Year 9 – IMC Geometry Exercises
Exercise 1 – Lengths
1. [Kangaroo Pink 2008 Q3] Four unit squares are placed edge to edge as shown. What
is the length of the line 𝑃𝑄?
A 5
B √13
C √5 + √2
D √5
E 13
2. Two identical squares are inscribed in a semicircle of radius√2. Find the length of
each square.
3. A square of length 2 is inscribed in a semicircle. Find the radius of the semicircle.
4. [Kangaroo Pink 2011 Q6] The diagram shows a shape made from a regular
hexagon of side one unit, six triangles and six squares. What is the perimeter of
the shape?
1
A 6(1 + √2)
B 6 (1 + 2 √3)
D 6 + 3√2
E 9
C 12
5. [IMC 2009 Q7] Four touching circles all have radius 1 and their centres are at the
corners of a square. What is the radius of the circle through the points of contact 𝑋, 𝑌, 𝑍
and 𝑇?
A
1
2
B
1
√2
2
C 1
D √2
E 2
6. [Based on IMC 2015 Q24] In square 𝑅𝑆𝑇𝑈 a quarter-circle arc with centre 𝑆 is drawn
from 𝑇 to 𝑅. A point 𝑃 on this arc is 1 unit from 𝑇𝑈 and 8 units from 𝑅𝑈. Suppose
we wanted to find the the length of the side of square 𝑅𝑆𝑇𝑈.
Let the length of the square be 𝑥.
Draw a line in your diagram horizontally right from the point 𝑃 to form a rightangled triangle. Find the lengths of the three sides of your triangle in terms of 𝑥 (you do not
at this stage need to find the value of 𝑥).
7. [Kangaroo Pink 2015 Q9] In the grid, each small square has side of length 1.
What is the minimum distance from ‘Start’ to ‘Finish’ travelling only on edges or
diagonals of the squares?
A 2√2
B √10 + √2 C 2 + 2√2
D 4√2
E 6
8.
[IMC 2011 Q19] Harrogate is 23km due north of Leeds, York is 30km due east of Harrogate,
Doncaster is 48km due south of York, and Manchester is 70km due west of Doncaster. To
the nearest kilometre, how far is it from Leeds to Manchester, as the crow flies?
A 38km
B 47km
C 56km
D 65km
E 74km
9. [IMC 2009 Q20] A square, of side two units, is folded in half to form a triangle. A second fold
is made, parallel to the first, so that the apex of this triangle folds onto a point on its base,
thereby forming an isosceles trapezium. What is the perimeter of this trapezium?
A 4 + √2
B 4 + 2√2
C 3 + 2√2
D 2 + 3√2
E 5
10. [Kangaroo Pink 2010 Q7] The diagram shows a square 𝑃𝑄𝑅𝑆 and two
equilateral triangles 𝑅𝑆𝑈 and 𝑃𝑆𝑇. 𝑃𝑄 has length 1. What is the length of 𝑇𝑈?
A √2
B
√3
2
C √3
D √5 − 1
E √6 − 1
11. Find the area of a regular octagon of side length 1.
12. [Based on Hamilton 2006 Q4] A circle is inscribed in a square and a rectangle
is placed inside the square but outside the circle. Two sides of the rectangle
lie along sides of the square and one vertex lies on the circle, as shown. The
rectangle is twice as high as it is wide. We wish to find the ratio of the area
of the square to the area of the rectangle.
a) Using the information, give suitable numeric lengths to the width and height of the
rectangle.
b) As per the usual advice for diagrams involving circles, add a suitable line to your diagram,
and form a right-angle triangle. If the radius of your circle is 𝑟, give expressions for the three
lengths of your triangle.
13. [IMC 2012 Q21] The parallelogram 𝑃𝑄𝑅𝑆 is formed by joining together four
equilateral triangles of side 1 unit, as shown. What is the length of the
diagonal 𝑆𝑄?
