Angles in Polygons

Topic

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Foundation June 2025 Paper 1 Q24

EdexcelCurrent spec4 marksAngles in Polygons

24 \(AB\), \(BC\), \(CD\) and \(DE\) are four sides of a regular polygon with \(n\) sides.

Sides AB, BC, CD and DE of the polygon, with equilateral triangle BCX and square CDYX drawn inside

\(BCX\) is an equilateral triangle.
\(CDYX\) is a square.

Work out the value of \(n\).
You must show all your working. (4)

Mark scheme

Answer	Mark	Mark scheme
12	P1	for (interior angle =) \((180 \div 3) + (360 \div 4)\ (= 150)\)
	P1	for (exterior angle =) \(180 - \text{``}150\text{''}\ (= 30)\) or for \(\text{``}150\text{''} = \dfrac{(n-2) \times 180}{n}\) oe
	P1	for \(360 \div \text{``}30\text{''}\) or for a correct process to solve the equation as far as \(\text{``}30\text{''}n = 360\)
	A1	cao

Additional guidance

150 may be seen on diagram.

May be seen on diagram
Exterior angle of 30 implies P1P1

A correct answer with no supportive working gets 0 marks

Minimum supportive working is P1P1
or
Trials of interior angle sum
Finding \(1800 \div 150 = 12\) oe with 12 as answer scores P1P1P1A1

Higher June 2025 Paper 1 Q6

EdexcelCurrent spec4 marksAngles in Polygons

6 \(AB\), \(BC\), \(CD\) and \(DE\) are four sides of a regular polygon with \(n\) sides.

\(BCX\) is an equilateral triangle.
\(CDYX\) is a square.

Work out the value of \(n\).
You must show all your working. (4)

Mark scheme

Answer	Mark	Mark scheme
12	P1	for (interior angle =) \((180 \div 3) + (360 \div 4)\ (= 150)\)
	P1	for (exterior angle =) \(180 - \text{``}150\text{''}\ (= 30)\) or for \(\text{``}150\text{''} = \dfrac{(n-2) \times 180}{n}\) oe
	P1	for \(360 \div \text{``}30\text{''}\) or for a correct process to solve the equation as far as \(\text{``}30\text{''}n = 360\)
	A1	cao

Additional guidance

150 may be seen on diagram.

May be seen on diagram
Exterior angle of 30 implies P1P1

A correct answer with no supportive working gets 0 marks

Minimum supportive working is P1P1
or
Trials of interior angle sum
Finding \(1800 \div 150 = 12\) oe with 12 as answer scores P1P1P1A1

Foundation November 2024 Paper 1 Q23

EdexcelCurrent spec4 marksAngles in Polygons

23 Here is a pentagon.

Pentagon with interior angles labelled a, b, c, d and e

Angle \(a\) = angle \(c\)
Angle \(b = 155^\circ\)
Angle \(d\) is three times the size of angle \(c\)
Angle \(e\) is two times the size of angle \(c\)

Work out the size of angle \(a\) (4)

Mark scheme

Answer	Mark	Mark scheme
55	P1	for process to find the sum of the interior angles of a pentagon, eg \(180 \times (5 - 2)\ (= 540)\) oe
	P1	for the start to a process of giving each angle in a common form, eg \(d = 3c\) or \(e = 2c\) or \(x\), \(3x\), \(2x\)
	P1	for process to find the value of \(c\), eg \(([540] - 155) \div 7\) oe or for a correct equation in one variable, eg \(c + 155 + c + 3c + 2c = [540]\) oe
	A1	cao

Additional guidance

Can be implied by the shape correctly divided into triangle and quadrilateral or three triangles with correct angle sums marked.

Can be implied by division by 7
or 1, 1, 3, 2 given in a ratio eg 1 : 2 : 1 : 3

Where [540] is what they believe to be the angle sum of the pentagon.

Higher November 2024 Paper 2 Q9

EdexcelCurrent spec3 marksAngles in Polygons

9 The diagram shows two congruent regular 9-sided polygons.
\(ABCD\) is a quadrilateral.

Two congruent regular nonagons sharing vertex B, with quadrilateral ABCD between them; A and C are vertices of the nonagons

Show that \(ABCD\) is not a square. (3)

