Height of Pentagon given Area using Interior Angle Calculator

Height of Pentagon Formula:

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

Height of Pentagon (h):

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1. What is the Height of Pentagon Formula?

The Height of Pentagon formula calculates the vertical distance from the base to the opposite vertex of a regular pentagon when the area is known. This formula uses trigonometric functions to derive the height based on the pentagon's area.

2. How Does the Calculator Work?

The calculator uses the following formula:

Where:

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

Explanation: The formula combines geometric properties of a regular pentagon with trigonometric functions to calculate the height from the given area.

3. Importance of Height Calculation

Details: Calculating the height of a pentagon is essential in geometry, architecture, and engineering applications where pentagonal shapes are used. It helps in determining spatial relationships and structural properties.

4. Using the Calculator

Tips: Enter the area of the pentagon in square meters. The value must be positive and greater than zero for accurate calculation.

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 degrees each).

Q2: Why are trigonometric functions used in this formula?
A: Trigonometric functions help relate the area of the pentagon to its height through the geometric properties and angles inherent in a regular pentagon.

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 units should I use for the area?
A: The calculator uses square meters (m²), but you can use any consistent unit system as long as the height will be in the corresponding linear units.

Q5: How accurate is this calculation?
A: The calculation is mathematically precise for regular pentagons, using exact trigonometric relationships derived from the pentagon's geometry.