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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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

複習筆記

## 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.

字卡

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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