Geometry SMC Questions
All the following questions were from the last 10 questions (i.e. Q16-25) of Senior Maths Challenge
papers from 2005-2012).
1. The diagram shows the ellipse whose equation is 𝑥 2 +
𝑦 2 − 𝑥𝑦 + 𝑥 − 4𝑦 = 12. The curve cuts the 𝑦-axis at
points 𝐴 and 𝐶 and cuts the 𝑥-axis at points 𝐵 and 𝐷.
What is the area of the inscribed quadrilateral 𝐴𝐵𝐶𝐷?
A 28
B 36
C 42
D 48
E 56
2. The diagram shows a pattern found on a floor tile in the cathedral in
Spoleto, Umbria. A circle of radius 1 surrounds four quarter circles, also of
radius 1, which enclose a square. The pattern has four axes of symmetry.
What is the side length of the square?
A
1
√2
B 2 − √2
C
1
√3
D
1
2
E √2 − 1
1
3. The diagram shows two squares, with sides of length 2, inclined at an
angle 2𝛼 to one another. What is the value of 𝑥?
A cos 𝛼
B
1
cos 𝛼
C sin 𝛼
D
1
sin 𝛼
E tan 𝛼
4. In trapezium 𝑃𝑄𝑅𝑆, 𝑆𝑅 = 𝑃𝑄 =25cm and 𝑆𝑃 is parallel to 𝑅𝑄. All four sides
of 𝑃𝑄𝑅𝑆 are tangent to a circle with centre 𝐶. The area of the trapezium is
600cm2. What is the radius of the circle?
A 7.5cm
B 8cm
C 9cm
D 10cm
E 12cm
5. A semicircle of radius 𝑟 is drawn with centre 𝑉 and diameter
𝑈𝑊. The line 𝑈𝑊 is then extended to the point 𝑋, such that
𝑈𝑊 and 𝑊𝑋 are of equal length. An arc of the circle with
centre 𝑋 and radius 4𝑟 is then drawn so that the line 𝑋𝑌 is a
tangent to the semicircle at 𝑍, as shown. What, in terms of 𝑟,
is the area of the triangle 𝑌𝑉𝑊?
A
4𝑟 2
9
B
2𝑟 2
3
C 𝑟2
D
4𝑟 2
3
E 2𝑟 2
6. The top diagram on the right shows a shape that tiles the plane, as shown in
the lower diagram. The tile has nine sides, six of which have length 1. It may
be divided into three congruent quadrilaterals as shown. What is the area
of the tile?
A
1+2√3
2
B
4√3
3
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C √6
D
3+4√3
4
E
3√3
2
1
7. 𝑃𝑄𝑅𝑆 is a rectangle. The area of triangle 𝑄𝑅𝑇 is 5 of the area of 𝑃𝑄𝑅𝑆,
1
and the area of triangle 𝑇𝑆𝑈 is 8 of the area of 𝑃𝑄𝑅𝑆. What fraction of
the area of rectangle 𝑃𝑄𝑅𝑆 is the area of triangle 𝑄𝑇𝑈?
A
27
40
B
21
40
C
1
2
D
19
40
E
23
60
8. The diagram shows a small regular octagram (an eight-sided star)
surrounded by eight squares (dark grey) and eight kites (light grey) to make a
large regular octagram. Each square has area 1. What is the area of one of
the light grey kites?
A 2
B √2 + 1
C
21
8
D 4√2 − 3
E
11
4
9. In the diagram ∠𝐴𝐵𝐸 = 10°; ∠𝐸𝐵𝐶 = 70°; ∠𝐴𝐶𝐷 = 50°; ∠𝐷𝐶𝐵 = 20°;
∠𝐷𝐸𝐹 = 𝛼.
Which of the following is equal to 𝑡𝑎𝑛 𝛼?
A
tan 10° tan 20°
tan 50°
B
tan 10° tan 20°
tan 70°
D
tan 20° tan 50°
tan 70°
E
tan 10° tan 70°
tan 50°
C
tan 10° tan 50°
tan 70°
10. Three circles and the lines 𝑃𝑄 and 𝑄𝑅 touch as shown. The
distance between the centres of the smallest and the biggest
circles is 16 times the radius of the smallest circle. What is the size
of ∠𝑃𝑄𝑅?
A 45°
B 60°
C 75°
D 90°
E 135°
11. A solid sculpture consists of a 4 × 4 × 4 cube with a 3 × 3 × 3 cube sticking
out, as shown. Three vertices of the smaller cube lie on edges of the larger
cube, the same distance along each.
What is the total volume of the sculpture?
