First Acute Angle of Concave Quadrilateral Calculator

Formula Used:

\[ \text{First Acute Angle} = \arccos\left(\frac{S_{\text{First Outer}}^2 + S_{\text{First Inner}}^2 - d_{\text{Inner}}^2}{2 \times S_{\text{First Outer}} \times S_{\text{First Inner}}}\right) \]

First Outer Side of Concave Quadrilateral:

First Inner Side of Concave Quadrilateral:

Inner Diagonal of Concave Quadrilateral:

First Acute Angle of Concave Quadrilateral:

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1. What is the First Acute Angle of Concave Quadrilateral?

The First Acute Angle of a Concave Quadrilateral is the angle formed between the first outer and first inner side of the quadrilateral. It is calculated using the law of cosines applied to the triangle formed by these sides and the inner diagonal.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{First Acute Angle} = \arccos\left(\frac{S_{\text{First Outer}}^2 + S_{\text{First Inner}}^2 - d_{\text{Inner}}^2}{2 \times S_{\text{First Outer}} \times S_{\text{First Inner}}}\right) \]

Where:

\( S_{\text{First Outer}} \) — First Outer Side of Concave Quadrilateral (m)
\( S_{\text{First Inner}} \) — First Inner Side of Concave Quadrilateral (m)
\( d_{\text{Inner}} \) — Inner Diagonal of Concave Quadrilateral (m)

Explanation: This formula applies the law of cosines to find the angle between two sides of a triangle when all three sides are known.

3. Importance of First Acute Angle Calculation

Details: Calculating this angle is important in geometry for understanding the properties of concave quadrilaterals, determining shape characteristics, and solving related geometric problems.

4. Using the Calculator

Tips: Enter all three values in meters. All values must be positive numbers. The calculator will compute the angle in degrees.

5. Frequently Asked Questions (FAQ)

Q1: What is a concave quadrilateral?
A: A concave quadrilateral is a four-sided polygon with at least one interior angle greater than 180°, causing an indentation in the shape.

Q2: Why use the law of cosines for this calculation?
A: The law of cosines is used when we know all three sides of a triangle and need to find one of its angles, which is exactly the case here.

Q3: What range of values is valid for the inputs?
A: All side lengths must be positive numbers. The inner diagonal must be less than the sum of the two sides but greater than their difference to form a valid triangle.

Q4: Can this calculator handle very small or very large values?
A: Yes, as long as the values are positive and the cosine calculation remains within the valid range of [-1, 1].

Q5: What if I get an "Invalid input" message?
A: This occurs when the input values don't satisfy the triangle inequality theorem, meaning they cannot form a valid triangle.