Question 1
The area of a circular plate is onesixteen the surface area of a ball. If the area of the plate is given as pcm^{2}, then the radius of the ball is (UME 1981, Question 35)
Solution
Area of circular plate = pcm^{2}
Let the area of ball be
Area of circular plate = qcm^{2}
Area of ball
Question 2
In the above diagram XZ is the diameter of a circle . If XY is 4cm then the area of the triangle XYZ is (UME 1981, Question 36)
Solution
Question 3
A square of cardboard is taped at the perimeter by a piece of ribbon 20 cm long. What is the area of the board (UME 1979, Question 16)
Solution
Perimeter of a square = 4l
4l =20cm
l = 5cm
Area of square = l^{2} = 5^{2} cm^{2} =25cm^{2}
Question 4
A steel ball of radius 1cm is dropped into a cylinder of radius 2cm and height 4cm. If the cylinder is now filled water. What is volume in the cylinder?
(UME 1979, Question 20)
Solution
Volume of the sphere =
Given radius of the sphere is 1 cm
Volume of the sphere will be =
Volume of the cylinder =
Given radius of the cylinder as 2cm and height as 4cm.
Volume of the cylinder will be
The sphere will displace water from the cylinder (Archimedes ‘s Principle)
The volume of water left will be
Question 5
A cylinder of height h and radius r is open at one end. Its surface area is
Question 6
In the parallelogram PQRS, PE is perpendicular to QR. Find area of parallelogram

Solution
Question 7
A Solid cylinder of radius 3cm has a total surface area of 36p cm^{2}. Find its height (UME 1980, Question9)
Solution
Total surface area of cylinder =
The height is 3cm
Question 8
In the below , , PS =7cm, PT =7cm, SR = 4cm. Find the ratio of QRST to the area of PQRS (UME1980, Question 11)
Solution
Area of QRST = b ´ h = 4 ´7cm^{2} =28cm^{2}
Area of PQRS = ½ ( 4 + 7)7 =77/2 cm^{2}
Ratio QRST : PQRS
28: 77/2
56: 77
Question 9
Find the area of curved surface of a cone whose base radius is 6cm and whose height is 8cm Take p= 22/7(UME 1980, Question 36)
Solution
Question 10
A cylinder motor of height 12cm has uniform thickness of 2cm. If the diameter of its outer cross –section is 10 cm. Find the volume of material (take p= 22/7) (UME 1980, Question 44)
Solution
Volume of cylinder =pr^{2}h
Volume of the inner cylinder =
Volume of the outer cylinder =
Volume of the composite cylinder =
Volume of the composite cylinder =
Question 11
The area of the curved surface of the cone generated by the circle radius 6cm and arc length 22cm is
Take p =22/ 7 (UME 1982, Question 39)
Solution
Radius of sector = l = 6cm
The length of arc AXB is the same as the circumference of the circular base of the cone
Question 12
The square base of a pyramid of side 3cm has height 8cm. If the pyramid is cut into two parts by a plane parallel to the base midway between the base and vertex.. The volumes of the two sections are (UME 1982, Question 40)
Solution
Total volume of square based pyramid
=
Using similar triangle
Volume of pyramid AHBCD =
The volume of the base section = (24 – 3) cm^{3}
The volume of the base section = 21cm^{3 }
The volume of the two section are 21cm^{3}, 3cm^{3}
Question 13
PQRS is a parallelogram with area 50 square cm and side PQ is 10 cm long. T is appoint on RS and TF is the attitude of the triangle, Find TF (UME 1982 Question 44)
Solution
Area of Parallelogram = base ´ height
50cm^{2} = 10h
h = 5cm
where h = TF, TF =5cm
Question 14
What is the volume of the regular threedimensional figure above? (UME 1984, Question 25)
Solution
Question 15
A cone is formed, by bending a sector having an angle of 120^{o}. Find the radius of the base of the cone. If the diameter of the circle is 12cm
Solution
Radius of the sector = l = d/2 =12/2 cm = 6cm
The length of arc AXB is the same as the circumference of the circular cone
Question 16
Find the area of regular hexagon inscribed in a circle of radius 8cm (UME 1985, Question 36)
Solution
Since the polygon is a regular polygon, all the sides AB = AO =DB = 8cm
Question17
(UME 1985, Question 42)
Solution
Area of trapezium =
Question 18
How many diameter disc can be cut out of sheet of cardboard long and wide (UME 1985 Question 45)
Solution
Area of the cardboard
=
Area of disc =
The number of disc that could be from the cardboard will be