A 79
B 81
C 82
D 84
E 85
12. 𝑃𝑄𝑅𝑆 is a quadrilateral inscribed in a circle of which 𝑃𝑅 is a diameter. The lengths of 𝑃𝑄,
𝑄𝑅 and 𝑅𝑆 are 60, 25 and 52 respectively. What is the length of 𝑆𝑃?
2
A 21 3
11
B 28 13
C 33
D 36
E 39
13. A solid cube of side 2cm is cut into two triangular prisms by a plane passing
through four vertices, as shown. What is the total surface area of these two
prisms?
A 8(3 + √2)
E 8√2
B 2(8 + √2)
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C 8(3 + 2√2)
D 16(3 + √2)
14. The diagrams show two different shaded rhombuses, each
inside a square with sides of length 6. Each rhombus is
formed by joining vertices of the square to midpoints of the
sides of the square. What is the difference between the
shaded areas?
A 4
B 3
C 2
D 1
E 0
15. The diagram shows a regular hexagon, with sides of length 1, inside a square.
Two vertices of the hexagon lie on a diagonal of the square and the other four
lie on the edges.
What is the area of the square?
A 2 + √3
B 4
C 3 + √2
D 1+
3√3
2
E
7
2
16. The diagram shows two different semicircles inside a square with sides of length
2. The common centre of the semicircles lies on a diagonal of the square.
What is the total shaded area?
A 𝜋
B 6𝜋(3 − 2√2)
C 𝜋√2
D 3𝜋(2 − √2)
E 8𝜋(2√2 − 3)
17. Three spheres of radius 1 are placed on a horizontal table and inside a vertical hollow
cylinder of height 2 units which is just large enough to surround them. What fraction of the
internal volume of the cylinder is occupied by the spheres?
A
2
7+4√3
B
2
2+√3
C
1
3
3
2+√3
D
E
6
7+4√3
18. A solid cube is divided into two pieces by a single rectangular cut. As a result, the total
surface area increases by a fraction 𝑓of the surface area of the original cube. What is the
greatest possible value of 𝑓?
A
1
3
B
√3
4
C
√2
3
D
1
2
E
1
√3
19. A point 𝑃 is chosen at random inside a square 𝑄𝑅𝑆𝑇. What is the probability that ∠𝑅𝑃𝑄 is
acute?
A
3
4
B √2 − 1
C
1
2
D
𝜋
4
𝜋
E 1−8
20. A frustum is the solid obtained by slicing a right-circular cone perpendicular to
its axis and removing the small cone above the slice. This leaves a shape with
two circular faces and a curved surface. The original cone has base radius 6cm
and height 8cm, and the curved surface area of the frustum is equal to the
area of the two circles. What is the height of the frustum?
A 3cm
B 4cm
C 5cm
D 6cm
E 7cm
21. 𝑀 and 𝑁 are the midpoints of sides 𝐺𝐻 and 𝐹𝐺 respectively, of parallelogram 𝐸𝐹𝐺𝐻. The
area of triangle 𝐸𝑁𝑀 is 12cm2. What is the area of the parallelogram 𝐸𝐹𝐺𝐻?
A 20cm2
B 24cm2
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C 32cm2
D 48cm2
E more information is required
22. A figure in the shape of a cross is made from five 1 × 1 squares, as shown.
The cross is inscribed in a large square whose sides are parallel to the dashed
square, formed by four of the vertices of the cross. What is the area of the
large outer square?
A 9
B
49
5
C 10
D
81
8
E
32
3
23. The shaded square of the lattice shown has area 1. What is the area of the
circle through the points 𝑋, 𝑌 and 𝑍?
A
9𝜋
2
B 8𝜋
C
25𝜋
2
D 25𝜋
E 50𝜋
24. The diagram shows four semicircles symmetrically placed between two
circles. The shaded circle has area 4 and each semicircle has area 18. What
is the area of the outer circle?
A 72√2
B 100
C 98
E 32√3
D 96
25. A pentagon is made by attaching an equilateral triangle to a square with the
same edge length. Four such pentagons are placed into the rectangle, as shown.
What is the ratio of the length of the rectangle to its width?
A √3: 1
B 2: 1
C √2: 1
D 3: 2
E 4: √3
26. The two triangles have equal areas and the four
marked lengths are equal. What is the value of
𝑥?
A 30
D 75
B 45
C 60
E more information needed
27. The largest circle which it is possible to draw inside triangle 𝑃𝑄𝑅
touches the triangle at 𝑆, 𝑇, 𝑈, as shown in the diagram. The size of
∠𝑆𝑇𝑈 = 55°. What is the size of ∠𝑃𝑄𝑅?
