Angles in polygons questions with answers

Questions

1. Question 1Interior angle sum of a polygon

   The diagram shows a hexagon $ABCDEF$. Find $x$.

   Show the worked solution

   1. Step 1: Sum of the interior angles of a hexagon$(6 - 2) \times 180^\circ = 720^\circ$
   2. Step 2: Take the other angles away$x = 720 - (124 + 128 + 111 + 116 + 118)$$x = 123$ (∠ sum of polygon)
2. Question 2Each interior angle of a regular polygon

   The diagram shows a regular pentagon. Find $x$.

   Show the worked solution

   1. Step 1: Sum of the interior angles of a pentagon$(5 - 2) \times 180^\circ = 540^\circ$
   2. Step 2: A regular polygon: all its angles are equal$x = \dfrac{540}{5}$$x = 108$ (∠ sum of polygon)
3. Question 3Each exterior angle of a regular polygon

   The diagram shows a regular 20-sided polygon with one side produced. Find $x$.

   Show the worked solution

   1. Step 1: The exterior angles are equal and go once round$x = \dfrac{360}{20}$$x = 18$ (ext. ∠s of polygon)
4. Question 4Number of sides from an angle

   Each interior angle of a regular polygon is $108^\circ$. The diagram shows part of the polygon. Find the number of sides of the polygon.

   Show the worked solution

   1. Step 1: Each exterior angle: the interior and exterior angles make a straight line$\text{Exterior angle} = 180^\circ - 108^\circ$$\text{Exterior angle} = 72^\circ$ (∠s on a str. line)
   2. Step 2: The exterior angles add up to 360°$\text{Number of sides} = \dfrac{360}{72}$$\text{Number of sides} = 5$
5. Question 5Exterior angles add up to 360°

   The diagram shows a quadrilateral $ABCD$ with each side produced. Find $x$.

   Show the worked solution

   1. Step 1: Find x and give the reason$74^\circ + x^\circ + 93^\circ + 96^\circ = 360^\circ$$x = 360 - 74 - 93 - 96$$x = 97$ (ext. ∠s of polygon)
6. Question 6Angles as expressions

   The diagram shows a quadrilateral $ABCD$ with each side produced. Find $x$.

   Show the worked solution

   1. Step 1: Write an equation in x$(2x - 10)^\circ + (x + 15)^\circ + 80^\circ + (2x + 10)^\circ = 360^\circ$ (ext. ∠s of polygon)
   2. Step 2: Collect like terms and solve$5x + 95 = 360$$5x = 265$$x = 53$
7. Question 7Interior and exterior angles in a ratio: find n

   Each interior angle of a regular polygon is $3$ times each of its exterior angles. Find each interior angle of the polygon.

   Show the worked solution

   1. Step 1: An interior angle and its exterior angle make a straight line: together 180°$3 + 1 = 4$ units $\rightarrow 180^\circ$$1$ unit $\rightarrow \dfrac{180}{4} = 45^\circ$
   2. Step 2: The interior angle is 3 unitsInterior angle $= 3 \times 45$$= 135^\circ$
8. Question 8Regular polygons meeting at a point

   In the diagram, an equilateral triangle, a regular nonagon and a regular polygon with $n$ sides fit together round the point $O$, with no gaps. Find $n$.

   Show the worked solution

   1. Step 1: Each interior angle of the two known polygons: (n − 2) × 180° ÷ nEquilateral triangle: $60^\circ$Regular nonagon: $140^\circ$
   2. Step 2: The three angles at O make one full turn$x = 360 - 60 - 140$$x = 160$ (∠s at a pt.)
   3. Step 3: The exterior angle of the third polygon, then n = 360° ÷ exterior angleExterior angle $= 180^\circ - 160^\circ$$= 20^\circ$$n = \dfrac{360}{20}$$n = 18$
9. Question 9Some angles given, the rest equal: find n

   A polygon has $n$ sides. 3 of its interior angles are $120^\circ$, $130^\circ$ and $110^\circ$. Each of the other interior angles is $168^\circ$. Find $n$.

   Show the worked solution

   1. Step 1: Work with exterior angles (180° − each interior angle): they add up to 360°Exterior angles of the given ones: $180^\circ - 120^\circ = 60^\circ$, $180^\circ - 130^\circ = 50^\circ$, $180^\circ - 110^\circ = 70^\circ$Their sum $= 180^\circ$
   2. Step 2: Each of the other exterior angles is 180° − 168°: how many fit into what is left of 360°?Each other exterior angle $= 180^\circ - 168^\circ$$= 12^\circ$Number of them $= \dfrac{360 - 180}{12}$$= 15$
   3. Step 3: Add the angles given$n = 15 + 3$$\phantom{n} = 18$
10. Question 10Can it be an interior angle of a regular polygon?

    Can $144^\circ$ be each interior angle of a regular polygon? Choose the answer with its reason.

    Show the worked solution

    1. Step 1: The exterior angle: 180° minus the interior angleExterior angle $= 180^\circ - 144^\circ$$= 36^\circ$
    2. Step 2: The number of sides is 360° ÷ the exterior angle: it must be a whole number$\dfrac{360}{36} = 10$It is a whole number: a regular 10-sided polygon has interior angles of $144^\circ$.
11. Question 11Diagonals and produced sides of regular polygons

    The diagram shows a regular nonagon $ABCD\ldots$. The sides $AB$ and $DC$ are produced to meet at $X$. Find $x$.

    Show the worked solution

    1. Step 1: Each exterior angle of the regular polygon: 360° ÷ nExterior angle $= \dfrac{360}{9}$$= 40^\circ$
    2. Step 2: ∠XBC and ∠XCB are exterior angles (sides produced): use the angle sum of triangle XBC$x = 180 - 40 - 40$$x = 100$ (∠ sum of △)
12. Question 12A regular polygon with a square or triangle on a side

    The diagram shows a regular hexagon $ABCDEF$ and a square $CDQP$ drawn inside it on the side $CD$. $B$ is joined to $P$. Find $x$.

    Show the worked solution

    1. Step 1: Each interior angle of the regular hexagon$\angle BCD = \dfrac{(6 - 2) \times 180^\circ}{6}$$\phantom{\angle BCD} = 120^\circ$
    2. Step 2: The angle of the square at C$\angle DCP = 90^\circ$ (angle of a square)
    3. Step 3: ∠BCP is what is left of ∠BCD$x = 120 - 90$$\phantom{x} = 30$
13. Question 13A polygon with one reflex angle

    A polygon has one reflex interior angle of $225^\circ$. Each of its other interior angles is $135^\circ$. Find the number of sides of the polygon.

    Show the worked solution

    1. Step 1: The interior angles of an n-sided polygon add up to (n − 2) × 180°$(n - 2) \times 180 = 225 + (n - 1) \times 135$ (∠ sum of polygon)
    2. Step 2: Expand both sides$180n - 360 = 135n + 90$
    3. Step 3: Collect the n terms on one side and solve$45n = 450$$n = 10$