Area of Pentagon given Height using Interior Angle Calculator

Pentagon Area Formula:

\[ A = \frac{5 \times \left( \frac{h \times \sin\left(\frac{3}{5}\pi\right)}{\left(\frac{3}{2} - \cos\left(\frac{3}{5}\pi\right)\right) \times \left(\frac{1}{2} - \cos\left(\frac{3}{5}\pi\right)\right)} \right)^2}{4 \times \tan\left(\frac{\pi}{5}\right)} \]

Area of Pentagon:

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1. What is the Pentagon Area Formula?

The formula calculates the area of a regular pentagon given its height using interior angle properties. It employs trigonometric functions to derive the area based on the pentagon's height measurement.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( A \) — Area of Pentagon (m²)
\( h \) — Height of Pentagon (m)
\( \pi \) — Archimedes' constant (≈3.14159)
\( \sin \) — Sine trigonometric function
\( \cos \) — Cosine trigonometric function
\( \tan \) — Tangent trigonometric function

Explanation: The formula uses trigonometric relationships derived from the pentagon's interior angles to calculate area from height.

3. Importance of Pentagon Area Calculation

Details: Calculating pentagon area is essential in geometry, architecture, engineering, and design applications where pentagonal shapes are used.

4. Using the Calculator

Tips: Enter the height of the pentagon in meters. The height must be a positive value greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a regular pentagon?
A: A regular pentagon is a five-sided polygon where all sides are equal in length and all interior angles are equal (108° each).

Q2: How is height defined for a pentagon?
A: The height of a pentagon is the perpendicular distance from any side to the opposite vertex.

Q3: Can this formula be used for irregular pentagons?
A: No, this formula is specifically designed for regular pentagons where all sides and angles are equal.

Q4: What are the units of measurement?
A: The height should be entered in meters, and the area result will be in square meters (m²).

Q5: How accurate is this calculation?
A: The calculation uses precise trigonometric functions and provides accurate results for regular pentagons.