Angles in Polygons - GCSE Maths

Name: Angles in Polygons - GCSE Maths
Uploaded: 2024-07-13T20:43:16.000Z
Duration: 40 min 41 s

What is a Polygon?

Definition and Characteristics

Polygons are defined as closed shapes with straight sides. Examples include various geometric figures, while shapes with gaps or curved sides do not qualify as polygons.

Types of Polygons

Polygons can be categorized into two types: regular and irregular. Regular polygons have equal side lengths and equal interior angles, whereas irregular polygons do not meet these criteria.

Understanding Regular vs Irregular Polygons

Examples of Regular Polygons

An equilateral triangle has three equal sides; a square has four equal sides; a pentagon has five sides; a hexagon has six; a heptagon has seven; an octagon has eight; a nonagon has nine; and a decagon has ten sides. The names for heptagons and nonagons are less commonly required for exams.

Interior Angles of Polygons

Angle Sum in Triangles and Quadrilaterals

The sum of the angles in a triangle is always 180°. For quadrilaterals, this sum is 360°, which can be understood by dividing the shape into two triangles (2 x 180°).

Extending to Other Shapes

For pentagons, the angle sum is calculated by splitting it into three triangles (3 x 180° = 540°). Hexagons can be divided into four triangles (4 x 180° = 720°). This pattern continues as more sides are added.

General Formula for Interior Angles

Deriving the Formula

The number of triangles formed from any polygon can be determined by subtracting two from the number of its sides (n - 2). Each triangle contributes an angle sum of 180°, leading to the formula: Interior Angle Sum = (n - 2) * 180°. This allows calculation for any polygon's interior angle sum based on its number of sides.

Applying the Interior Angle Sum Formula

Example Calculations

To find the interior angle sum of a decagon (10-sided), use: 10 - 2 multiplied by 180, resulting in 1,440°. For a polygon with 24 sides, apply 24 - 2 multiplied by 180, yielding 3,960°. These calculations illustrate practical applications of the formula in exam scenarios.

Solving Specific Problems

Understanding Angles in Polygons

Finding Angles in Irregular Polygons

To find the size of angle X in an irregular polygon, subtract known angles from the total interior angle sum. For example, 540° - 410° = 130°, indicating angle X is 130°.

Solving for Unknown Angles

In a scenario where angle ABC is three times angle BCD (let's call it X), we express ABC as 3X. The interior angle sum for a six-sided polygon is calculated as (6 - 2) * 180 = 720°.

Setting Up Equations

Combine algebraic expressions with numerical values to form an equation: 4X + 496 = 720. This leads to solving for X by isolating it on one side of the equation.

Calculating Specific Angle Values

After determining that X = 56, calculate angle ABC as 3X: 3 * 56 = 168°. Thus, the answer to the question regarding angle ABC is confirmed as 168°.

Regular Polygon Interior Angles

For regular polygons like pentagons and hexagons, use the formula for interior angles: (n - 2) * 180 / n. For a pentagon, this results in each angle being 108°; for a hexagon, each is 120°.

Calculating Interior Angles of Regular Polygons

Applying Formulas to Find Angle Sizes

The interior angle sum can be calculated using (n - 2) * 180. For example, with a decagon (10 sides), each interior angle equals 144°.

Example Calculation with Different Sides

When calculating for polygons with varying sides such as a polygon with thirty sides, replace n in the formula accordingly. Here, it yields an interior angle of 168°.

Understanding Exterior Angles

Defining Exterior Angles

An exterior angle is formed by extending one side of a polygon and measuring the adjacent internal line's extension. This differs from simply considering angles outside the shape.

Summation of Exterior Angles

All exterior angles combined equal 360°. For instance, in a regular pentagon with five equal exterior angles: dividing gives each exterior angle as 72°.

Application to Other Shapes

Similar calculations apply to other shapes; e.g., for a regular hexagon (six sides), each exterior angle measures 60°.

Formula for Exterior Angle Calculation

Understanding Angles in Regular Polygons

Calculating Exterior Angles

The exterior angle of a regular polygon can be calculated using the formula: 360° divided by the number of sides (n). For an octagon, this results in 45° as shown here: .

For a polygon with 15 sides, the exterior angle is found similarly: 360° / 15 = 24°. This method simplifies finding exterior angles for any regular polygon. .

Interior and Exterior Angle Relationship

The interior angle of a regular pentagon is 108°, while its exterior angle is 72°. Their sum equals 180°, illustrating that interior plus exterior angles always equal this value for any regular polygon: .

To find both angles for an 18-sided polygon, calculate the exterior first (20°) and then use the relationship to find the interior (160°): .

Formulas for Interior Angles

The formula for calculating the interior angle is given by: (n - 2) * 180 / n. For an 18-sided polygon, substituting gives us an interior angle of 160° after confirming via the exterior angle calculation: .

Alternatively, knowing one can quickly derive the other since they add up to 180°. Thus, if you know the exterior angle (20°), subtracting from 180° yields the interior angle directly. .

Finding Number of Sides from Exterior Angles

Rearranging the formula for finding exterior angles allows us to determine how many sides a polygon has if we know its exterior angle. Specifically, n = 360/textexterior: .

If given an exterior angle of 10°, applying this gives us 36 sides. Conversely, if provided with an interior angle of 171°, we first find its corresponding exterior (9°) before determining it has 40 sides using our rearranged formula: .

Application in Exam Questions

In practical scenarios like exam questions involving polygons, understanding these formulas becomes crucial. For instance, when analyzing two pentagons and another unknown shape X with known angles around a point summing to 360°, we can deduce missing values effectively: .

By calculating known angles (both pentagons at 108°), we find that shape X's internal angle must be 144°. From there, deriving its external leads us to conclude it has 10 sides through established formulas: .

Further Examples and Practice