GCSE
Mathematics
Targeting Grade C
Shape and Space
Unit 4
Interior/Exterior Angles
Can you…
If not you need
TOP
• Methods to calculate the
Interior Angle Sum
interior/exterior angles of any polygon
• Work out angles in regular polygons
Try a test
Calculate angles in polygons
Try a test
Practice 1: Regular Polygons
TAIL 1
Practice 2: All polygons
TAIL 2
TOP
Remember – There are 2 ways of doing this
Method 1
To find the Interior Angle Sum (IAS) for an ‘n’ sided polygon
x = exterior angle
y = interior angle
The exterior angles of any
polygon sum to 360°
Each Exterior Angle
x = 360 ÷ n
So… For a regular hexagon:
Each exterior angle = 360 ÷ 6 = 60°
Each interior angle = 180 - 60 = 120°
Each Interior Angle
y = 180 - x
Next
TOP
Remember – There are 2 ways of doing this
Method 2
To find the Interior Angle Sum (IAS) for an ‘n’ sided polygon
Any polygon can be
split into (n-2) triangles
The IAS is the sum of all
the triangles:180(n-2)
So… For a regular pentagon:
IAS = 180(5-2) = 180 x 3 = 540°
Each interior angle = 540 ÷ 5 = 108°
Lesson
Each Interior Angle
= IAS ÷ n
Each Exterior Angle
= 180 - Interior angle
Practice 1: Regular Ploygons
1.
Are you ready for
the answers ?
A decagon is a polygon with 10 sides.
Work out the size of each exterior angle of a regular decagon
2.
360 ÷ 10 = 36°
The size of each exterior angle of a regular polygon is 40°.
Work out the number of sides of the polygon
3.
360 ÷ 40 = 9 sides
Diagram NOT accurately drawn
Work out the size of each interior angle of a regular octagon.
360 ÷ 8 = 45
4.
180 – 45 = 135°
The size of each exterior angle of an regular polygon is 24°.
Work out the number of sides the polygon has.
Lesson
360 ÷ 24 = 15 sides
Are you ready for
the answers ?
TAIL 1
1.
The size of each exterior angle of a regular polygon is 30°.
Work out the number of sides of the polygon.
360 ÷ 30 = 12 sides
2.
Diagram NOT
accurately drawn
(a)
Work out the size of an exterior angle of a regular pentagon. a) 360 ÷ 5 = 72°
The area of the pentagon is 8560 mm2.
(b) Change 8560 mm2 to cm2.
b) 8560 ÷ (10 x 10) = 85.6
(2)
(2)
Each side of another regular pentagon has a length of 101 mm, correct to the nearest millimetre.
(c) (i) Write down the least possible length of each side.
c) (i) 100.5
(ii) Write down the greatest possible length of each side.
(2)
(ii) 101.5
(Total 6 marks)
Lesson
Are you ready for
the answers ?
Practice 2:
1.
The diagram shows a 5 sided shape.
All of the sides of the shape are equal in length.
x°
y°
Diagram NOT
accurately drawn
(a)
(i)
60°
(ii)) Top triangle is equilateral
(a)
(b)
(i)
Find the value of x.
(ii) Give a reason for your answer.
Work out the value of y.
(b)
90 + 60 = 150°
(Total 4 marks)
Next
Are you ready for
the answers ?
Practice 2:
2.
Diagram NOT
accurately drawn
x°
y°
(a)
(b)
(a)
60°
(b)
360 – 90 – 90 – 60 = 120°
(c)
6 x 2 = 12cm²
This is part of the design of a pattern found at the theatre of Diana at Alexandria.
It is made up of a regular hexagon, squares and equilateral triangles.
Write down the size of the angle marked x°.
Work out the size of the angle marked y°.
The area of each equilateral triangle is 2 cm2.
(c) Work out the area of the regular hexagon.
Lesson
(1)
(2)
(2)
(Total 5 marks)
Are you ready for
the answers ?
TAIL 2
1.
The diagram shows a shape.
The shape is a 6-sided polygon.
Diagram NOT
accurately drawn
(a) Write down the mathematical name for a 6-sided
polygon.
(1 mark)
Hexagon
The diagram below shows how the shape tessellates.
The size of each of the angles marked x is 135°.
(b) Give reasons why.
(2 marks)
x x
Diagram NOT
accurately drawn
360 – 90 = 270
270 ÷ 2 = 135°
Angles at a point add up to 360°
Next
TAIL 2
Are you ready for
the answers ?
1. contd
30 cm
8 cm
Diagram NOT
accurately drawn
The diagram shows the lengths of two of the sides of the shape.
(c) Work out the perimeter of the shape.
P = (30 x 4) + (8 x 2) = 136 cm
Lesson
(2 marks)
(Total 5 marks)