MENSURATION

Question 1

The area of a circular plate is one-sixteen the surface area of a ball. If the area of the plate is given as pcm², then the radius of the ball is (UME 1981, Question 35)

Solution

Area of circular plate = pcm²

Let the area of ball be

Area of circular plate = qcm²

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 = l² = 5² cm² =25cm²

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 cm². 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 ´7cm² =28cm²

Area of PQRS = ½ ( 4 + 7)7 =77/2 cm²

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 =pr²h

Volume of the inner cylinder =

Volume of the outer 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) cm³

The volume of the base section = 21cm³

The volume of the two section are 21cm³, 3cm³

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

50cm² = 10h

h = 5cm

where h = TF, TF =5cm

Question 14

What is the volume of the regular three-dimensional 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 ´ 12cm³

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 68cm² 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 cm² =6hcm²

Area of BRSF = 2 ´ 3h = 6h cm²

Area of ABCD =(2h – 4) ´ 2cm² =(4h –8)cm²

Total Area of the shaded portion =(6h + 6h +4h – 8 )cm² =72cm²

16h = 80, Þ 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 35m². How wide is the lawn

Solution

Perimeter = 2(l + w) =24

l+ w =12 – – – –(1)

Area of the rectangular lawn A =lw = 35m² – – – (2)

From (1) l =12 – w

Substitute (12 – w) for l in equation (2)

(12 – w)w = 35

w² – 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 73.5cm² 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

4cm

Q

U

Question 35

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 100cm². 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 154cm² ( (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 35cm²(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 3080m³. 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 21cm². 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 cm², 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) 12.5cm² (b) 75.0cm² (c) 78.5cm² (d) 8.8cm²

(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: 1m³ =1 litre

44m³ = 44 litres

Answer: Option D

Question 63

Find the radius of a sphere whose surface area is 154cm², take

(a) 7.00cm (b) 3.50cm (c) 3.00cm (d) 1.75cm

(UME 2009, Question 29, Set A U03)

Solution

Answer: Option B

Solution

Solution

Question 3

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Question 4

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Question 5

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Question 8

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Question 9

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Solution

Volume of the inner cylinder = MPSetEqnAttrs('eq0020','',3,[[93,10,0,-1,-1],[124,14,0,-1,-1],[156,17,0,-1,-1],[139,15,0,-1,-1],[186,19,0,-1,-1],[233,24,1,-2,-2],[387,41,1,-3,-3]]); MPEquation();

Question 11

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Question 12

Question 13

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Question 14

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Question 16

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Question 18

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Question 19

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Question 21

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Question 22

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Question 24

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Question 26

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Question 28

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Question 29

The perimeter of rectangular lawn is 24cm, if the area of the lawn is 35m2. How wide is the lawn

Solution

Question 30

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Question 32

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Question 33

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U

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Question 36

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Question 39

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Question 41

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Question 43

Question 44

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Question 47

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Question 50

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Question 51

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Solution

Volume of the inner cylinder =

The perimeter of rectangular lawn is 24cm, if the area of the lawn is 35m². How wide is the lawn