Worksheet with six geometry problems on exterior and interior angles of regular polygons.

Let’s solve each problem one by one, step by step.

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Problem 1:
The diagram shows a regular nonagon. Calculate x.

A regular nonagon has 9 equal sides and 9 equal angles.

For any polygon, the sum of all exterior angles is always 360°, no matter how many sides it has.

Since it’s regular, each exterior angle = 360° ÷ number of sides
→ 360° ÷ 9 = 40°

So, x = 40

✔ Check: 9 × 40° = 360° → correct.

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Problem 2:
The exterior angle of a regular polygon is 45°. What is the size of 1 of its interior angles?

Interior angle + Exterior angle = 180° (they form a straight line)

So, Interior angle = 180° - 45° = 135°

✔ Check: Makes sense — if exterior is small, interior is large.

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Problem 3:
Calculate x.

We are given a pentagon (5-sided shape) with some exterior angles marked: 75°, 45°, x°, 95°, 98°

Sum of all exterior angles of ANY polygon = 360°

So add them up:

75 + 45 + x + 95 + 98 = 360

Add known values:

75 + 45 = 120
120 + 95 = 215
215 + 98 = 313

So: 313 + x = 360
→ x = 360 - 313 = 47

✔ Check: 75+45+47+95+98 = let’s verify:
75+45=120; 120+47=167; 167+95=262; 262+98=360 → perfect.

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Problem 4:
The diagram shows part of a regular polygon and its exterior angle = 15°. How many sides does the complete polygon have?

Again, total exterior angles = 360°

Each exterior angle = 15°

Number of sides = 360° ÷ 15° = 24

✔ Check: 24 × 15 = 360 → yes.

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Problem 5:
What is the exterior angle of a regular quadrilateral?

Quadrilateral = 4 sides

Exterior angle = 360° ÷ 4 = 90°

✔ Check: Square is a regular quadrilateral — each corner turns 90° when walking around → makes sense.

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Problem 6:
I’m thinking of a regular polygon. The interior angle is exactly half of the exterior angle. What polygon is it?

Let exterior angle = E
Then interior angle = I = E/2

But we know: I + E = 180° (straight line)

Substitute:
E/2 + E = 180
→ (3/2)E = 180
Multiply both sides by 2: 3E = 360
→ E = 120°

Wait — that can’t be right. If exterior angle is 120°, then interior is 60°, but 60° is half of 120°? Yes, mathematically it works... BUT hold on!

Actually, in polygons, exterior angles must be less than 180°, which 120° is okay, but let’s check what polygon has exterior angle 120°.

Number of sides = 360° ÷ exterior angle = 360 ÷ 120 = 3

That’s a triangle!

Check: For equilateral triangle, each interior angle = 60°, each exterior = 120° → and 60 is half of 120 → YES!

✔ So it’s a triangle.

Wait — let me double-check the logic.

If interior = half of exterior → I = E/2
And I + E = 180 → E/2 + E = 180 → 3E/2 = 180 → E = 120 → n = 360/120 = 3 → triangle.

Yes, correct.

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Final Answer:

1. 40
2. 135
3. 47
4. 24
5. 90
6. Triangle