Polygons and their angles

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What is a Polygon?

A polygon is a closed plane figure made from straight line segments joined end to end.
Each straight segment is called a side (or edge).
The points where two sides meet are the vertices (corners).
A polygon with n sides is called an n-gon; for example, a triangle is a 3-gon.
A simple polygon does not cross itself — the only meeting points of sides are shared vertices.
A self-intersecting polygon crosses over itself, creating shapes like star polygons.

Naming Polygons by Number of Sides

Polygons are named by how many sides they have.
3 sides: triangle; 4 sides: quadrilateral; 5 sides: pentagon.
6 sides: hexagon; 7 sides: heptagon; 8 sides: octagon.
9 sides: nonagon; 10 sides: decagon.
The number of sides always equals the number of vertices.

Regular and Irregular Polygons

A regular polygon has all sides equal in length and all interior angles equal.
An irregular polygon does not have all sides equal or all angles equal (or both).
Equilateral means all sides are the same length.
Equiangular means all corner angles are equal.
A polygon is regular if and only if it is both equilateral and equiangular.

Convex and Concave Polygons

A convex polygon has all interior angles less than 180°.
In a convex polygon, any line segment between two boundary points lies entirely inside the polygon.
A concave polygon has at least one interior angle greater than 180°.
All convex polygons are simple, but not all simple polygons are convex.

Triangles and Special Quadrilaterals

A triangle has 3 sides and 3 angles; its interior angles sum to 180°.
Special quadrilaterals include square, rectangle, parallelogram, rhombus, trapezium, and kite.
A square has 4 equal sides, 4 right angles, and 4 lines of symmetry.
A rectangle has opposite sides equal, 4 right angles, and 2 lines of symmetry.
A parallelogram has opposite sides parallel and equal, and opposite angles equal.
A rhombus has 4 equal sides, opposite sides parallel, and diagonals that bisect at right angles.

Sum of Interior Angles

The sum of the interior angles of a simple n-gon is (n - 2) imes 180^\circ.
This works because any simple n-gon can be split into (n - 2) triangles, each summing to 180°.
For a triangle (n = 3): (3 - 2) imes 180^\circ = 180^\circ.
For a quadrilateral (n = 4): (4 - 2) imes 180^\circ = 360^\circ.
For a pentagon (n = 5): (5 - 2) imes 180^\circ = 540^\circ.
For a hexagon (n = 6): (6 - 2) imes 180^\circ = 720^\circ.

Interior Angles of a Regular Polygon

Each interior angle of a regular n-gon is 180^\circ - rac{360^\circ}{n}.
Equivalently, each interior angle is rac{(n - 2) imes 180^\circ}{n}.
For a regular pentagon (n = 5): each interior angle is 180^\circ - rac{360^\circ}{5} = 108^\circ.
For a regular hexagon (n = 6): each interior angle is 180^\circ - rac{360^\circ}{6} = 120^\circ.
For a regular octagon (n = 8): each interior angle is 180^\circ - rac{360^\circ}{8} = 135^\circ.

Exterior Angles

An exterior angle is formed by extending one side of the polygon at a vertex.
The exterior angle and its adjacent interior angle add up to 180° (they are on a straight line).
The sum of the exterior angles of any convex polygon is always 360°.
For a regular n-gon, each exterior angle is rac{360^\circ}{n}.
Example: a regular hexagon has exterior angles of rac{360^\circ}{6} = 60^\circ.

Finding the Number of Sides from an Angle

If you know one interior angle of a regular polygon, first find the exterior angle: exterior = 180^\circ - interior.
Then use n = rac{360^\circ}{ ext{exterior angle}} to find the number of sides.
Example: if each interior angle is 150°, the exterior angle is 30°, so n = rac{360^\circ}{30^\circ} = 12 sides.
Alternatively, solve 180^\circ - rac{360^\circ}{n} = interior angle for n.
Check your answer: the number of sides must be a whole number greater than 2.

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Übungsfragen

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1.What is the name of a polygon with 5 sides?
Easy
- APentagon
- BHexagon
- CQuadrilateral
- DHeptagon
2.A regular polygon has all sides equal in length and all interior angles equal.
Easy
True or false?
3.What is the sum of the interior angles of a hexagon?
Easy
- A720°
- B540°
- C900°
- D360°
4.In a regular octagon, what is the size of each exterior angle?
Medium
- A45°
- B135°
- C40°
- D60°
5.Which of the following statements are true for any triangle? (select all that apply)
Medium
- AThe sum of interior angles is 180°.
- BThe sum of exterior angles is 360°.
- CAll three sides are equal.
- DIt has four vertices.
- EEach interior angle is less than 180°.
6.Match each polygon to its number of sides.
Medium
- Triangle
- Quadrilateral
- Pentagon
- Hexagon
- 3
- 4
- 5
- 6
7.Arrange these polygons in order of increasing number of sides.
Medium
- Triangle
- Pentagon
- Heptagon
- Nonagon
8.Each interior angle of a regular polygon is 150°. How many sides does the polygon have?
Medium
- A12
- B10
- C15
- D8

