What do the exterior angles of a convex 8-sided polygon add up to? A. 180° B. 1440° C. 1080° D. 360° Choose the right answer

D. 360°. However many sides, a convex polygon's exterior angles add to 360° (sum of ext. ∠s of polygon).

Each angle of a regular polygon is 4 times each of its exterior angles. Each exterior angle = □°, and it has □ sides. Fill in each blank

36, 10. Interior + exterior = 180° (adj. ∠s on st. line), so 4 exterior angles and one more, 5 in all, make 180°: exterior = 180° ÷ 5 = 36°, and sides = 360° ÷ 36° = 10.

Polygons · Ages 13–14

Exterior angles of a polygon

Exterior angles always make 360°; sides = 360° ÷ one exterior angle.

How to do it

Method

Exterior angles always make 360°; sides = 360° ÷ one exterior angle

Interior 140°: exterior 180° − 140° = 40°, n = 360° ÷ 40° = 9

Another way

An equation from the angle sum

(n − 2) × 180 = 140n, n = 9

In full

Extend each side of a convex polygon the same way round and each vertex gets an exterior angle: it lies on a straight line with the interior one, so exterior = 180° − interior. However many sides there are, a convex polygon's exterior angles add up to 360° (reason: sum of ext. ∠s of polygon): walk round the edge, turning by one exterior angle at each vertex, and you are back where you began after exactly one full turn. A regular n-sided polygon's exterior angles are 360° ÷ n each; the other way round, the number of sides n = 360° ÷ one exterior angle. Interior angles of 140°: exterior = 180° − 140° = 40°, n = 360° ÷ 40° = 9. Another way: an equation from the angle sum, (n − 2) × 180 = 140n, 180n − 360 = 140n, n = 9. The same answer, but the exterior angles are much quicker.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
Every side of the hexagon is extended, giving six exterior angles. x =
Show the answer and working
Answer: 59°
However many sides, the exterior angles add to 360° (sum of ext. ∠s of polygon): x = 360° − 301° = 59°.
Example 2
What do the exterior angles of a convex 8-sided polygon add up to?
- A. 180°
- B. 1440°
- C. 1080°
- D. 360°
Choose the right answer
Show the answer and working
Answer: D. 360°
However many sides, a convex polygon's exterior angles add to 360° (sum of ext. ∠s of polygon).
Example 3
Each angle of a regular polygon is 4 times each of its exterior angles. Each exterior angle = °, and it has sides.
Fill in each blank
Show the answer and working
Answer: 36, 10
Interior + exterior = 180° (adj. ∠s on st. line), so 4 exterior angles and one more, 5 in all, make 180°: exterior = 180° ÷ 5 = 36°, and sides = 360° ÷ 36° = 10.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Each exterior angle of a regular 10-sided polygon =
Answer:
2.What do the exterior angles of a convex 5-sided polygon add up to?
- A. 180°
- B. 360°
- C. 900°
- D. 540°
Choose the right answer
Answer:
3.A regular polygon's exterior angles can each be 25°.
True or false?
Answer:
4.Every side of the hexagon is extended, giving six exterior angles. x =
Answer:
5.One side of the hexagon is extended. x =
Answer:

All the answers

1. 36°
2. B. 360°
3. False
4. 72°
5. 47°

Free worksheet: Exterior angles of a polygonWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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