The figure is a solid with the trapezium PQRS as its uniform cross –section. Find its volume
Solution
Volume of block =Area of PQRS ´ 12cm^{3}
Area of PQRS =
With the construction of figure above
Area of PQRS =
Volume of the block is
Question 19
Find the total surface area of a solid cone of radius .(UME 1986, Question 41)
Solution
Question 20
An open rectangular box externally measures 4m ´ 3m ´ 4m. Find the total cost of painting the box externally. If it cost N2.00 to paint one square metre (UME 1986, Question 44)
Solution
Total area of the rectangular box =
Total cost of painting 68cm^{2} will be 68 ´ N2.00 = N 136.00
Question 21
In the figure above PQRS is rectangle, if the shaded area is 72sq.cm. Find h
Solution
Area of QAEP = 2´3h cm^{2} =6hcm^{2}
Area of BRSF = 2 ´ 3h = 6h cm^{2}
Area of ABCD =(2h – 4) ´ 2cm^{2} =(4h –8)cm^{2}
Total Area of the shaded portion =(6h + 6h +4h – 8 )cm^{2} =72cm^{2}
16h = 80, Ãž h =5 cm
Question 22
The base of a pyramid is a square of side 8cm. If its vertex is directly above the centre, find the height, given hat the edge is (UME 1987,Question 43)
Solution
Using Pythagoras theorem
Question
In the figure, PS =7cm and RY = 9cm. If the area of parallelogram PQRS is 56cm. Find the area of trapezium. (UME 1988, Question 33)
Area of parallelogram = b ´ h
Question 23
A quadrant of a circle of radius 6cm is cut away from each corner of a rectangle 25cm long and 18cm wide. Find the perimeter of the remaining figure. (UME 1988 Question 34)
Solution
Perimeter of the rectangle = 2(18 +25) cm = 86cm
Perimeter of the one quadrant =
Perimeter of the four quadrants =
Perimeter of the remaining figure is
= (86 – 12p)cm
Question 24
If a metal pipe 10cm has an external diameter of 12cm and thickness of 1cm. Find the volume of the metal used in making the pipe (UME 1988, Question 44)
Solution
Volume of the cylinder =
Volume of internal cylinder=
Volume of external cylinder =
Volume of the pipe =
Question 25
In the figure above, a solid consist of hemisphere by a right circular cone with radius 3cm and height 6.0cm.Find the volume of the solid (UME 1988 Question 45)
Solution
Volume of cone =
Volume of hemisphere =
Volume of the solid shape
Question 26
A square tile has side 30cm. How many of these tiles will cover a rectangular floor of length 7.2m and width 4.2m? (UME 1989, Question 43)
Solution
Question 27
A cylindrical metal pipe 1m long has an outer diameter of 7.2cm and an inner diameter of 2.8cm. Find the volume of the metal used of the cylinder.(UME 1989, Question 44)
Solution
Volume of the metal =
R =3.6cm, r =1.4cm, h =100cm
Volume =
Question 28
OXYZW is a pyramid with a square base such that OX = OY = OZ = OW = 5m and XY =XW =YZ = WZ = 6cm. Find the height of OT (UME 1989 Question 45)
Solution
Question 29
The perimeter of rectangular lawn is 24cm, if the area of the lawn is 35m^{2}. How wide is the lawn
Solution
Perimeter = 2(l + w) =24
l+ w =12 – – – –(1)
Area of the rectangular lawn A =lw = 35m^{2} – – – (2)
From (1) l =12 – w
Substitute (12 – w) for l in equation (2)
(12 – w)w = 35
w^{2} – 12w + 35 = 0
(w – 7)(w – 5) =0
w = 7 or w = 5
The width should normally be less to the length of the lawn. So width will be 5m
Question 30
In the diagram, the area PQRS is 73.5cm^{2} and its height is 10.5cm. Find the length of PS. If QR is one–third of PS (UME 1990, Question 31)
Solution
Question 31
Solution
Using Pythagoras theorem
Question 32
A cylinder pipe made of metal is 3cm thick. If the internal radius is 10cm. Find the volume of the metal used in making 3m of the pipe (UME 1990, Question 42)
Solution
Radius of internal pipe = r =10cm
Radius of external pipe = R =13cm
Volume of the pipe =
Question
If the height of two circular are in the ratio 2:3 and their bases radii are in the ratio 9:8. What is the ratio of their volume (UME 1990, Question 43)
Solution
Question 33
Find the curve surface area of the frustum in the figure above (UME 1990 Question 44)
Solution
Using similar triangles



In the figure above PQRS is a square of sides’ 8cm. What is the area of (UME 1991, Question 44)
Solution
Using Pythagoras
Question 36