Mehr Fragen

9.Match each term to its correct definition.
Easy
- Regular polygon
- Equilateral polygon
- Equiangular polygon
- Convex polygon
- All sides are equal in length and all interior angles are equal
- All sides are equal in length
- All interior angles are equal
- All interior angles are less than 180°
10.Which of the following statements are true for any polygon? (select all that apply)
Medium
- AThe number of vertices equals the number of sides.
- BThe sum of the exterior angles is always 360°.
- CAll interior angles are less than 180°.
- DThe sum of the interior angles is (n − 2) × 180°.
- EAll sides are equal in length.
11.Arrange these steps in the correct order to find the size of each interior angle of a regular 12-sided polygon.
Easy
- Use the formula (n − 2) × 180° to find the sum of interior angles.
- Substitute n = 12 into the formula.
- Calculate (12 − 2) × 180° = 1800°.
- Divide the sum by 12 to get each interior angle: 1800° ÷ 12 = 150°.
12.Which of the following are properties of a regular hexagon? (select all that apply)
Medium
- AIt has 6 lines of symmetry.
- BEach interior angle is 120°.
- CEach exterior angle is 60°.
- DIt has 9 diagonals.
- EAll interior angles are equal.
13.Match each polygon to the sum of its interior angles.
Medium
- Triangle
- Quadrilateral
- Pentagon
- Hexagon
- 180°
- 360°
- 540°
- 720°
14.What is the sum of the interior angles of a decagon (10-sided polygon)?
Medium
- A1440°
- B1620°
- C1800°
- D1260°
15.Each exterior angle of a regular polygon is 20°. How many sides does the polygon have?
Medium
- A18
- B20
- C16
- D15
16.The sum of the exterior angles of any convex polygon is always 360°.
Easy
True or false?
17.A concave polygon has at least one interior angle greater than 180°.
Easy
True or false?
18.What is the sum of the interior angles of a quadrilateral?
Easy
- A180°
- B270°
- C360°
- D540°
19.In a regular pentagon, what is the size of each exterior angle?
Medium
- A60°
- B72°
- C90°
- D108°
20.Each exterior angle of a regular polygon is 30°. How many sides does the polygon have?
Medium
- A6
- B10
- C12
- D15
21.In a regular polygon, each interior angle is 140°. How many sides does it have?
Medium
- A7
- B8
- C9
- D10
22.Which of the following statements are true? (select all that apply)
Medium
- AA regular polygon is both equilateral and equiangular.
- BA polygon with 8 sides is called a heptagon.
- CThe sum of the exterior angles of any convex polygon is 360°.
- DA concave polygon has all interior angles less than 180°.
- EA triangle is a 3-gon.
23.The sum of the interior angles of a triangle is 180°.
Easy
True or false?
24.Match each regular polygon to the size of one of its exterior angles.
Medium
- Equilateral triangle
- Square
- Regular hexagon
- Regular octagon
- 120°
- 90°
- 60°
- 45°
25.A regular polygon has an interior angle of 156°. How many sides does it have?
Medium
- A15
- B12
- C18
- D24
26.A regular polygon has 20 sides. What is the size of each exterior angle?
Medium
- A18°
- B20°
- C36°
- D9°
27.Which of the following statements about a regular polygon with 18 sides are true? (select all that apply)
Medium
- AEach exterior angle is 20°.
- BEach interior angle is 160°.
- CThe sum of the interior angles is 2880°.
- DEach interior angle is 140°.
- EThe sum of the exterior angles is 180°.
28.A polygon with 7 sides is called a heptagon.
Easy
True or false?
29.Match each regular polygon to the sum of its interior angles.
Medium
- Pentagon
- Heptagon
- Nonagon
- Decagon
- 540°
- 900°
- 1260°
- 1440°
30.Arrange these regular polygons in order of increasing size of each interior angle.
Medium
- Hexagon
- Octagon
- Decagon
- Dodecagon
31.In a regular polygon, the interior angle is 8 times the exterior angle. How many sides does the polygon have?
Hard
- A18
- B20
- C16
- D24
32.What is the name of the two-dimensional region enclosed by the sides of a polygon?
Easy
- APolygon body
- BPolygon edge
- CPolygon vertex
- DPolygon chain
33.Which of the following statements about convex polygons is true?
Medium
- AA convex polygon can have an interior angle greater than 180°.
