Here we will learn about exterior angles of polygons including how to calculate the sum of exterior angles for a polygon, a single exterior angle and how to use this knowledge to solve problems. Show
There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck. What are exterior angles?Exterior angles are angles between a polygon and the extended line from the vertex of the polygon. Sum of exterior angles of a polygon = 360º Interior and exterior angles form a straight line – they add to 180º. Check out our lessons on interior angles of polygons and sum of the interior angles to find out more. What are exterior angles?Keywords
How to solve problems involving exterior anglesIn order to solve problems involving exterior angles following these steps:
How to solve problems involving exterior angles.Interior and exterior angles worksheetGet your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Interior and exterior angles worksheetGet your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Exterior angles of polygons examplesExample 1: finding the size of a single exterior angle for a regular polygonCalculate the size of a single exterior angle for a regular hexagon.
A hexagon has 6 sides. Regular – therefore all exterior angles are equal. 2 Identify what the question is asking you to find. The size of one exterior angle. We know the sum of exterior angles for a polygon is 360°. 3 Solve the problem using the information you have already gathered. \[360 \div 6 = 60\] The size of each exterior angle is 60º. Example 2: finding an exterior angle given an interior angle for an irregular polygonAn irregular octagon has one interior angle of size 130º. What is the size of the adjacent exterior angle? Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons. 8 sides and irregular – (irregular octagon). Identify what the question is asking you to find. As adjacent means next to we are being asked to find the size of the exterior angle which is on an straight line with the interior angle. Solve the problem using the information you have already gathered. We know that angles on a straight line add to 180º, so if the interior angle is 130º then the exterior angle will be 50º . Example 3: interior + exterior angle = 180ºCalculate angle x. Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons. 6 sides – irregular hexagon. The interior angles of a hexagon add to 720 degrees. Identify what the question is asking you to find. Find the exterior angle x.
Solve the problem using the information you have already gathered. We can work out the missing interior angle of the polygon. \begin{aligned} 150+120+90+160+130+a&=720 \\\\ 650+a&=720 \\\\ a&=70 \end{aligned} The interior angle + the exterior angle must equal 180º
Example 4: finding the number of sides given the exterior angle of a regular polygonAn exterior angle of a regular polygon is 20º. How many sides does the polygon have? Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons. Unknown number of sides.
Identify what the question is asking you to find. We need to find the number of sides.
Solve the problem using the information you have already gathered. \begin{aligned} 20 \times \text { number of sides }&=360\\\\ 20n&=360 \\\\ n&=18 \end{aligned} Therefore the polygon has 18sides. Example 5: finding the number of sides given the interior angle of a regular polygonThe size of each interior angle of a regular polygon is 150º. How many sides does the polygon have? Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons. Unknown number of sides.
Identify what the question is asking you to find. We need to find the number of sides.
Solve the problem using the information you have already gathered. If the interior angle is 150º then the exterior angle will be 30º.
Example 6: multi step problem involving interior and exterior anglesThe size of each interior angle of a regular polygon is 11 times the size of each exterior angle. Work out the number of sides the polygon has. Identify the number of sides in any polygon/s given in the question. Note whether these are regular or irregular polygons. Unknown sides.
Identify what the question is asking you to find. Number of sides of the polygon. Other Information we know: Total of Exterior Angles =360ºInterior +Exterior Angle =180º11 ×Interior Angle =Exterior Angle Solve the problem using the information you have already gathered. We will call each of the interior angles x.
\begin{aligned} x+11 x&=180 \\\\ 12 x&=180 \\\\ x&=15 \end{aligned} The size of one exterior angle is 15º.
Common misconceptions
E.g. The exterior angle of a triangle is the angle between the side and the extension of an adjacent side. Here the interior angle (internal angle) is 60º, so the exterior angle (external angle) must be 120º.
Practice exterior angles of a polygon questionsExterior angles of a polygon add up to 360 . A regular quadrilateral has 4 interior angles equal in size, so the four exterior angles are equal. This means we can divide 360 by 4 to get the solution. Exterior angles of a polygon add up to 360 . A regular octagon has 8 interior angles equal in size, so the eight exterior angles are equal. This means we can divide 360 by 8 to get the solution. Exterior angles of a polygon add up to 360 . A regular nonagon has 9 interior angles equal in size, so the nine exterior angles are equal. This means we can divide 360 by 9 to get the solution. Exterior angles of a polygon add up to 360 . This means we can divide 360 by 12 to get the solution. Exterior angles of a polygon add up to 360 . This means we can divide 360 by 20 to get the solution. The four known exterior angles will be 55^{\circ} , since angles on a straight line sum to 180 . This means the fifth exterior angle will be 140^{\circ} because exterior angles add up to 360 . Using angles on a straight line once more means that the missing angle is 40^{\circ} . Exterior angles of a polygon GCSE questions1. A regular polygon has 15 sides. Calculate the size of one exterior angle. (1 mark) Show answer 360 ÷ 15 = 24 = 24^{\circ} (1) 2. (a) The diagram below shows part of a regular polygon. Calculate the size of the exterior angle of the polygon. (b) Work out how many sides this polygon has. (3 marks) Show answer (a) 180 – 162=18 = 18^{\circ} (1) (b) 360 ÷ 18 (1) 20 (1) 3. Shown below are parts of two regular polygons. Polygon A has 9 sides and an exterior angle of x. Polygon B has an interior angle of 3x. How many sides does polygon B have? (4 marks) Show answer 360 ÷ 9 = 40 (1) x = 40, 3x = 120 (1) Polygon B: Interior angle is 120^{\circ} , exterior angle is 60^{\circ} (1) 360 \div 60 = 6 (1) Learning checklistYou have now learned how to:
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