Find the total area of the surface of a solid cylinder whose base radius is 4cm and height is 5cm (UME 1991, Question 45)
Solution
Question
Find the volume of the figure above (UME 1991, Question 46)
Solution
Question 38
The above figure is a circle with center O. Find the area of the shaded portion (UME 1992 , Question30)
Area of Triangle =
Area of a quadrant =
Area of the shaped portion =
Question 39
In the figure above, the area of the square PQRS is 100cm^{2}. If the ratio of the area of the square TUYS to the area of the square XQVU is 1:16, Find YR (UME 1993 Question 35)
Solution
Question 40
Find the radius of a sphere whose surface area is 154cm^{2} ( (UME 1993 Question 36)
Solution
Question 41
An open rectangular box is made of wood of 2cm thick. If the internal dimension of the box are 50cm long, 36cm wide and 20 cm deep. The volume of the box is (UME 1994, Question 29)
Solution
Internal dimensions are 50cm, 36cm and 20cn
External dimension are 54cm, 40cm and 21cm
Question 43
In the frustum of a cone shown above, the top diameter. If the height of the frustum is h centimeters, find the height of the cone
Using similar triangle
Question 44
In the diagram above, the base diameter is 14cm, whilke the height is 12cm. Calculate the total surface area if the cylinder has bothe a base and a top [p =22/7] (UME 1995, Question 33)
Solution
Question 46
In the diagram above, find PQ, If the area of the triangle PQR is 35cm^{2 }(UME 1995, Question 34)
Solution
Area of triangle =
PQ =14cm
Question 47
The figure above shows circles radii 3cm and 2cm with center at X and Y respectively. The circle have a transverse common tangent of length 25cm. Calculate XY (UME 1997 Question 27)
Solution
Question 48
A cone with the sector angle of 45^{o} is cut out of a circle of radius r cm. Find the base radius of the cone. (UME 1997, Question 31)
Solution
Question 49
In the figure above PQST is a parallelogram and TSR is a straight line. If the area of QRS is 20cm^{2.} Find the area of trapezium PQRT
Solution
^{}
Question 50
A cylindrical drum of diameter 56cm contain 123.2 litres of oil when full. Find the height of the drum in centimeters
Solution
Question 51
Find the value of l in the diagram above (UME 1999, Question 32)
Solution
Question 52
A cylindrical tank has a capacity of 3080m^{3}. What is the depth of the tank if the diameter of its base is 14m (UME 2001, Question 23)
Solution
Question 53
In the diagram above are two concentric circles of radii r and R respectivelywith centre O. If . Express the area of the shaded portion in terms of
Solution
Question 54
A bucket is 12cm in diameter at the top, 8cm in diameter at the bottom and 4cm deep. Calculate its volume
Solution
Question 55
In the diagram above, a cylinder is surmounted by a hemispherical bowl. Calculate the volume of the solid
Solution
Question 56
A trapezium has two parallel sides of length 5cm and 9cm. If the area is 21cm^{2}. Find the distance between the parallel sides (UME 2003, Question 26)
Solution
Question 57
In the diagram above, PQ = 4cm and TS = 6cm , if the area of parallelogram PQTU is 32 cm^{2}, Find the area of trapezium PQRU
Solution 58
Area of parallelogram = bh
Question 59
In the diagram above is the diameter of the semicircle QR. Find the area of the figure to the nearest whole number (UME 2006, Question 27)
Solution
Question 60
The volume of a hemispherical bowl is . Find its radius.
Solution
Question 61
Find the area of figure above (a) 12.5cm^{2} (b) 75.0cm^{2} (c) 78.5cm^{2} (d) 8.8cm^{2}
(UME 2008, Question 28, Set A U03)
Solution
Area of rectangle ACDE =
Area of semicircle =
Area of the figure =
Answer: Option D
Question 62
Find the capacity in litres of a cylindrical well of radius 1metre and depth 14metres
(a) 44,000litres (b) 4400litres (c) 440litres (d) 44litres
(UME 2008, Question 29, Set A U03)
Solution
Given r =1 metre, depth = h =14 metres
Volume of cylinder =
Note: 1m^{3} =1 litre
44m^{3} = 44 litres
Answer: Option D
Question 63
Find the radius of a sphere whose surface area is 154cm^{2}, take
(a) 7.00cm (b) 3.50cm (c) 3.00cm (d) 1.75cm
(UME 2009, Question 29, Set A U03)
Solution
Answer: Option B
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