- BA convex polygon is always self-intersecting.
- CA convex polygon has all interior angles less than 180°.
- DA convex polygon is always a star polygon.
34.Which of the following are true about a regular polygon? (select all that apply)
Medium
- AAll sides are equal in length.
- BAll interior angles are equal.
- CIt is both equilateral and equiangular.
- DIt must be concave.
- EIt cannot be cyclic.
35.A polygon with 3 sides is called a triangle.
Easy
True or false?
36.Match each term to its definition.
Easy
- Equilateral polygon
- Equiangular polygon
- Regular polygon
- All sides equal in length
- All interior angles equal
- Both equilateral and equiangular
37.Arrange these steps in the correct order to find the size of each exterior angle of a regular polygon with 10 sides.
Medium
- Recall that exterior angles sum to 360°.
- Divide 360° by the number of sides.
- Calculate 360° ÷ 10 = 36°.
- State the size of each exterior angle.
38.The sum of the interior angles of a polygon is 1260°. How many sides does the polygon have?
Medium
- A7
- B8
- C9
- D10
39.Which of the following is a property of a regular pentagon?
Easy
- AIt has 5 sides of equal length and 5 equal interior angles.
- BIt has 5 sides but all interior angles are different.
- CIt has 6 sides and 6 equal angles.
- DIt is a concave polygon.
40.The sum of the interior angles of a regular polygon is 1080°. How many sides does the polygon have?
Medium
41.Complete the sentence about polygon angles.
Easy
The sum of the exterior angles of any convex polygon is ____ degrees.
42.What name is given to a polygon in which all interior angles are less than 180°?
Medium
43.Complete the sentence about the angles of a regular polygon.
Medium
In a regular polygon, each interior angle is equal to 180° minus ____ divided by the number of sides.
44.A regular polygon has 15 sides. Calculate the size of each exterior angle, in degrees.
Medium
45.Calculate the size of each exterior angle of a regular decagon (10 sides), in degrees.
Easy
46.A regular polygon has 12 sides. Calculate the size of each interior angle, in degrees.
Medium
47.Complete the sentence about the exterior angles of a regular polygon.
Easy
Each exterior angle of a regular polygon with n sides is found by dividing ____ by n.
48.Complete the sentence about the interior angles of a regular polygon.
Medium
Each interior angle of a regular polygon with n sides is found by calculating 180° − ____.
49.A regular polygon has 8 sides. Calculate the size of each exterior angle, in degrees.
Easy
50.A regular polygon has 5 sides. Calculate the size of each interior angle, in degrees.
Medium
51.The sum of the interior angles of a polygon is 1440°. How many sides does the polygon have?
Medium
52.Explain why the sum of the interior angles of a polygon with n sides is (n − 2) × 180°.
Medium
53.Explain why the exterior angles of any convex polygon always sum to 360°.
Hard
54.A regular polygon has 9 sides. Calculate the size of each interior angle. Show your method.
Medium
55.Explain why the sum of the exterior angles of any convex polygon is always 360°.
Medium
56.The sum of the interior angles of a regular polygon is 1080°. How many sides does the polygon have?
Medium
- A10 sides
- B9 sides
- C8 sides
- D7 sides
57.A regular polygon has 15 sides. Calculate the size of each exterior angle, in degrees.
Medium
- A22 °
- B26 °
- C24 °
- D28 °
58.Calculate the size of each exterior angle of a regular decagon (10 sides), in degrees.
Easy
- A44 °
- B40 °
- C36 °
- D32 °
59.A regular polygon has 12 sides. Calculate the size of each interior angle, in degrees.
Medium
- A135 °
- B150 °
- C165 °
- D180 °
60.A regular polygon has 8 sides. Calculate the size of each exterior angle, in degrees.
Easy
- A55 °
- B45 °
- C40 °
- D50 °
61.A regular polygon has 5 sides. Calculate the size of each interior angle, in degrees.
Medium
- A119 °
- B97 °
- C108 °
- D130 °
62.The sum of the interior angles of a polygon is 1440°. How many sides does the polygon have?
Medium
- A10 sides
- B9 sides
- C11 sides
- D12 sides

