Question

In case of girth measurement the basal area of the tree is estimated using the formula

• A. $\frac{G}{\pi}$
• B. $\frac{G^2}{4\pi}$
• C. $\frac{G}{4\pi}$
• D. $\frac{G}{2\pi}$

Hint:

If the question uses Diameter ($d$), the formula is \( \frac{\pi d^2}{4} \). Since \( G = \pi d \), both formulas are mathematically identical.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical formula used in forest mensuration to calculate the cross-sectional area (Basal Area) of a tree stem based on its girth ($G$).

Step 2: Key Formula or Approach:
Tree stems are assumed to be circular in cross-section.
1. Let Girth ($G$) = Circumference of the circle.
2. Circumference \( G = 2\pi r \), where $r$ is the radius.
3. Therefore, radius \( r = \frac{G}{2\pi} \).
4. The formula for the Area of a circle is \( A = \pi r^2 \).

Step 3: Detailed Explanation:
Substitute the value of $r$ into the area formula:
\[ A = \pi \left( \frac{G}{2\pi} \right)^2 \]
\[ A = \pi \left( \frac{G^2}{4\pi^2} \right) \]
Cancelling one $\pi$:
\[ A = \frac{G^2}{4\pi} \]
This Basal Area is usually calculated at breast height ($1.37$ m) and is used to estimate the volume of wood in the tree or the stand.

Step 4: Final Answer:
The formula used is $\frac{G^2}{4\pi}$.