Step 2: Perimeter, Closed Polygon Boundaries & Altitude Basics
Measuring physical boundaries is the foundational skill of geometry. Before exploring area, we must precisely quantify perimeter.
1. The Metric of Perimeter ($P$)
- Definition: The perimeter of a plane figure is the total length of its continuous outer boundary.
- General Polygon: For an $n$-sided polygon with edge lengths $e_1, e_2, \dots, e_n$:
$$P = \sum_{i=1}^n e_i$$
- Rectangle: With length $l$ and breadth $b$:
$$P = 2(l + b)$$
- Square & Rhombus: With side $a$:
$$P = 4a$$
2. Standard Area Formulas for Triangles
A triangle is defined by vertices $A, B, C$ and opposite side lengths $a, b, c$.
- Standard Formula (Base & Altitude):
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} b h$$
- Right-Angled Triangle: The two perpendicular legs directly serve as base and height:
$$\text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$$
- Equilateral Triangle: With side $a$, altitude $h = \frac{\sqrt{3}}{2}a$:
$$\text{Area} = \frac{\sqrt{3}}{4} a^2$$
- Isosceles Triangle: With equal sides $a, a$ and distinct base $b$:
$$h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2} = \frac{1}{2}\sqrt{4a^2 - b^2} \implies \text{Area} = \frac{b}{4}\sqrt{4a^2 - b^2}$$
3. The Altitude Limitation
In real-world field surveys, measuring internal altitudes requires specialized transit levels. When only boundary lengths are accessible, standard base-height formulas fail, establishing the absolute necessity for Heron's formula.