IGCSE · Mathematics 0580 · Geometry

Polygons: IGCSE Mathematics questions with answers

Free Polygons questions for IGCSE Mathematics (0580), part of Geometry. Each question has the answer and the marking points an examiner would look for.

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Polygons practice questions

Question 1Structured · 14 marksA* challenge

Some regular polygons fit exactly around a point, with no gaps or overlaps, so that the point is a vertex of each polygon.

(a)

Three regular polygons fit exactly around a point. One of them is an equilateral triangle.
Find all the possible pairs of polygons that complete the arrangement. Give the number of sides of each.

[6]
(b)

A regular polygon X and two identical regular polygons fit exactly around a point.
Find every possible polygon X, and the two polygons that go with it.

[4]
(d)

Four regular polygons, which are not all the same, fit exactly around a point. Find all the possible sets of four polygons.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(7, 42), (8, 24), (9, 18), (10, 15), (12, 12)
interior angle of an n-sided regular polygon = 180 − 360n B1
60 + (180 − 360b) + (180 − 360c) = 360 M1
1b + 1c = 16 A1
with b ≤ c, 16 < 2b, so 7 ≤ b ≤ 12 M1
at least three correct pairs A1
all five pairs and no others A1
6
(b)(3; 12, 12), (4; 8, 8), (6; 6, 6), (10; 5, 5)
1x + 2y = 12 M1
y = 4xx − 2 = 4 + 8x − 2 M1
x − 2 is a factor of 8: x = 3, 4, 6, 10 A1
3 with 12, 12; 4 with 8, 8; 6 with 6, 6; 10 with 5, 5 A1
4
(d){3, 3, 4, 12}, {3, 3, 6, 6}, {3, 4, 4, 6}
1a + 1b + 1c + 1d = 1 B1
the smallest polygon must be a triangle (four squares are all the same) M1
systematic search on the second polygon M1
all three sets and no others A1
4
Question 2Structured · 4 marksA* challenge

The interior angle of a regular polygon is 8 times its exterior angle.
Find the number of sides of the polygon and the sum of its interior angles.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
18 sides, 2880°
ext + 8 × ext = 180 M1
exterior angle = 20° A1
n = 18 A1
2880 B1
4
Question 3Structured · 3 marksA* challenge

At a point, an equilateral triangle and two squares and one regular polygon P fit together exactly with no gaps.
Find the number of sides of P.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
6
interior angles 60, 90, 90 seen B1
angle of P = 360 − 240 = 120 M1
exterior angle 60, n = 6 A1
3
Question 4Structured · 4 marksA* challenge

The interior angle of a regular polygon is 7 times its exterior angle.
Find the number of sides of the polygon and the sum of its interior angles.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
16 sides, 2520°
ext + 7 × ext = 180 M1
exterior angle = 22.5° A1
n = 16 A1
2520 B1
4
Question 5Structured · 3 marks

Each interior angle of a regular polygon is 144°.
Work out the number of sides of the polygon.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
10
180 − 144 = 36 M1
360 ÷ 36 M1
10 A1
3
Question 6Structured · 3 marksA* challenge

At a point, four equilateral triangles and one regular polygon P fit together exactly with no gaps.
Find the number of sides of P.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
6
interior angles 60, 60, 60, 60 seen B1
angle of P = 360 − 240 = 120 M1
exterior angle 60, n = 6 A1
3

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in