Lernnotizen

## Polygons and their angles A **polygon** is a closed plane figure made of straight line segments. The segments are **sides** (edges); where two sides meet is a **vertex** (corner). An **n-gon** has n sides, so a triangle is a 3-gon. ### Naming and classifying - Named by number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10). - **Simple**: boundary does not cross itself. **Convex**: all interior angles < 180°. **Concave** (non-convex, simple): at least one interior angle > 180°. **Self-intersecting**: boundary crosses itself. - **Equilateral**: all sides equal. **Equiangular**: all angles equal. **Regular**: both equilateral and equiangular. - **Cyclic**: all vertices lie on one circle (the circumcircle). **Tangential**: all sides touch an inscribed circle. ### Triangles and special quadrilaterals - Triangle angle sum = 180°. - Quadrilateral angle sum = 360°. - Special quadrilaterals (e.g. square, rectangle, parallelogram, rhombus, trapezium) have their own side, angle, symmetry and diagonal properties. ### Angle sums - **Interior angles** of a simple n-gon: \((n - 2) \times 180^\circ\). - Reason: the n-gon splits into (n − 2) triangles, each summing to 180°. - **Exterior angles** of any polygon sum to **360°**. - For a **regular** n-gon, each interior angle is \(180^\circ - \frac{360^\circ}{n}\), and each exterior angle is \(\frac{360^\circ}{n}\). ### Finding the number of sides If a regular polygon has each exterior angle e°, then \(n = \frac{360}{e}\). If each interior angle is i°, then exterior = 180° − i°, so \(n = \frac{360}{180 - i}\). ### Quick checks - Sum of interior angles grows with n: triangle 180°, quadrilateral 360°, pentagon 540°, hexagon 720°. - A regular polygon's interior and exterior angles at a vertex add to 180°. - Diagonals: a polygon's diagonals join non-adjacent vertices; their number and symmetry depend on the shape.

Polygon: A plane figure made of line segments connected to form a closed polygonal chain.
Side (edge): One of the line segments making up a polygon.
Vertex (corner): A point where two sides of a polygon meet.
n-gon: A polygon with n sides; e.g. a triangle is a 3-gon.
Simple polygon: A polygon whose boundary does not cross itself.
Convex polygon: A polygon in which all interior angles are less than 180°.
Concave polygon: A simple non-convex polygon with at least one interior angle greater than 180°.
Equilateral: All edges are of the same length.
Equiangular: All corner angles are equal.
Regular polygon: A polygon that is both equilateral and equiangular.
Cyclic polygon: A polygon whose corners all lie on a single circle (the circumcircle).
Tangential polygon: A polygon whose sides are all tangent to an inscribed circle.
Interior angle sum: For a simple n-gon, the sum of interior angles is (n − 2) × 180°.
Exterior angle sum: The exterior angles of any polygon add up to 360°.
Regular interior angle: Each interior angle of a convex regular n-gon is 180° − 360°/n.

Karteikarten

What is a polygon?

A plane figure made of line segments joined end-to-end to form a closed chain.

What are the sides of a polygon?

The line segments that make up the closed chain.

What are the vertices of a polygon?

The points where two sides meet (also called corners).

What is an n-gon?

A polygon with n sides. For example, a triangle is a 3-gon.

What is a simple polygon?

A polygon whose boundary does not cross itself; the only intersections are shared endpoints of consecutive sides.

What is a convex polygon?

A polygon where any line through it (not tangent to an edge or corner) meets the boundary exactly twice; all interior angles are less than 180°.

What is a concave polygon?

A simple, non-convex polygon with at least one interior angle greater than 180°.

What is a regular polygon?

A polygon that is both equilateral (all sides equal) and equiangular (all angles equal).

What is an irregular polygon?

A polygon that is not regular; its sides and/or angles are not all equal.

What is the sum of the interior angles of a simple n-gon?

The sum is (n − 2) × 180°.

Why is the sum of interior angles of an n-gon (n − 2) × 180°?

Any simple n-gon can be divided into (n − 2) triangles, each with angle sum 180°.

What is the sum of the exterior angles of any polygon?

The exterior angles of any polygon add up to 360°.

For a regular n-gon, how do you find each interior angle?

Each interior angle = 180° − 360°/n.

For a regular n-gon, how do you find each exterior angle?

Each exterior angle = 360°/n.

How can you find the number of sides of a regular polygon if you know one interior angle?

Use n = 360° / (180° − interior angle).

How can you find the number of sides of a regular polygon if you know one exterior angle?

Use n = 360° / exterior angle.

What is the sum of the interior angles of a triangle?

180°.

What is the sum of the interior angles of a quadrilateral?

360°.

What is the sum of the interior angles of a pentagon?

540°.

What is the sum of the interior angles of a hexagon?

720°.

What is the sum of the interior angles of an octagon?

1080°.

What is the sum of the interior angles of a decagon?

1440°.

What is a cyclic polygon?

A polygon whose vertices all lie on a single circle (the circumcircle).

What is a tangential polygon?

A polygon whose sides are all tangent to an inscribed circle.

What is a star polygon?

A polygon that self-intersects in a regular way.

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