#23 Angles in Polygons - Series 2 Edexcel IGCSE Exam Questions

Summary

In this EXL IGCSE Series 2 tutorial, Mr. Asprey guides viewers through a range of polygon problems, focusing on interior and exterior angles, sums of angles, and properties of regular polygons. Using step-by-step explanations, he demonstrates how to calculate unknown angles in pentagons, hexagons, octagons, and decagons, incorporating techniques such as using exterior angles, isosceles triangle properties, symmetry, and trigonometry. Each problem is broken down clearly, showing multiple approaches where applicable. By the end, viewers gain a strong understanding of polygon angle relationships and problem-solving strategies, all reinforced with practical examples and visual reasoning.

Takeaways

• 😀 The sum of exterior angles of any regular polygon is always 360°, and each exterior angle can be found by dividing 360° by the number of sides.
• 😀 Interior and exterior angles of a polygon are supplementary, meaning they always add up to 180°.
• 😀 The sum of all interior angles of a polygon is calculated as (n - 2) × 180°, where n is the number of sides.
• 😀 To find the number of sides of a polygon when given an interior angle, subtract the interior angle from 180° to get the exterior angle, then divide 360° by that exterior angle.
• 😀 When calculating unknown angles inside polygons, it is often easier to first find exterior angles or use symmetry properties.
• 😀 Shared sides between regular polygons ensure all corresponding sides are equal, which can simplify calculations and form isosceles triangles.
• 😀 For polygons around a point, the sum of angles at that point must equal 360°, which can be used to solve for unknown angles.
• 😀 Trigonometry (tan θ = opposite/adjacent) is useful for finding angles in non-right-angle triangles within polygons.
• 😀 In complex shapes formed by multiple polygons, using interior angles, exterior angles, and symmetry can systematically solve for unknown angles.
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• 😀 Regular polygons allow predictable relationships between sides and angles, which simplifies finding interior, exterior, and combined angles.

Q & A

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

The sum of exterior angles of any regular polygon is always 360°.

How can you calculate one exterior angle of a regular polygon?

Divide 360° by the number of sides of the polygon: Exterior angle = 360 ÷ n.

What is the relationship between interior and exterior angles of a polygon?

Interior and exterior angles are supplementary, meaning Interior angle + Exterior angle = 180°.

How do you calculate the sum of interior angles of an n-sided polygon?

The sum of interior angles = (n − 2) × 180°, where n is the number of sides.

Given an interior angle of 140° for a regular polygon, how do you find the number of sides?

First find the exterior angle: 180 − 140 = 40°. Then divide 360 by the exterior angle: 360 ÷ 40 = 9 sides.

How can you find an unknown angle y in a hexagon when some interior angles are given?

Sum the known angles and subtract from the total interior sum: y = (Total sum − sum of known angles) ÷ number of unknowns.

When a pentagon and an octagon share a side, how do you determine angles in the formed isosceles triangle?

Calculate the interior angles of the pentagon and octagon, then use the isosceles triangle property: X = (180 − difference between octagon and pentagon angles) ÷ 2.

How do you calculate an angle in a polygon formed by joining a pentagon and a hexagon?

Find the interior angles of both polygons, then subtract the known angles from 360° (angles around a point) to determine the unknown angle.

How do you find an angle using trigonometry in a right-angled triangle inside a polygon?

Use tan θ = opposite ÷ adjacent, then apply inverse tan to find the angle. Subtract this angle from the known interior angle to find the required angle.

How can symmetry be used to determine unknown angles in a regular 10-sided shape?

By identifying symmetrical sections, assign equal angles, then use the sum of interior angles to solve for unknown angles. This can show X = Y in symmetric cases.

What is the interior angle of a regular pentagon and a regular hexagon?

A regular pentagon has an interior angle of 108°, and a regular hexagon has an interior angle of 120°.

How can you solve for angles in polygons with multiple connected shapes sharing sides?

Calculate interior angles for each polygon, consider angles around shared vertices (sum to 360°), use isosceles triangle properties where applicable, and apply symmetry if present.