Measuring the Interior
The Area is the amount of 2D space inside a shape. In Calculus (Year 2), we will find the area of any shape, but we must start with the standard ones.
Key Formulas
- Rectangle: \(A = l \cdot w\)
- Triangle: \(A = \frac{1}{2}bh\)
- Circle: \(A = \pi r^2\)
- Trapezoid: \(A = \frac{a+b}{2} \cdot h\)
Worked Examples
Example 1: Area of a Triangle
Find the area of a triangle with a base of 10 and height of 4.
- \(A = \frac{1}{2}(10)(4) = 20\).
- Result: 20
Example 2: Change in Area
If you triple the radius of a circle, what happens to the area?
- Original: \(\pi r^2\).
- New: \(\pi (3r)^2 = \pi (9r^2) = 9\pi r^2\).
- Result: The area increases by a factor of 9 (it is squared!).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we don't know exactly where a particle is, so we talk about the Probability Density (the height of the wave). The chance of finding the particle in a certain region is equal to the Area under the curve of that wave. If the area under the wave is 0.5, there is a 50% chance the particle is there. Calculating areas is the only way to turn "abstract waves" into "real probabilities." All of Integral Calculus is based on this single geometric concept.