Find The Sum Of Interior Angles Of A Polygon Calculator

Enter the number of sides of the polygon to calculate the sum of its interior angles.

Results

Formula used: Sum = (n – 2) × 180°

Polygon Angle Sums and Visualization

Sum of Interior Angles for Common Polygons

Number of Sides (n)	Polygon Name	Sum of Interior Angles	Each Interior Angle (if Regular)
3	Triangle	180°	60°
4	Quadrilateral	360°	90°
5	Pentagon	540°	108°
6	Hexagon	720°	120°
7	Heptagon	900°	~128.57°
8	Octagon	1080°	135°
9	Nonagon	1260°	140°
10	Decagon	1440°	144°

Chart: Sum of Interior Angles vs. Number of Sides

What is a Sum of Interior Angles of a Polygon Calculator?

A sum of interior angles of a polygon calculator is a tool used to determine the total measure of all the angles inside a simple polygon, given the number of its sides. An interior angle is an angle formed inside a polygon by two adjacent sides. The “sum” refers to adding up all these interior angles together. This calculator is based on a fundamental geometric formula and is useful for students, teachers, architects, engineers, and anyone working with geometric shapes. It specifically helps find the sum for any polygon, from a triangle (3 sides) to polygons with many more sides, using the sum of interior angles of a polygon calculator.

Anyone studying geometry or needing to understand the properties of polygons can use this calculator. It’s particularly helpful for quickly finding the sum without manual calculation, especially for polygons with a large number of sides. A common misconception is that the sum of interior angles is the same for all polygons, but it actually increases with the number of sides, as our sum of interior angles of a polygon calculator clearly demonstrates.

Sum of Interior Angles of a Polygon Formula and Mathematical Explanation

The formula to find the sum of the interior angles of any simple (non-self-intersecting) polygon with n sides is:

Sum of Interior Angles = (n – 2) × 180°

Where ‘n’ is the number of sides of the polygon.

Derivation:

1. Take any convex polygon with ‘n’ sides.
2. Choose one vertex (corner) of the polygon.
3. Draw diagonals from this vertex to all other non-adjacent vertices.
4. This divides the polygon into (n – 2) triangles. For example, a quadrilateral (n=4) is divided into 2 triangles, a pentagon (n=5) into 3 triangles, and so on.
5. The sum of the interior angles of any triangle is always 180°.
6. Therefore, the sum of the interior angles of the polygon is the sum of the interior angles of these (n – 2) triangles, which is (n – 2) × 180°.

The sum of interior angles of a polygon calculator uses this exact formula.

Variables Table

Variable	Meaning	Unit	Typical Range
n	Number of sides of the polygon	None (integer)	n ≥ 3
Sum	Sum of the interior angles	Degrees (°)	≥ 180°

If the polygon is a regular polygon (all sides and angles are equal), the measure of each interior angle can be found by dividing the sum by the number of sides:

Each Interior Angle (Regular Polygon) = [(n – 2) × 180°] / n

Practical Examples (Real-World Use Cases)

Example 1: Pentagon (5 sides)

Let’s find the sum of interior angles of a pentagon using the sum of interior angles of a polygon calculator or the formula:

• Number of sides (n) = 5
• Sum = (5 – 2) × 180° = 3 × 180° = 540°

The sum of the interior angles of any pentagon is 540°. If it’s a regular pentagon, each interior angle is 540°/5 = 108°.

Example 2: Octagon (8 sides)

Let’s find the sum of interior angles of an octagon:

• Number of sides (n) = 8
• Sum = (8 – 2) × 180° = 6 × 180° = 1080°

The sum of the interior angles of any octagon is 1080°. If it’s a regular octagon, each interior angle is 1080°/8 = 135°.

Using the sum of interior angles of a polygon calculator above, you can verify these results quickly.

How to Use This Sum of Interior Angles of a Polygon Calculator

1. Enter the Number of Sides: Input the number of sides (n) of your polygon into the “Number of Sides (n)” field. Remember, a polygon must have at least 3 sides.
2. Calculate: The calculator will automatically update the results as you type or you can click the “Calculate” button.
3. View Results: The “Results” section will display:
   • The Sum of Interior Angles.
   • The measure of Each Interior Angle (if the polygon were regular).
   • The number of triangles the polygon can be divided into from one vertex.
   • The common name of the polygon if it’s one of the well-known types.
4. Reset: Click “Reset” to clear the input and results, setting the number of sides back to the default (5).
5. Copy: Click “Copy Results” to copy the main results and formula to your clipboard.

The sum of interior angles of a polygon calculator provides immediate feedback, making it easy to explore different polygons.

Key Factors That Affect the Sum of Interior Angles

1. Number of Sides (n): This is the *only* factor that determines the sum of the interior angles of a simple polygon. The formula (n-2) * 180° directly depends on ‘n’. As ‘n’ increases, the sum of the interior angles increases linearly.
2. Polygon Type (Simple vs. Complex): The formula (n-2) * 180° applies to *simple* polygons (those that do not intersect themselves). Complex or self-intersecting polygons (star polygons) have different angle sum properties. Our sum of interior angles of a polygon calculator is for simple polygons.
3. Convexity vs. Concavity: The formula (n-2) * 180° works for both convex and concave simple polygons. A convex polygon has all interior angles less than 180°, while a concave polygon has at least one interior angle greater than 180° (a reflex angle). The sum remains the same.
4. Regularity of the Polygon: Whether a polygon is regular (all sides and angles equal) or irregular (sides and/or angles are different) does NOT affect the *sum* of its interior angles. It only affects the measure of *each individual* interior angle if you were to average them or if it’s regular.
5. Dimensions/Side Lengths: The lengths of the sides of the polygon do not influence the sum of its interior angles, as long as the number of sides remains the same.
6. The Plane: The formula assumes the polygon lies in a Euclidean plane (flat surface). Polygons on curved surfaces (like a sphere) have different angle sum rules (e.g., spherical triangles can have angle sums greater than 180°). This sum of interior angles of a polygon calculator is for planar geometry.

Frequently Asked Questions (FAQ)

1. What is the minimum number of sides a polygon can have?

A polygon must have at least 3 sides, which forms a triangle.

2. Does the sum of interior angles formula work for all polygons?

It works for all *simple* polygons (convex and concave) in a plane. It does not apply to complex (self-intersecting) polygons or spherical polygons in the same way. Our sum of interior angles of a polygon calculator is for simple, planar polygons.

3. How do I find the measure of each interior angle of a regular polygon?

First, find the sum using (n-2) * 180°, then divide the sum by the number of sides, n. The formula is [(n-2) * 180°] / n.

4. What is the sum of exterior angles of a convex polygon?

The sum of the exterior angles of any convex polygon (one at each vertex) is always 360°.

5. Can an interior angle be greater than 180°?

Yes, in a concave polygon, at least one interior angle is a reflex angle (greater than 180°).

6. Why is the formula (n-2) * 180°?

Because any simple polygon with n sides can be divided into (n-2) triangles by drawing diagonals from one vertex, and each triangle has an angle sum of 180°.

7. What if the input to the sum of interior angles of a polygon calculator is less than 3?

The calculator will show an error or not compute, as a polygon cannot have fewer than 3 sides.

8. Does the calculator work for irregular polygons?

Yes, the sum of interior angles is the same for regular and irregular polygons with the same number of sides. The calculator gives the sum, and also what each angle *would be* if it were regular.

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