Here are the step-by-step solutions for the visible questions on the worksheet.
Question 1
Task: Work out the size of an exterior angle of a regular hexagon.
1.
Rule: The sum of all exterior angles of any polygon is always $360^\circ$.
2.
Identify sides: A hexagon has 6 sides. Since it is "regular," all exterior angles are equal.
3.
Calculation: Divide the total sum by the number of sides.
$$360^\circ \div 6 = 60^\circ$$
Final Answer: $60^\circ$
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Question 2
Task: Work out the size of each interior angle in a regular octagon.
1.
Identify sides: An octagon has 8 sides.
2.
Find Exterior Angle: First, find the exterior angle because it is easier to calculate.
$$360^\circ \div 8 = 45^\circ$$
3.
Find Interior Angle: Interior and exterior angles lie on a straight line, so they add up to $180^\circ$.
$$180^\circ - 45^\circ = 135^\circ$$
*(Alternative Method: Total interior angles = $(8-2) \times 180 = 1080$. Then $1080 \div 8 = 135$)*
Final Answer: $135^\circ$
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Question 3
Task: Work out the size of each interior angle in a regular pentagon.
1.
Identify sides: A pentagon has 5 sides.
2.
Find Exterior Angle:
$$360^\circ \div 5 = 72^\circ$$
3.
Find Interior Angle: Subtract the exterior angle from $180^\circ$.
$$180^\circ - 72^\circ = 108^\circ$$
Final Answer: $108^\circ$
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Question 4
Task: The size of each exterior angle in a regular polygon is $20^\circ$. Work out how many sides the polygon has.
1.
Rule: Number of sides = $360^\circ \div$ Exterior Angle.
2.
Calculation:
$$360 \div 20 = 18$$
Final Answer: 18 sides
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Question 5
Task: The size of each exterior angle in a regular polygon is $18^\circ$. Work out how many sides the polygon has.
1.
Rule: Number of sides = $360^\circ \div$ Exterior Angle.
2.
Calculation:
$$360 \div 18 = 20$$
Final Answer: 20 sides
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Question 8
Task: $ABCDEF$ is a hexagon. Given angles are $A=129^\circ$, $B=125^\circ$, $F=144^\circ$, $E=121^\circ$. Also, Angle $CDE = 2 \times$ Angle $BCD$. Work out the size of angle $CDE$.
1.
Total Sum of Angles: Find the sum of interior angles for a hexagon (6 sides).
Formula: $(n - 2) \times 180$
$$(6 - 2) \times 180 = 4 \times 180 = 720^\circ$$
2.
Sum of Known Angles: Add the angles given in the diagram.
$$129 + 125 + 144 + 121 = 519^\circ$$
3.
Remaining Angles: Subtract the known sum from the total sum to find what is left for angles $C$ and $D$.
$$720 - 519 = 201^\circ$$
So, Angle $BCD$ + Angle $CDE = 201^\circ$.
4.
Use Algebra:
Let Angle $BCD = x$.
The problem states Angle $CDE = 2x$ (because it is twice the size).
$$x + 2x = 201$$
$$3x = 201$$
$$x = 201 \div 3$$
$$x = 67^\circ$$
5.
Calculate Final Angle: We need Angle $CDE$, which is $2x$.
$$2 \times 67 = 134^\circ$$
Final Answer: $134^\circ$
Parent Tip: Review the logic above to help your child master the concept of angles of a polygon worksheet.
