\) | \(5\) |
Two sides and the included angle \(\Rightarrow\) \(\dfrac12 ab\sin C\).
| \(\text{Area}\) | \(=\) | \(\dfrac12(40)(55)\sin 105^\circ\) |
| \(\approx\) | \(1062\ \text{ft}^2\) |
Common pitfalls
Frequently asked questions
How do you find the area with two sides and an angle?
Use \(\text{Area}=\dfrac12 ab\sin C\), where \(C\) is the angle included between sides \(a\) and \(b\).
What is Heron's formula?
\(\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\), with \(s=\dfrac{a+b+c}{2}\). It gives the area from the three side lengths.
When do you use each area formula?
Use \(\dfrac12 ab\sin C\) for two sides and the included angle (SAS); use Heron's formula when you know all three sides (SSS).
Why does 1/2 a b sin C work?
Because \(b\sin C\) is the triangle's height on base \(a\), so \(\dfrac12 ab\sin C\) is just \(\dfrac12\,\text{base}\times\text{height}\).