A 55°
B 60°
C 65°
D 70°
E 75°
28. A triangle is cut from the corner of a rectangle. The resulting pentagon has sides of length 8,
10, 13, 15 and 20 units, though not necessarily in that order. What is the area of the
pentagon?
A 252.5
B 260
C 270
D 275.5
E 282.5
29. In triangle 𝑃𝑄𝑅, 𝑆 and 𝑇 are the midpoints of 𝑃𝑅 and 𝑃𝑄
respectively; 𝑄𝑆 is perpendicular to 𝑅𝑇; 𝑄𝑆 = 8; 𝑅𝑇 = 12.
What is the area of triangle 𝑃𝑄𝑅?
A 24
B 32
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C 48
D 64
E 96
30. The sum of the lengths of the 12 edges of a cuboid is 𝑥 cm. The distance from one corner of
the cuboid to the furthest corner is 𝑦 cm. What, in cm2, is the total surface area of the
cuboid?
A
𝑥 2 −2𝑦 2
2
B 𝑥2 + 𝑦2
C
𝑥 2 −4𝑦 2
4
D
𝑥𝑦
2
E
𝑥 2 −16𝑦 2
16
31. A paperweight is made from a glass cube of side 2 units by first shearing off the eight
tetrahedral corners which touch at the midpoints of the edges of the cube. The remaining
inner core of the cube is discarded and replaced by a sphere. The eight corner pieces are
now stuck onto the sphere so that they have the same positions relative to each other as
they did originally. What is the diameter of the sphere?
A √8 − 1
B √8 + 1
C
1
(6 +
3
√3)
D
4
√3
3
E 2√3
32. If 𝛼 < 𝛽, how many different values are there among the following expressions?
sin 𝛼 sin 𝛽
A 1
B 2
C 3
sin 𝛼 cos 𝛽
cos 𝛼 sin 𝛽
𝑐𝑜𝑠𝛼 cos 𝛽
E It depends on the value of 𝛼
D 4
33. A trapezium is bounded by four lines, the equations of which are 𝑥 = 0, 𝑥 = 4, 4𝑦 = 3𝑥 + 8
and 𝑦 = 𝑘, where 𝑘 < 2.
For which values of 𝑘 is he numerical value of the perimeter of the trapezium equal to the
numerical value of the area of the trapezium?
A
3
2
B 1
C
1
2
1
D −2
E −1
34. A toy pool table is 6 feet long and 3 feet wide. It has pockets at each of
the four corners 𝑃, 𝑄, 𝑅 and 𝑆. When a ball hits a side of the table, it
bounces off the side at the same angle as it hit that side. A ball, initially 1
foot to the left of pocket 𝑃, is hit from the side 𝑆𝑃 twards the side 𝑃𝑄 as
shown.
How many feet from 𝑃 does the ball hit side 𝑃𝑄 if it lands in pocket 𝑆 after two bounces?
A 1
B
6
7
C
3
4
D
2
3
E
3
5
35. In the diagram, the circle and the two semicircles have radius 1. What is
the perimeter of the square?
A 6 + 4√2
B 2 + 4√2 + 2√3
D 4 + 2√2 + 2√6
E 12
C 3√2 + 4√3
36. A solid red plastic cube, volume 1cm3, is painted white on its outside. The
cube is cut by a plane passing through the mid-points of various edges, as
shown.
What, in cm2, is the total red area exposed by the cut?
A
3√3
2
B 2
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C
9√2
5
D 3
E
3(√3+√2)
4
37. Eight identical octagons are placed edge to edge in a ring in such a way
that a symmetrical star shape is formed by the interior edges. If each
octagon has sides of length 1, what is the area of the star?
A 5 + 10√2
D 16 − 4√2
B 8√2
C 9 + 4√2
E 8 + 4√2
38. A cube exactly fits inside a sphere and another sphere exactly fits inside this cube. What is
the ratio of the volume of the smaller sphere to the volume of the larger sphere?
A 1: 3√3
B 1: 4
C 1: 3
D 2: 3
E 1: 2
39. It takes two weeks to clean the 3312 panes of glass in the 6000m2 glass roof of the British
Museum, a task performance once every two years. Assuming that all the panes are
equilateral triangles of the same size, roughly how long is the side of each pane?
A 50cm
B 1m
C 2m
D 3m
E 4m
40. The diagram shows four touching circles, each of which also touches the
sides of an equilateral triangle with sides of length 3.
What is the area of the shaded region?
A
11𝜋
12
B 𝜋
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C
(4+√3)𝜋
6
D
(3+√3)𝜋
4
E
37𝜋
36