Polygons
A A polygon is a 2-dimensional closed shape with straight sides, eg triangle, hexagon, etc. is a Having only two dimensions, usually length (or height) and width. shape with straight sides. Some polygons have special names, for example, triangles and quadrilaterals.
Types of polygon
Polygons can be regular or irregular. If the angles are all equal and all the sides are equal length it is a regular polygon.
Interior angles of polygons
To find the sum of interior angles in a polygon divide the polygon into triangles.
The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.
Example
Calculate the sum of interior angles in a pentagon.
A pentagon has 5 sides so it contains 3 triangles. The sum of the interior angles is:
\(180^\circ \times 3 = 540^\circ\)
The number of triangles in each polygon is two less than the number of sides.
The formula for calculating the sum of interior angles is:
\((n - 2) \times 180^\circ\) (where \(n\) is the number of sides)
Question
Calculate the sum of interior angles in an octagon.
Using \((n - 2) \times 180^\circ\) where \(n\) is the number of sides:
An octagon has 8 sides so \(n = 8\)
\((8 - 2) \times 180 = 1,080^\circ\)
Calculating the interior angles of regular polygons
All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is:
\(\text{interior angle of a polygon} = \text{sum of interior angles} \div \text{number of sides}\)
Question
Calculate the size of the interior angle of a regular A polygon with six sides..
The sum of interior angles is \((6 - 2) \times 180^\circ = 720^\circ\)
One interior angle is \(720^\circ \div 6 = 120^\circ\).
Exterior angles of polygons
If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle.
The sum of the exterior angles of a polygon is 360°.
Calculating the exterior angles of regular polygons
The formula for calculating the size of an exterior angle is:
\(\text{exterior angle of a polygon} = 360 \div \text{number of sides}\)
Remember the interior and exterior angle add up to 180°.
Question
Calculate the size of the exterior and interior angle in a regular A polygon with five sides..
Method 1
The sum of exterior angles is 360°.
The exterior angle is 360° ÷ 5 = 72°.
The interior and exterior angles add up to 180°.
The interior angle is 180° - 72° = 108°.
Method 2
The sum of interior angles is (5 - 2) × 180° = 540°.
The interior angle is 540° ÷ 5 = 108°.
The interior and exterior angles add up to 180°.
The exterior angle is 180° - 108° = 72°.
More guides on this topic
- NEW: Angles
- NEW: Angles in parallel lines
- NEW: Polygons
- NEW: Line and rotational symmetry
- NEW: Constructing triangles
- NEW: Ruler and compass constructions
- NEW: Loci
- NEW: Bearings
- NEW: Perimeter
- NEW: Area
- NEW: Volume of a prism
- NEW: Surface area of a prism
- NEW: Pyramids, cones and spheres
- NEW: Nets, plans and elevations
- NEW: Circumference and arc length
- NEW: Area of circles and sectors
- NEW: Higher – Calculating angles using circles
- NEW: Higher – Using the alternate segment theorem, tangents and chords
- NEW: Reflection
- NEW: Rotation
- NEW: Translation
- NEW: Enlargement
- NEW: Higher − Negative enlargements
- NEW: Higher – Combined transformations and invariant points
- NEW: Congruent and similar shapes
- NEW: Higher – Similarity in 2D and 3D shapes
- NEW: Pythagoras' theorem
- NEW: Solving 2D and 3D problems using Pythagoras' theorem
- NEW: Right-angled trigonometry
- NEW: Higher – Sine rule
- Loci and constructions - AQA
- 2-dimensional shapes - AQA
- 3-dimensional shapes - AQA
- Circles, sectors and arcs - AQA
- Circle theorems - Higher - AQA
- Transformations - AQA
- Pythagoras' theorem - AQA
- Units of measure - AQA
- Trigonometry - AQA
- Vectors - AQA