A √7
B √8
C 3
D √6
E √5
14. [IMC 2004 Q18] In the triangle 𝑃𝑄𝑅, there is a right angle at 𝑄 and angle 𝑄𝑃𝑅 is
60°. The bisector of the angle 𝑄𝑃𝑅 meets 𝑄𝑅 at 𝑆, as shown. What is the ratio
𝑄𝑆: 𝑆𝑅?
A 1: 1
B 1: √2
C 1: (3 − √3)
D 1: √3
E 1: 2
15. [Cayley 2014 Q4] The square 𝐴𝐵𝐼𝐽 lies inside the regular octagon 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻. The sides of
the octagon have length 1. Find length 𝐶𝐽.
16. [JMO Mentoring May2012 Q4] A triangle has two angles which measure 30° and 105°. The
side between these angles has length 2 cm. What is the perimeter of the triangle?
17. [IMC 2004 Q23] In the diagram, the letter 𝑆 is made from two arcs, 𝐾𝐿 and 𝑀𝑁,
which are each five-eighths of the circumference of a circle of radius 1, and the line
segment 𝐿𝑀, which is tangent to both circles. At points 𝐾 and 𝑁, common tangents
to the two circles touch one of the circles. What is the length 𝐿𝑀?
A
3
2
B 3 − √2
C 2
D
3√2
2
E 1 + √2
18. [IMC 1999 Q25] A rectangular sheet of paper with sides 1 and √2 has
been folded once as shown, so that one corner just meets the opposite
long edge. What is the value of the length 𝑑?
A
1
2
B √2 − 1
C
7
16
D √3 − √2
E
√2
3
19. [Kangaroo Pink 2008 Q24] In the diagram, 𝐾𝐿𝑀𝑁 is a unit square. Arcs of radius
one unit are drawn using each of the four corners of the squares as centres. The
arcs centred at 𝐾 and 𝐿 intersect at 𝑄; the arcs centred at 𝑀 and 𝑁 intersect at
𝑃. What is the length of 𝑃𝑄?
A 2 − √2
E √3 − 1
B
3
4
C √5 − √2
D
√3
3
Exercise 2 – Using Variables and Pythagoras
1. [SMC 2014 Q19] The diagram shows a quadrant of radius 2, and two touching
semicircles. The larger semicircle has radius 1. What is the radius of the smaller
semicircle?
A
𝜋
6
B
√3
2
C
1
2
D
1
√3
E
2
3
2. [IMC 2015 Q24] In square 𝑅𝑆𝑇𝑈 a quarter-circle arc with centre 𝑆 is drawn
from 𝑇 to 𝑅. A point 𝑃 on this arc is 1 unit from 𝑇𝑈 and 8 units from 𝑅𝑈. What
is the length of the side of square 𝑅𝑆𝑇𝑈?
A 9
B 10
C 11
D 12
E 13
3. [IMC 2011 Q23] A window frame in Salt’s Mill consists of two equal semicircles
and a circle inside a large semicircle with each touching the other three as shown.
The width of the frame is 4m. What is the radius of the circle in metres?
A
2
3
B
√2
2
C
3
4
D 2√2 − 1
E 1
4. [IMC 2010 Q23] The diagram shows a pattern of eight equal shaded squares inside
a circle of area 𝜋 square units. What is the area (in square units) of the shaded
region?
A 1
1
3
B 1
3
5
C 1
2
3
D 1
7
9
E 2
5. [Cayley 2010 Q5] A square sheet of paper ABCD is folded along FG, as shown, so
that the corner B is folded onto the midpoint M of CD. Prove that the sides of
triangle GCM have lengths in the ratio 3: 4: 5.
6. [Cayley 2008 Q5] A kite has sides AB and AD of
length 25cm and sides CB and CD of length 39cm.
The perpendicular distance from B to AD is 24cm.
The perpendicular distance from B to CD is h cm.
Find the value of h.
