Find Measure Of Each Angle Calculator

Easily calculate the measure of each interior angle of a regular polygon or find the third angle of a triangle using our Find Measure of Each Angle Calculator. Input the number of sides or two angles and get instant results with formulas.

Regular Polygon Angle Calculator

Triangle Angle Calculator

Common Regular Polygons

Name	Sides (n)	Sum of Interior Angles	Each Interior Angle	Each Exterior Angle
Equilateral Triangle	3	180°	60°	120°
Square	4	360°	90°	90°
Regular Pentagon	5	540°	108°	72°
Regular Hexagon	6	720°	120°	60°
Regular Heptagon	7	900°	~128.57°	~51.43°
Regular Octagon	8	1080°	135°	45°
Regular Nonagon	9	1260°	140°	40°
Regular Decagon	10	1440°	144°	36°

Table of interior and exterior angles for common regular polygons.

Interior Angle vs. Number of Sides

Chart showing how the measure of each interior angle changes as the number of sides of a regular polygon increases.

What is a Find Measure of Each Angle Calculator?

A find measure of each angle calculator is a tool designed to determine the size of angles within geometric shapes, particularly regular polygons and triangles. For regular polygons, where all sides and angles are equal, it calculates the measure of each individual interior angle based on the number of sides. For triangles, given two angles, it calculates the third angle, knowing the sum of angles in a triangle is always 180 degrees. This calculator is invaluable for students, teachers, engineers, and anyone working with geometry.

People who should use this tool include students learning geometry, architects designing structures, engineers calculating forces, and hobbyists working on projects involving regular shapes or triangles. A common misconception is that all polygons with the same number of sides have the same interior angles; however, this is only true for *regular* polygons.

Find Measure of Each Angle Formula and Mathematical Explanation

The formulas used by the find measure of each angle calculator depend on the shape.

Regular Polygons

For a regular polygon with 'n' sides:

1. Sum of Interior Angles (S): The sum of all interior angles is given by the formula: S = (n – 2) * 180°. This is derived by dividing the polygon into (n-2) triangles from one vertex.
2. Measure of Each Interior Angle (A_i): Since it's a regular polygon, all interior angles are equal. So, each interior angle is: A_i = S / n = ((n – 2) * 180°) / n.
3. Measure of Each Exterior Angle (A_e): The sum of exterior angles of any convex polygon is 360°. For a regular polygon, each exterior angle is: A_e = 360° / n. Also note that A_i + A_e = 180°.

Triangles

For any triangle, the sum of its three interior angles (A, B, C) is always 180°. If you know two angles, say A and B, the third angle C is found by: C = 180° – (A + B).

Variables Table (Regular Polygon)

Variable	Meaning	Unit	Typical Range
n	Number of sides	None (integer)	3, 4, 5, … (≥ 3)
S	Sum of interior angles	Degrees (°)	180°, 360°, 540°, …
Ai	Measure of each interior angle	Degrees (°)	60° up to <180°
Ae	Measure of each exterior angle	Degrees (°)	>0° up to 120°

Variables Table (Triangle)

Variable	Meaning	Unit	Typical Range
A, B, C	Angles of the triangle	Degrees (°)	>0° and <180°

Practical Examples (Real-World Use Cases)

Example 1: Regular Hexagon

An architect is designing a floor tiling pattern using regular hexagons.

• Input: Number of sides (n) = 6
• Calculation:
  • Sum of interior angles = (6 – 2) * 180° = 4 * 180° = 720°
  • Each interior angle = 720° / 6 = 120°
  • Each exterior angle = 360° / 6 = 60°
• Output: Each interior angle is 120°. This helps the architect ensure the tiles fit together perfectly.

Example 2: Finding the Third Angle of a Triangle

A surveyor measures two angles of a triangular plot of land as 45° and 70°.

• Input: Angle A = 45°, Angle B = 70°
• Calculation: Angle C = 180° – (45° + 70°) = 180° – 115° = 65°
• Output: The third angle is 65°. Knowing all angles is crucial for area calculations and mapping. Our find measure of each angle calculator can quickly give this result.

How to Use This Find Measure of Each Angle Calculator

1. For Regular Polygons:
   • Enter the number of sides (n) of the regular polygon into the "Number of Sides" field. It must be 3 or greater.
   • The calculator will instantly display the measure of each interior angle, the sum of interior angles, and the measure of each exterior angle.
   • The chart below the calculator also updates to show the interior angle for polygons from 3 sides up to 'n' sides.
2. For Triangles:
   • Enter the measures of two known angles (Angle A and Angle B) into their respective fields. Ensure their sum is less than 180°.
   • The calculator will show the measure of the third angle (Angle C).
3. Reading Results: The primary result is highlighted, with intermediate values shown below, along with the formulas used.
4. Resetting: Click the "Reset Polygon" or "Reset Triangle" buttons to clear inputs and results for the respective calculator section and restore default values.
5. Copying Results: Click "Copy Results" to copy the main calculated values and formulas to your clipboard.

Using the find measure of each angle calculator helps in understanding the geometric properties of shapes quickly and accurately.

Key Factors That Affect Angle Measures

The measure of angles in polygons and triangles is determined by fundamental geometric principles.

1. Number of Sides (for regular polygons): This is the primary determinant. As the number of sides increases, each interior angle of a regular polygon also increases, approaching 180°.
2. Polygon Regularity: The formulas used here for "each angle" apply *only* to regular polygons (all sides and angles equal). For irregular polygons, angles can vary.
3. Sum of Angles in a Triangle: The fact that the sum of interior angles in any Euclidean triangle is always 180° is a fundamental constant used for the triangle calculation.
4. Convexity of the Polygon: The formulas (n-2)*180 assume a simple, convex polygon.
5. Type of Geometry: These calculations are based on Euclidean geometry. In non-Euclidean geometries (like spherical or hyperbolic), the sum of angles in a triangle is not 180°.
6. Accuracy of Input: For the triangle calculator, the accuracy of the input angles directly affects the accuracy of the calculated third angle.

Understanding these factors is key to correctly interpreting the results from a find measure of each angle calculator.

Frequently Asked Questions (FAQ)

Q: What is a regular polygon? A: A regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).

Q: Can I use this calculator for irregular polygons? A: No, the "each interior angle" formula ((n-2)*180/n) is only for regular polygons. For irregular polygons, the sum is still (n-2)*180, but individual angles vary. The triangle part works for any triangle if you know two angles.

Q: What is the minimum number of sides a polygon can have? A: A polygon must have at least 3 sides (a triangle).

Q: What happens to the interior angle as the number of sides gets very large? A: As the number of sides (n) of a regular polygon increases, the measure of each interior angle approaches 180 degrees, and the polygon looks more and more like a circle.

Q: Why is the sum of angles in a triangle always 180 degrees? A: This is a fundamental theorem in Euclidean geometry.

Q: What are exterior angles? A: An exterior angle of a polygon is the angle between one side of a polygon and the line extended from an adjacent side. The sum of exterior angles of any convex polygon is 360°.

Q: Can I input angles in radians for the triangle calculator? A: No, this calculator expects angles in degrees. You would need to convert radians to degrees (multiply by 180/π) before using the triangle calculator.

Q: What if the two angles I enter for the triangle add up to 180° or more? A: The calculator will show an error because a valid triangle cannot be formed with two angles summing to 180° or more.