Lesson 3: Area of Triangles and Heron's Formula
Estimated time: 25-30 minutes
Learning Objectives
- Find area using A = (1/2)ab sinC (SAS formula)
- Apply Heron's formula when all three sides are known
- Choose the appropriate area formula for a given situation
SAS Area Formula
Area = (1/2) a b sin C
where a and b are two sides and C is the included angle.
Example 1
a=8, b=12, C=30°. Find the area.
Area = (1/2)(8)(12)sin30 = (1/2)(96)(1/2) = 24 square units
Example 2
b=15, c=20, A=110°. Find the area.
Area = (1/2)(15)(20)sin110 = 150 sin110 ≈ 150(0.9397) ≈ 140.95 sq units
Heron's Formula
Heron's Formula: Given sides a, b, c, let s = (a+b+c)/2 (semi-perimeter). Then:
Area = √[s(s−a)(s−b)(s−c)]
Example 3
a=7, b=8, c=9.
s = (7+8+9)/2 = 12
Area = √[12(12−7)(12−8)(12−9)] = √[12·5·4·3] = √720 = 12√5 ≈ 26.83
Example 4
a=3, b=4, c=5 (right triangle).
s=6. Area = √[6·3·2·1] = √36 = 6 (matches (1/2)(3)(4)=6).
Choosing the Right Formula
- Know two sides and included angle (SAS)? Use A = (1/2)ab sinC.
- Know all three sides (SSS)? Use Heron's formula.
- Know base and height? Use A = (1/2)bh.
Check Your Understanding
1. a=10, b=14, C=65°. Find the area.
A=(1/2)(10)(14)sin65 = 70sin65 ≈ 63.44.
2. Sides 5, 12, 13. Find area using Heron's.
s=15. √[15·10·3·2]=√900=30.
3. When is (1/2)ab sinC = 0?
When C = 0° or 180° (degenerate triangle — the vertices are collinear).
Key Takeaways
- SAS area: A = (1/2)ab sinC works when you know two sides and the included angle.
- Heron's formula: uses only the three side lengths and the semi-perimeter.
- Both formulas give the same result; choose based on what information you have.