7. [Hamilton 2010 Q5] The diagram shows three touching circles, whose radii are a, b and c,
and whose centres are at the vertices Q, R and S of a rectangle QRST. The fourth vertex T of
the rectangle lies on the circle with centre S. Find the ratio 𝑎: 𝑏: 𝑐.
8. [Kangaroo Pink 2004 Q24] The shaded area is equal to 2𝜋. What is the length of 𝑃𝑄?
A 1
B 2
C 3
D 4
E impossible to determine
9. [Maclaurin 2011 Q5] Three circles touch the same straight line and touch
each other, as shown. Prove that the radii a, b, and c, where c is smallest,
satisfy the equation:
1
1
1
+
=
√𝑎 √𝑏 √𝑐
Exercise 3 – Areas and Circles
1. [JMO 2015 A5] Two circles of radius 1 cm fit exactly between two parallel
lines, as shown in the diagram. The centres of the circles are 3 cm apart. What
is the area of the shaded region bounded by the circles and the lines?
2. [Kangaroo Grey 2005 Q9] In the diagram, the five circles have the same radii and
touch as shown. The square joins the centres of the four outer circles. The ratio of
the area of the shaded parts of all five circles to the area of the unshaded parts of all
five circles is
A 5: 4
B 2: 3
C 2: 5
D 1: 4
E 1: 3
3. [IMC 2014 Q20] The diagram shows a regular pentagon and five circular arcs. The
sides of the pentagon have length 4. The centre of each arc is a vertex of the
pentagon, and the ends of the arc are the midpoints of the two adjacent edges.
What is the total shaded area?
A 8𝜋
B 10𝜋
C 12𝜋
D 14𝜋
E 16𝜋
4. [IMC 2014 Q10] The diagram shows five touching semicircles, each with radius 2.
What is the length of the perimeter of the shaded shape?
A 5𝜋
B 6𝜋
C 7𝜋
D 8𝜋
E 9𝜋
5.
[Hamilton 2011 Q4] A square just fits within a circle, which itself just fits within another
square, as shown in the diagram. Find the ratio of the two shaded areas.
6. [IMC 1997 Q23] The square 𝐴𝐵𝐶𝐷 is inscribed in a circle of radius 1.
Semicircles are drawn with diameters 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, 𝐷𝐴 as shown, and the parts
of these semicircles which lie outside the original circle are shaded. What is
the total area of these four shaded regions?
A 3𝜋 + 2
B 2
C 𝜋
D 1
E
𝜋
2
7. [Cayley 2009 Q2] The boundary of a shaded figure consists of four
semicircular arcs whose radii are all different. The centre of each arc lies on
the line AB, which is 10cm long.
What is the length of the perimeter of the figure.
8. [IMC 2008 Q17] The shaded region is bounded by eight equal circles with centres at the
corners and midpoints of the sides of a square. The perimeter of the square has length
8. What is the length of the perimeter of the shaded region?
A 𝜋
B 2𝜋
C 8
D 3𝜋
E 4𝜋
9. [Kangaroo Pink 2007 Q18] A coin with diameter 1 cm rolls around the outside of a
regular hexagon with edges of length 1cm until it returns to its original position. In
centimetres, what is the length of the path traced out by the centre of the coin?
𝜋
A 6+2
B 12 + 𝜋
C 6+𝜋
D 12 + 2𝜋
E 6 + 2𝜋
10. [IMC 2006 Q21] The diagram shows two semicircular arcs, 𝑃𝑄𝑅𝑆 and 𝑄𝑂𝑅. The
diameters, 𝑃𝑆 and 𝑄𝑅, of the two semicircles are parallel; 𝑃𝑆 is of length 4 and
is a tangent to semicircular arc 𝑄𝑂𝑅. What is the area of the shaded region?
A 2𝜋 − 2
B 3𝜋
C 𝜋
D 4
E 2𝜋 − 4
11. [STMC Regional 2007/08 Q5] The square in the diagram below has sides of
length two units. The shaded sections are enclosed by 4 semi-circles. Calculate
the exact value of the total area of the unshaded regions.
