1. What is a Scalene Triangle?
Imagine a triangle where none of the sides are the same length. This is called a Scalene Triangle.
- Different Sides: Every side (a, b, and c) has a unique measurement.
- Different Angles: Because the sides are different, all three inside angles are different too.
2. Area Using Base & Height
The simplest way to find the area is if you know the base and the height.
Basic Formula
Area = 12
× base (b) × height (h)
Look at the diagram in Section 1 again. The line AD is the height (h), and the side BC is the base (a). If you multiply them and divide by 2, you get the area!
3. Heron's Formula
What if you don't know the height? You can still find the area using only the three sides (a, b, and c). This is called Heron's Formula.
Step A: Find the Semi-perimeter (s)
First, add all sides and divide by 2:
Step B: Use Heron's Magic Formula
Plug 's' and the side lengths into this square root formula:
Area = √[ s (s - a) (s - b) (s - c) ]
4. Proving Heron's Formula
Let's see how we get that big formula using the Pythagorean Theorem.
1 Split the triangle into two right-angled triangles using the height h.
In Triangle 1 (ADB): h² + (c - x)² = a²
In Triangle 2 (ADC): h² + x² = b²
2 Since h² is in both, we can set the equations equal to each other:
a² - (c - x)² = b² - x²
3 After a lot of algebra and substituting s for the perimeter, the height h turns into the Heron's expression we use today!