Angles in Polygons
The sum of the interior angles of a polygon with n sides is given by (n - 2) x 180 degrees, and for a regular polygon each interior angle is found by dividing this sum by n. The exterior angles of any polygon, regular or irregular, always add up to 360 degrees, and at each vertex the interior angle and the exterior angle add up to 180 degrees.
Method
- Count the number of sides, n, of the polygon.
- To find the sum of the interior angles, use the formula (n - 2) x 180.
- For a regular polygon, divide this sum by n to find the size of each interior angle.
- Remember the exterior angles of any polygon always add up to 360 degrees, whether or not the polygon is regular.
- For a regular polygon, find each exterior angle using 360 / n, and remember that the interior angle and the exterior angle at any vertex add up to 180 degrees.
- For an irregular polygon, use the interior angle sum and any angles you are given to form an equation, then solve it for the unknown angle(s).
Worked example
A regular polygon has 9 sides. Find (a) the sum of its interior angles, (b) the size of each interior angle, (c) the size of each exterior angle.
- Sum of interior angles = (n - 2) x 180 = (9 - 2) x 180 = 7 x 180 = 1260 degrees.
- Each interior angle (regular) = 1260 / 9 = 140 degrees.
- Each exterior angle = 180 - 140 = 40 degrees (this also matches 360 / 9 = 40).
Practice questions
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Q1Find the sum of the interior angles of a hexagon (6 sides).Show answer
Answer: (6 - 2) x 180 = 4 x 180 = 720 degrees
Q2Find the size of each interior angle of a regular pentagon (5 sides).Show answer
Answer: Sum = (5 - 2) x 180 = 540, each angle = 540 / 5 = 108 degrees
Q3Find the size of each exterior angle of a regular octagon (8 sides).Show answer
Answer: 360 / 8 = 45 degrees
Q4The sum of the exterior angles of any polygon is always the same number. State this number.Show answer
Answer: 360 degrees
Q5A regular polygon has an exterior angle of 20 degrees. How many sides does it have?Show answer
Answer: 360 / 20 = 18 sides
Q6A quadrilateral has interior angles of 90, 90, x and 100 degrees. Find x.Show answer
Answer: Sum = (4 - 2) x 180 = 360, so 90 + 90 + 100 + x = 360, 280 + x = 360, and x = 80
Q7Find the size of each interior angle of a regular decagon (10 sides).Show answer
Answer: Sum = (10 - 2) x 180 = 1440, each angle = 1440 / 10 = 144 degrees
Q8A regular polygon has an interior angle of 156 degrees. How many sides does it have?Show answer
Answer: Exterior angle = 180 - 156 = 24 degrees, so the number of sides = 360 / 24 = 15
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A polygon has 7 sides. Work out the sum of its interior angles.
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A regular polygon has an exterior angle of 15 degrees. (a) Work out how many sides the polygon has. (b) Work out the size of each interior angle. (c) A second, irregular pentagon has interior angles of 100, 100, 100, x and 2x degrees. Work out the value of x.
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Free printable worksheet
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This topic is chapter 10 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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