Mark scheme

Answer	Mark	Mark scheme
shown	M1	for method to find the interior angle eg \((9 - 2) \times 180 \div 9\ (= 140)\) or exterior angle eg \(360 \div 9\ (= 40)\) of a regular nonagon
	M1	for a complete method to find the interior angle \(ABC\), eg \(360 - \text{``}140\text{''} - \text{``}140\text{''}\ (= 80)\) or \(\text{``}40\text{''} \times 2\ (= 80)\) or finds the angle sum assuming a square, eg \(90 + \text{``}140\text{''} + \text{``}140\text{''}\ (= 370)\)
	A1	for complete solution with correct conclusion, eg interior angle of a square is 90, not 80 or angles around a point sum to 360 not 370

Additional guidance

Angles may be shown on the diagram
Any angle labelled correctly as 140 or 40 and not contradicted scores this mark

Foundation November 2024 Paper 3 Q7

EdexcelCurrent spec2 marksAngles in Polygons

7 Here is a quadrilateral.

A shaded trapezium drawn on a square grid, with a horizontal top side of 2 squares and a horizontal bottom side of 5 squares

(a) Write down the special name for this quadrilateral. (1)

(b) On the grid below, draw a right-angled triangle.

An empty square grid, 9 squares wide and 6 squares high

(1)

Answer	Mark	Mark scheme
Trapezium	B1	cao

Answer	Mark	Mark scheme
Drawn	B1	for a right-angled triangle drawn

Higher November 2024 Paper 1 Q6

EdexcelCurrent spec4 marksAngles in Polygons

6 Here is a pentagon.

Angle \(a\) = angle \(c\)
Angle \(b = 155^\circ\)
Angle \(d\) is three times the size of angle \(c\)
Angle \(e\) is two times the size of angle \(c\)

Work out the size of angle \(a\) (4)

Mark scheme

Answer	Mark	Mark scheme
55	P1	for process to find the sum of the interior angles of a pentagon, eg \(180 \times (5 - 2)\ (= 540)\) oe
	P1	for the start to a process of giving each angle in a common form, eg \(d = 3c\) or \(e = 2c\) or \(x\), \(3x\), \(2x\)
	P1	for process to find the value of \(c\), eg \(([540] - 155) \div 7\) oe or for a correct equation in one variable, eg \(c + 155 + c + 3c + 2c = [540]\) oe
	A1	cao

Additional guidance

Can be implied by the shape correctly divided into triangle and quadrilateral or three triangles with correct angle sums marked

Can be implied by division by 7
or 1, 1, 3, 2 given in a ratio eg 1 : 2 : 1 : 3

Where [540] is what they believe to be the angle sum of the pentagon

Foundation November 2024 Paper 2 Q4

EdexcelCurrent spec1 markAngles in Polygons

4 Write down the mathematical name of this polygon. (1)

Answer	Mark	Mark scheme
hexagon	B1	cao

Foundation November 2023 Paper 1 Q13

AQACurrent spec1 markAngles in Polygons

13 A shape has

an even number of sides
more sides than a square
fewer sides than a decagon

Write down the name of one shape this could be. [1 mark]

Answer	Mark	Comments
hexagon or octagon	B1

Foundation November 2023 Paper 3 Q3

AQACurrent spec2 marksAngles in Polygons Circles & Sectors

(a) Write down the name of a triangle with three equal sides. [1 mark]

(b) Write down the name for the straight line inside this circle.

A circle with a straight line joining two points on its circumference, not passing through the centre

[1 mark]

Answer	Mark	Comments
Equilateral	B1	ignore spelling, mark intention

Answer	Mark	Comments
Chord	B1	ignore spelling, mark intention

Foundation June 2017 Paper 1 Q23

AQACurrent spec2 marksAngles in Polygons

23 A regular polygon has an exterior angle of \(20^\circ\)

Work out the number of sides of the polygon. [2 marks]

Answer	Mark	Comments
\(360 \div 20\) or \(20 \times 18 = 360\)	M1	oe
18	A1

Foundation June 2017 Paper 3 Q22

AQACurrent spec2 marksAngles in Polygons

22 This shape is made from two triangles and four congruent parallelograms.

Two rows, one above the other. Each row is a triangle between two parallelograms, the three shapes together forming a trapezium

Not drawn accurately

For each statement, tick the correct box.

(a) The triangles are equilateral. [1 mark]

Must be true
Could be true
Must be false

(b) The triangles are congruent. [1 mark]

Must be true
Could be true
Must be false

Answer	Mark	Comments
Could be true	B1

Answer	Mark	Comments
Must be true	B1

Foundation June 2017 Paper 1 Q3

AQACurrent spec1 markAngles in Polygons

3 Which of these shapes has the most sides?

Circle your answer. [1 mark]

Hexagon
Octagon
Rhombus
Trapezium