Height of Pentagon given Inradius using Interior Angle Calculator

Formula Used:

\[ h = r_i \times \left(1 + \frac{1}{\frac{1}{2} - \cos\left(\frac{3}{5}\pi\right)}\right) \]

Height of Pentagon (h):

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1. What is the Height of Pentagon given Inradius using Interior Angle?

The Height of Pentagon given Inradius using Interior Angle is a geometric calculation that determines the vertical height of a regular pentagon based on its inradius (the radius of the inscribed circle) using the interior angle properties of the pentagon.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ h = r_i \times \left(1 + \frac{1}{\frac{1}{2} - \cos\left(\frac{3}{5}\pi\right)}\right) \]

Where:

\( h \) — Height of Pentagon (m)
\( r_i \) — Inradius of Pentagon (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)
\( \cos \) — Cosine trigonometric function

Explanation: This formula derives from the geometric relationships between the inradius, interior angles, and height of a regular pentagon, utilizing trigonometric functions to establish the connection.

3. Importance of Height Calculation

Details: Calculating the height of a pentagon is essential in various geometric applications, architectural design, and engineering projects where precise dimensional relationships are required for regular pentagonal structures.

4. Using the Calculator

Tips: Enter the inradius value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding height of the pentagon.

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: What is the inradius of a pentagon?
A: The inradius is the radius of the circle that can be inscribed inside the pentagon, touching all five sides.

Q3: Why use the interior angle in this calculation?
A: The interior angle properties help establish trigonometric relationships between the inradius and other dimensions of the pentagon.

Q4: Are there other methods to calculate pentagon height?
A: Yes, height can also be calculated using side length or circumradius, but this specific formula uses the inradius and interior angle properties.

Q5: What applications use pentagon height calculations?
A: Architecture, engineering design, geometric modeling, and various mathematical applications involving regular polygons.