5th Math Challengers Team Training Worksheet
1. A convex hexagon has one internal right angle. The other five internal angles are all equal to each
other. How many degrees are in the measure of one of these five angles?
(126 degrees)
2. Three gold spheres have, respectively, diameter 9 cm, 12 cm, and 15 cm. They are melted down
and made into a single gold sphere. What is the diameter of that sphere?
(18 cm)
3. What is the area of the triangle whose sides are 17, 17, and 16? (120)
3
4. A circle is inscribed in a square that has perimeter π . What is the circumference of the circle?
Express the answer as a common fraction.
𝟑
(𝟒)
5. The two circles below have the same radius, and have centres O and O’. The line segment joining O
and O’ meets the circle with centre O’ at the point P. Point A is one of the intersection points of the
circles, and the line OA meets the circle with centre O’ at a second point X. Given that ∠AOP is 28
degrees, how many degrees are in ∠XPO’ ? (42 degrees)
5th Math Challengers Team Training Worksheet
6. ABCDEF is a regular hexagon with side 1. What is the area of equilateral triangle ACE? Express the
answer as
𝑎√𝑑
𝑏
, where 𝑎 and 𝑏 are positive integers with no common factor greater than 1, and 𝑑 is
an integer which is not divisible by any perfect square > 1.
𝟑√𝟑
(
𝟒
)
7. Both triangles in the picture are equilateral and have the same centre. The sides of the inner
triangle are parallel to the sides of the outer triangle. The distance between corresponding edges
of the two triangles is equal to one-tenth of the height of the outer triangle. What is the ratio of the
area of the inner triangle to the area of the outer triangle? Express the answer as a common
fraction.
𝟒𝟗
(𝟏𝟎𝟎)
8. A circle is inscribed in a right triangle with legs √2 and 2 √2. What is the area of the circle? Write
the answer in the form (A − √𝐵)π, where A and B are integers.
(7 − √𝟒𝟓)π
9. Triangle ABC is isosceles, with CA = CB. Point P is on AB. Given that AP = 3, PB = 7, and CP = 5,
what is the area of Triangle ABC ? Express the answer in simplest radical form.
(𝟓√𝟐𝟏)
10. What is the radius of a circle inscribed in a triangle with sides of length 5, 12 and 13 units?
(2)
5th Math Challengers Team Training Worksheet
11. Each edge of a regular hexagon has length
4
√π
. The hexagon is inscribed in a circle. What is the
area of the circle, in square units?
(16)
12. The area of equilateral triangle ABC is nine times the area of equilateral triangle APQ. What is the
ratio of the perimeter of the trapezoid P BCQ to the perimeter of Triangle ABC? Express the
answer as a common fraction.
𝟖
(𝟗)
13. Rectangle ABCD has base 20. A semicircle is drawn that has the base AB as a diameter. This
semicircle meets side CD in the points P and Q, where DP = CQ = 7 and P Q = 6. What is the height
of the rectangle (that is, what is the length of line segment BC)? Express the answer in simplest
radical form.
(√𝟗𝟏)
14. A rectangle of perimeter 22 cm is inscribed in a circle of area 16π cm2. What is the area of the
rectangle? Express your answer as a decimal to the nearest tenth. (28.5)
15. Point E lies within rectangle ABCD. If AE = 7, BE = 5 and CE = 8, what is DE? Express your answer
in simplest radical form.
(𝟐√𝟐𝟐)
16. In square ABCD, shown here, sector BCD was drawn with a center C and BC
= 24 cm. A semicircle with diameter AE is drawn tangent to the sector BCD.
If points A, E and D are collinear, what is AE?
(12)