Finding Angles in Regular Polygons Worksheet, perfect angles ... - Free Printable

Here are the solutions to the problems on the worksheet, broken down by section.

Section A

1) Find the size of each interior angle in a regular hexagon.
* Step 1: A hexagon has 6 sides ($n=6$).
* Step 2: Calculate the sum of interior angles: $(6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}$.
* Step 3: Divide by the number of angles to find one angle: $720^{\circ} \div 6 = 120^{\circ}$.
* Answer: $120^{\circ}$

2) Find the size of each interior angle in a regular nonagon.
* Step 1: A nonagon has 9 sides ($n=9$).
* Step 2: Calculate the sum of interior angles: $(9 - 2) \times 180^{\circ} = 7 \times 180^{\circ} = 1260^{\circ}$.
* Step 3: Divide by the number of angles: $1260^{\circ} \div 9 = 140^{\circ}$.
* Answer: $140^{\circ}$

3) Find the size of each exterior angle in a regular dodecagon.
* Step 1: A dodecagon has 12 sides ($n=12$).
* Step 2: The sum of exterior angles for any polygon is always $360^{\circ}$.
* Step 3: Divide by the number of sides: $360^{\circ} \div 12 = 30^{\circ}$.
* Answer: $30^{\circ}$

4) Write an expression for the sum of the interior angles in a polygon with n sides.
* Rule: Sum $= (n - 2) \times 180$.
* Answer: $(n - 2) \times 180$

5) Write an expression for the size of each interior angle in a polygon with n sides.
* Rule: Take the sum from question 4 and divide by $n$.
* Answer: $\frac{(n - 2) \times 180}{n}$

6) Write an expression for the size of each exterior angle in a polygon with n sides.
* Rule: Divide total exterior degrees ($360$) by $n$.
* Answer: $\frac{360}{n}$

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Section B

1) Regular Pentagon
* Angle a (Interior):
* Sides $n=5$. Interior angle $= \frac{(5-2) \times 180}{5} = \frac{540}{5} = 108^{\circ}$.
* a = $108^{\circ}$
* Angle b (Exterior):
* Exterior angle $= \frac{360}{5} = 72^{\circ}$ (or $180 - 108$).
* b = $72^{\circ}$

2) Regular Pentagon with a diagonal
* Angle c:
* The triangle at the top is formed by two sides of the pentagon and a diagonal. It is an isosceles triangle because the two sides are equal length.
* The top angle is an interior angle of the pentagon ($108^{\circ}$).
* The base angles ($c$ and the other one) are equal.
* Calculation: $(180 - 108) \div 2 = 72 \div 2 = 36^{\circ}$.
* c = $36^{\circ}$
* Angle d:
* Angle $d$ plus angle $c$ make up the full interior angle of the pentagon ($108^{\circ}$).
* Calculation: $108 - 36 = 72^{\circ}$.
* d = $72^{\circ}$

3) Regular Hexagon with diagonals
* Angle e:
* This angle is inside a triangle formed by three vertices of the hexagon.
* The triangle is isosceles. The top angle is an interior angle of the hexagon ($120^{\circ}$).
* Base angles calculation: $(180 - 120) \div 2 = 30^{\circ}$.
* e = $30^{\circ}$
* Angle f:
* This angle is part of the corner vertex. The full interior angle is $120^{\circ}$.
* The diagonal cuts off a triangle identical to the one used for angle $e$, so the "slice" taken away is $30^{\circ}$.
* Calculation: $120 - 30 = 90^{\circ}$.
* f = $90^{\circ}$

4) Regular Hexagon divided into triangles
* Angle g (Exterior):
* Exterior angle of a hexagon $= \frac{360}{6} = 60^{\circ}$.
* g = $60^{\circ}$
* Angle h:
* The lines meet at the center. The angle around the center is $360^{\circ}$, split into 6 equal triangles.
* Calculation: $360 \div 6 = 60^{\circ}$.
* h = $60^{\circ}$

5) Regular Octagon
* Angle i:
* Interior angle of an octagon ($n=8$): $\frac{(8-2) \times 180}{8} = \frac{1080}{8} = 135^{\circ}$.
* i = $135^{\circ}$
* Angle j:
* This angle is inside a triangle formed by connecting vertices. The triangle spans 3 sides of the octagon.
* The vertex angle of this triangle (at the top right) corresponds to an interior angle of $135^{\circ}$.
* The triangle is isosceles. The two base angles are equal. One base angle is $j$.
* Wait, let's look closer. The chord connects vertex 1 to vertex 4. The triangle involved has vertices 1, 2, and 4? No, usually these diagrams imply symmetry.
* Let's use the circle method. An octagon fits in a circle. Each side subtends $\frac{360}{8} = 45^{\circ}$ at the center.
* Angle $j$ is an inscribed angle subtending 2 sides (from bottom-right to top-left via the right side).
* Inscribed angle = Half the central angle. Central angle for 2 sides $= 2 \times 45 = 90^{\circ}$.
* Angle $j = 90 \div 2 = 45^{\circ}$.
* j = $45^{\circ}$

6) Regular Decagon (10 sides)
* Angle k:
* Interior angle of decagon ($n=10$): $\frac{(10-2) \times 180}{10} = \frac{1440}{10} = 144^{\circ}$.
* Angle $k$ is inside an isosceles triangle formed by two sides and a short diagonal.
* Top angle is $144^{\circ}$. Base angles $= (180 - 144) \div 2 = 18^{\circ}$.
* k = $18^{\circ}$
* Angle l:
* This angle is formed by two intersecting diagonals.
* Using the circle rule again: Each side is $36^{\circ}$ ($\frac{360}{10}$).
* Angle $l$ faces an arc of 3 sides (looking at the geometry, it spans 3 segments of the perimeter).
* Central angle for 3 sides $= 3 \times 36 = 108^{\circ}$.
* Inscribed angle $l = 108 \div 2 = 54^{\circ}$.
* l = $54^{\circ}$

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Final Answer:
Section A
1) 120°
2) 140°
3) 30°
4) (n - 2) × 180
5) ((n - 2) × 180) / n
6) 360 / n

Section B
1) a = 108°, b = 72°
2) c = 36°, d = 72°
3) e = 30°, f = 90°
4) g = 60°, h = 60°
5) i = 135°, j = 45°
6) k = 18°, l = 54°