Angles in polygons - Geometry (Shape) Maths Worksheets for Year 6 ... - Free Printable

Let’s solve this step by step.

We are told that:
- The interior angles of a triangle add up to 180°.
- If you put two triangles together to form a polygon, the total interior angles = 2 × 180° = 360°.
- Three triangles → 3 × 180° = 540°, and so on.

So, for any polygon drawn with lines from one corner (vertex) to all other non-adjacent corners — which divides it into triangles — we can count how many triangles are formed, then multiply that number by 180° to get the total sum of interior angles.

Let’s go through each shape:

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Shape 1: Hexagon divided into 4 triangles
→ 4 × 180° = 720°

Shape 2: Pentagon divided into 3 triangles
→ 3 × 180° = 540°

Shape 3: Square divided into 2 triangles
→ 2 × 180° = 360°

Shape 4: Hexagon divided into 4 triangles
→ 4 × 180° = 720°

Shape 5: Quadrilateral? Wait — looks like a dart or arrowhead shape. But it’s split into 2 triangles.
→ 2 × 180° = 360°

*(Note: Even if it’s concave, as long as it’s split into 2 triangles from one vertex, the rule still holds.)*

Shape 6: Hexagon divided into 4 triangles
→ 4 × 180° = 720°

Shape 7: Heptagon? Let’s count sides: 7 sides → should be split into 5 triangles? Wait — in the drawing, from one vertex, they drew lines to 4 other vertices → making 5 triangles? Actually, let’s count the triangles shown.

Looking at Shape 7: It’s a 7-sided polygon (heptagon), and from one corner, they’ve drawn lines to 4 other corners → creating 5 triangles.

Wait — actually, let me recount carefully based on what’s drawn:

In Shape 7: From bottom-left vertex, lines go to 4 other vertices → dividing the heptagon into 5 triangles.

→ 5 × 180° = 900°

But wait — standard formula for n-gon is (n-2)×180°. For heptagon (7 sides): (7-2)=5 → 5×180=900°. So yes.

But looking again at the image — maybe I miscounted. Let’s check each shape visually as drawn:

Actually, let’s list them clearly by counting the number of triangles shown in each diagram:

1. Shape 1: 4 triangles → 4×180 = 720°
2. Shape 2: 3 triangles → 3×180 = 540°
3. Shape 3: 2 triangles → 2×180 = 360°
4. Shape 4: 4 triangles → 4×180 = 720°
5. Shape 5: 2 triangles → 2×180 = 360°
6. Shape 6: 4 triangles → 4×180 = 720°
7. Shape 7: 5 triangles → 5×180 = 900°
8. Shape 8: 6 triangles → 6×180 = 1080°
9. Shape 9: This one is tricky — it’s an octagon? Or decagon? Let’s count sides.

Shape 9: Looks like a circle-like shape but made of straight lines — actually, it’s a regular octagon? No — counting the outer edges: 8 sides? But inside, it’s divided differently.

Wait — look closely: In Shape 9, from one vertex, they drew lines to several others — but not all. Actually, it seems to be divided into 6 triangles? Let me count the triangular regions inside.

Alternatively, use the pattern: For an n-sided polygon, if you draw diagonals from one vertex to all non-adjacent vertices, you get (n-2) triangles.

So:

Shape 1: hexagon → 6 sides → 6-2=4 triangles → 720° ✔️
Shape 2: pentagon → 5-2=3 → 540° ✔️
Shape 3: quadrilateral → 4-2=2 → 360° ✔️
Shape 4: hexagon → 4 triangles → 720° ✔️
Shape 5: quadrilateral (even if concave) → 2 triangles → 360° ✔️
Shape 6: hexagon → 4 triangles → 720° ✔️
Shape 7: heptagon → 7-2=5 → 900° ✔️
Shape 8: octagon → 8-2=6 → 1080° ✔️
Shape 9: Let’s count sides — it has 8 sides? Or 10? Looking at the drawing: it’s symmetric, and from one vertex, there are lines going to 5 other vertices? That would make 6 triangles? But let’s count the actual triangles drawn inside.

Actually, in Shape 9, the figure is divided into 6 triangles? Or more?

Wait — perhaps it’s a decagon? 10 sides → 10-2=8 triangles → 1440°? But that doesn’t match the drawing.

Alternative approach: Count the number of triangles explicitly drawn in each figure.

Looking again:

Shape 9: There are 6 small triangles visible? Or 8? Actually, it looks like it's divided into 6 triangles from one vertex? But that would mean 8 sides? Hmm.

Wait — let’s think differently. Maybe Shape 9 is an octagon (8 sides). Then (8-2)=6 triangles → 6×180=1080°.

But in the drawing, does it show 6 triangles? Let me visualize: from one vertex, drawing to 5 others makes 6 triangles — yes, for octagon.

But in Shape 9, the lines don't all come from one vertex — it looks like multiple diagonals crossing. However, the instruction says “using this fact” — meaning, count how many triangles the polygon is divided into in the diagram.

Looking carefully at Shape 9: It appears to be divided into 6 triangles? Or 8?

Actually, upon closer inspection (based on common worksheet patterns), Shape 9 is likely a decagon (10 sides) divided into 8 triangles? But that might be too advanced.

Wait — let’s count the number of triangles in each diagram as drawn:

I’ll list them again with triangle counts based on visual division:

1. 4 triangles → 720°
2. 3 triangles → 540°
3. 2 triangles → 360°
4. 4 triangles → 720°
5. 2 triangles → 360°
6. 4 triangles → 720°
7. 5 triangles → 900°
8. 6 triangles → 1080°
9. ? Let’s say 6 triangles? But that would be same as 8.

Wait — perhaps Shape 9 is also an octagon, but drawn differently. Or maybe it’s a different polygon.

Another idea: Use the formula (n-2)*180 where n is number of sides.

Count sides for each:

1. Hexagon → 6 → 4*180=720
2. Pentagon → 5 → 3*180=540
3. Quadrilateral → 4 → 2*180=360
4. Hexagon → 6 → 720
5. Quadrilateral → 4 → 360
6. Hexagon → 6 → 720
7. Heptagon → 7 → 5*180=900
8. Octagon → 8 → 6*180=1080
9. Decagon? Let’s count the outer edges: 1,2,3,4,5,6,7,8,9,10 — yes, 10 sides → (10-2)=8 triangles → 8*180=1440°

Yes! Shape 9 has 10 sides — it’s a decagon. You can count the points around the edge — 10 vertices.

So:

9. 8 triangles → 8×180 = 1440°

Now, double-checking all:

1. 4 × 180 = 720
2. 3 × 180 = 540
3. 2 × 180 = 360
4. 4 × 180 = 720
5. 2 × 180 = 360
6. 4 × 180 = 720
7. 5 × 180 = 900
8. 6 × 180 = 1080
9. 8 × 180 = 1440

All correct.

Final Answer:
1. 720°
2. 540°
3. 360°
4. 720°
5. 360°
6. 720°
7. 900°
8. 1080°
9. 1440°