12. [Hamilton 2005 Q2] The region shown shaded in the diagram is bounded
by three touching circles of radius 1 and the tangent to two of the circles.
Calculate the perimeter of the shaded region.
13. [Hamilton 2010 Q4] The diagram shows a quarter-circle with centre O and two
semicircular arcs with diameters OA and OB. Calculate the ratio of the area of
the region shaded grey to the area of the region shaded black.
14. [Kangaroo Grey 2005 Q18] Two rectangles 𝐴𝐵𝐶𝐷 and 𝐷𝐵𝐸𝐹 are shown in the
diagram. What is the area of the rectangle 𝐷𝐵𝐸𝐹?
A 10cm2
B 12cm2
C 13cm2
D 14cm2
E 16cm2
15. [IMC 2003 Q19] What is the area of the pentagon shown?
A
D
1
𝑎(𝑏 − 𝑐)
2
1
𝑏(𝑐 − 𝑎)
2
B
E
1
𝑏(𝑎 + 𝑐)
2
1
𝑐(𝑎 + 𝑏)
2
C
1
𝑎(𝑏
2
+ 𝑐)
16. [IMC 2005 Q18] Three-quarters of the area of the rectangle has been shaded. What
is the value of 𝑥?
A 2
B 2.4
C 3
D 3.6
E 4
17. [Cayley 2015 Q5] The diagram shows a right-angled triangle and three circles. Each
side of the triangle is a diameter of one of the circles. The shaded region R is the
region inside the two smaller circles but outside the largest circle. Prove that the
area of R is equal to the area of the triangle.
18. [Kangaroo Grey 2012 Q17] In the diagram, 𝑊𝑋𝑌𝑍 is a square, 𝑀 is the midpoint of
𝑊𝑍 and 𝑀𝑁 is perpendicular to 𝑊𝑌. What is the ratio of the area of the shaded
triangle 𝑀𝑁𝑌 to the area of the square?
A 1: 6
B 1: 5
C 7: 36
D 3: 16
E 7: 40
19. [IMC 1998 Q24] Each of the sides of this regular octagon has length 2cm. What
is the difference between the area of the shaded region and the area of the
unshaded region (in cm2)?
A 0
B 1
C 1.5
D 2
E 2√2
Exercise 4: Cutting and Reforming
Test Your Understanding: The figure shows two shapes that fit together exactly.
Each shape is formed by four semicircles of radius 1. What is the total shaded area?
Test Your Understanding: The diagram shows a design formed by drawing six lines in a
regular hexagon. The lines divide each edge of the hexagon into three equal parts.
What fraction of the hexagon is shaded?
1
2
A5
1
B9
3
C4
D 10
5
16
E
Question 1: The diagram contains six equilateral triangles with sides of length 2 and a
regular hexagon with sides of length 1.
What fraction of the whole shape is shaded?
A
1
8
B
1
7
C
1
6
D
1
5
E
1
4
Question 2: The diagram shows an equilateral triangle with its corners at the mid-points of
alternate sides of a regular hexagon. What fraction of the area of the hexagon is shaded?
A
1
2
B
1
3
C
3
8
D
4
9
E
7
12
Question 3: Find the fraction of the shape which is shaded.
Question 4: The diagram on the right shows a rectangle with sides of length 5cm and
4cm. All the arcs are quarter-circles of radius 2cm. What is the total shaded area in cm2.
Question 5: The diagram shows a ceramic design by the Catalan architect Antoni Gaudi. It is
formed by drawing eight lines connecting points which divide the edges of the outer regular
octagon into three equal parts, as shown.
What fraction of the octagon is shaded?
A 1/5
B 2/9
C 1/4
D 3/10
E 5/16
Question 6: The diagram shows a square with two lines from a corner to the middle of an
opposite side. The rectangle fits exactly inside these two lines and the square itself. What
fraction of the square is occupied by the shaded rectangle?
A
1
3
B
2
5
C
3
10
D
1
2
E
3
8