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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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 62 câu hỏi. Đăng ký để xem tất cả.

  1. 1.What is the name of a polygon with 5 sides?

    Easy
    • APentagon
    • BHexagon
    • CQuadrilateral
    • DHeptagon

  2. 2.A regular polygon has all sides equal in length and all interior angles equal.

    Easy

    True or false?

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

    Easy
    • A720°
    • B540°
    • C900°
    • D360°

  4. 4.In a regular octagon, what is the size of each exterior angle?

    Medium
    • A45°
    • B135°
    • C40°
    • D60°

  5. 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. 6.Match each polygon to its number of sides.

    Medium
    • Triangle
    • Quadrilateral
    • Pentagon
    • Hexagon
    • 3
    • 4
    • 5
    • 6

  7. 7.Arrange these polygons in order of increasing number of sides.

    Medium
    • Triangle
    • Pentagon
    • Heptagon
    • Nonagon

  8. 8.Each interior angle of a regular polygon is 150°. How many sides does the polygon have?

    Medium
    • A12
    • B10
    • C15
    • D8

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