Geometric Mathematics
Ratios in Geometry & Scale Factors
Geometric similarity relies entirely on constant ratios between corresponding linear lengths, surface areas, and 3D volumes.
1. The Linear Scale Factor (k) vs. Area (k²) and Volume (k³)
When you scale a geometric shape by a linear factor k:
| Dimension | Scale Ratio | Example (Scale Factor k = 3) |
|---|---|---|
| Linear (Length / Perimeter) | 1 : k | Side lengths triple (1 : 3). |
| 2D Surface Area | 1 : k² | Area increases 9-fold (1 : 9 = 3²). |
| 3D Volume / Capacity | 1 : k³ | Volume increases 27-fold (1 : 27 = 3³). |
2. Indirect Measurement Using Shadow Proportions
By comparing the height of a known object to its shadow, you can compute the height of a tall tree or building using similar triangles:
Example Problem:
A 2-meter tall pole casts a 3-meter shadow. At the same moment, a building casts a 24-meter shadow. How tall is the building?
- Set up proportion:
(Height of Pole ÷ Shadow of Pole) = (Height of Building ÷ Shadow of Building) 2 / 3 = H / 24- Cross-multiply:
3 × H = 2 × 24 = 48 - Solve for H:
H = 48 ÷ 3 = 16 meters tall.