ONE-EIGHTH STAGE SPEED RACE
2018
Prempeh College: 62 points
St. Francis Xavier: 29 points
Saviour SHS: 18 points
QUESTION
Identify the polygon whose sum of interior angles is $5\pi$ radians.
ANSWER: Heptagon (7 sides)
SOLUTION 1
Just add 2 to the 5 to get 7 sides
A polygon with 7 sides is a heptagon.
SOLUTION 2
Convert the sum of interior angles from radians to degrees:
$5\pi \text{ radians} = 5 \times 180^\circ = 900^\circ$
Use the formula for the sum of interior angles of an $n$-sided polygon:
$(n-2) \times 180^\circ = 900^\circ$
Divide both sides by $180^\circ$:
$n-2 = 5$
Add 2 to both sides:
$n = 7$
A polygon with 7 sides is a heptagon.
PRACTICE QUESTIONS
1. Identify the polygon whose sum of interior angles is $5\pi$ radians.
ANSWER: Heptagon (7 sides)
SOLUTION 1
Add $2$ to the coefficient of $\pi$: $5+2=7$ sides.
A polygon with $7$ sides is a heptagon.
SOLUTION 2
Convert the sum to degrees: $5\pi\text{ radians}=5\times180^\circ=900^\circ$.
$(n-2)\times180^\circ=900^\circ$
$n-2=5$
$n=7$
A polygon with $7$ sides is a heptagon.
2. Identify the polygon whose sum of interior angles is $3\pi$ radians.
ANSWER: Pentagon (5 sides)
SOLUTION 1
Add $2$ to the coefficient of $\pi$: $3+2=5$ sides.
A polygon with $5$ sides is a pentagon.
SOLUTION 2
Convert the sum to degrees: $3\pi\text{ radians}=3\times180^\circ=540^\circ$.
$(n-2)\times180^\circ=540^\circ$
$n-2=3$
$n=5$
A polygon with $5$ sides is a pentagon.
3. Identify the polygon whose sum of interior angles is $7\pi$ radians.
ANSWER: Nonagon (9 sides)
SOLUTION 1
Add $2$ to the coefficient of $\pi$: $7+2=9$ sides.
A polygon with $9$ sides is a nonagon.
SOLUTION 2
Convert the sum to degrees: $7\pi\text{ radians}=7\times180^\circ=1260^\circ$.
$(n-2)\times180^\circ=1260^\circ$
$n-2=7$
$n=9$
A polygon with $9$ sides is a nonagon.
4. Identify the polygon whose sum of interior angles is $10\pi$ radians.
ANSWER: Dodecagon (12 sides)
SOLUTION 1
Add $2$ to the coefficient of $\pi$: $10+2=12$ sides.
A polygon with $12$ sides is a dodecagon.
SOLUTION 2
Convert the sum to degrees: $10\pi\text{ radians}=10\times180^\circ=1800^\circ$.
$(n-2)\times180^\circ=1800^\circ$
$n-2=10$
$n=12$
A polygon with $12$ sides is a dodecagon.
5. Identify the polygon whose sum of interior angles is $6\pi$ radians.
ANSWER: Octagon (8 sides)
SOLUTION 1
Add $2$ to the coefficient of $\pi$: $6+2=8$ sides.
A polygon with $8$ sides is a octagon.
SOLUTION 2
Convert the sum to degrees: $6\pi\text{ radians}=6\times180^\circ=1080^\circ$.
$(n-2)\times180^\circ=1080^\circ$
$n-2=6$
$n=8$
A polygon with $8$ sides is a octagon.
6. Identify the polygon whose sum of interior angles is $8\pi$ radians.
ANSWER: Decagon (10 sides)
SOLUTION 1
Add $2$ to the coefficient of $\pi$: $8+2=10$ sides.
A polygon with $10$ sides is a decagon.
SOLUTION 2
Convert the sum to degrees: $8\pi\text{ radians}=8\times180^\circ=1440^\circ$.
$(n-2)\times180^\circ=1440^\circ$
$n-2=8$
$n=10$
A polygon with $10$ sides is a decagon.