Area Of Concave Regular Pentagon Given Distance Of Tips Calculator

Formula Used:

\[ A = \left(\frac{d_{Tips}}{1+\sqrt{5}}\right)^2 \times \left(\sqrt{25+10\sqrt{5}}-\sqrt{10+2\sqrt{5}}\right) \]

Area of Concave Regular Pentagon:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Area of Concave Regular Pentagon?

The Area of Concave Regular Pentagon is the total quantity of plane enclosed by the boundary of the Concave Regular Pentagon. It represents the two-dimensional space occupied by this geometric shape.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ A = \left(\frac{d_{Tips}}{1+\sqrt{5}}\right)^2 \times \left(\sqrt{25+10\sqrt{5}}-\sqrt{10+2\sqrt{5}}\right) \]

Where:

\( A \) — Area of Concave Regular Pentagon (m²)
\( d_{Tips} \) — Distance of Tips of Concave Regular Pentagon (m)
\( \sqrt{5} \) — Square root of 5 (approximately 2.23607)

Explanation: This formula calculates the area of a concave regular pentagon based on the distance between its two upper tips, using mathematical constants derived from the golden ratio properties.

3. Importance of Area Calculation

Details: Calculating the area of geometric shapes is fundamental in mathematics, engineering, architecture, and various design fields. For concave regular pentagons, this calculation helps in material estimation, space planning, and structural analysis.

4. Using the Calculator

Tips: Enter the distance between the tips of the concave regular pentagon in meters. The value must be positive and greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is a concave regular pentagon?
A: A concave regular pentagon is a five-sided polygon with equal sides and angles, but with at least one interior angle greater than 180 degrees, causing it to cave inward.

Q2: How is this formula derived?
A: The formula is derived from geometric properties of regular pentagons and the mathematical relationships involving the golden ratio (φ = (1+√5)/2).

Q3: What are the practical applications of this calculation?
A: This calculation is used in architectural design, artistic patterns, tiling problems, and various engineering applications where pentagonal shapes are employed.

Q4: Can this formula be used for convex pentagons?
A: No, this specific formula is designed for concave regular pentagons. Convex regular pentagons have a different area calculation formula.

Q5: What is the relationship between tip distance and area?
A: The area increases with the square of the tip distance, following a quadratic relationship as shown in the formula.