Solid mensuration

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1. 1Prisms and cylindersVolume as cross-section × length, surface area from the net, the cylinder (curved surface 2πrh, open and closed ends), hollow pipes and boxes, and the diagonal of a cuboid.Done22 min
2. 2Cones and pyramidsVolume as one third of base × height, the slant height from Pythagoras, the curved surface πrl, and a cone made by bending a sector.Done18 min
3. 3Spheres, composite solids and capacitySpheres and hemispheres, solids made of two shapes, similar solids, and liquid problems: litres, cubic metres and how far a level rises or falls.Done18 min

Past questions (118)

From 1983–2025: 60 WAEC, 11 NECO, 47 JAMB. Theory questions first, then objective, newest first.

Theory (42)

• NECO 2024 · Paper 2 · Q8A cone has a radius of $10\text{ cm}$ and a slant height of $26\text{ cm}$. Find the: $\left[\pi = \frac{22}{7}\right]$
• WAEC 2024 · Paper 2 · Q13In the diagram, $O$ is the centre of the circle, $PQ$ is a tangent to the circle at $T$ and $ABC$ is a straight line. $TC$ bisects $\angle BTQ$, …
• NECO 2023 · Paper 2 · Q10A bucket full of water is $40\text{ cm}$ in diameter at the open end, $24\text{ cm}$ in diameter at the bottom and $32\text{ cm}$ deep. The bucket …
• WAEC 2023 · Paper 2 · Q9A cylindrical tank of diameter $3.5\text{ m}$ is full of water. Three tankers, each of capacity $6{,}500$ litres, draw water from the tank. Calculate, …
• NECO 2022 · Paper 2 · Q10A composite solid consists of a cone on top of a cylinder. The cone is 8 cm high and its slant height is 10 cm. If the solid is 18 cm high, calculate …
• NECO 2022 · Paper 2 · Q8If $x$ and $y$ are the surface area and volume of a sphere respectively and $\frac{x}{y} = \frac27$, find the values of $x$ and $y$. $\left(\text{Take } \pi = \frac{22}{7}\right)$
• WAEC 2022 · Paper 2 · Q12The probability that an athlete will not win any of the three races is $\frac14$. If the athlete runs in all the races, what is the probability …
• WAEC 2022 · Paper 2 · Q13The diameter of a cylinder closed at both ends is $7\text{ cm}$. If the total surface area is $209\text{ cm}^2$, calculate the height. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2022 · Paper 2 · Q10A cylindrical fuel storage tank of diameter $7\text{ m}$ is full of gasoline. If four tankers, each of capacity $13\,500$ litres, draw gasoline …
• WAEC 2022 · Paper 2 · Q4The volume of a cylinder of radius $14\text{ cm}$ is $9240\text{ cm}^3$. Calculate the curved surface area of the cylinder. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2022 · Paper 2 · Q8In a class of 60 students, 30 read Literature and 35 read Government. All the students read at least one of the subjects. If a student is …
• WAEC 2022 · Paper 2 · Q5The total surface area of a closed cone of radius $3.8\text{ cm}$ is $374\text{ cm}^2$. Calculate, correct to two decimal places, the volume …
• WAEC 2021 · Paper 2 · Q6The diagram shows the net of a rectangular pyramid: a $10\text{ cm} \times 4\text{ cm}$ rectangle with a triangle on each side; the triangles …
• WAEC 2021 · Paper 2 · Q3The curved surface area of a cone is $242\text{ cm}^2$. If the slant height is $4\text{ cm}$ more than the radius, calculate, correct to one …
• WAEC 2020 · Paper 2 · Q4The total surface area of a cone of slant height $l\text{ cm}$ and base radius $r\text{ cm}$ is $224\pi\text{ cm}^2$. If $r : l = 2 : 5$, find: …
• WAEC 2020 · Paper 2 · Q7The diagram shows a wooden structure in the form of a cone, mounted on a hemispherical base. The vertical height of the cone is $48\text{ m}$ …
• WAEC 2020 · Paper 2 · Q10The ratio of the radius ($r$) of the base of a cone to the height ($h$) is $2 : 3$. If the height of the cone is $8.1\text{ cm}$, calculate, …
• WAEC 2019 · Paper 2 · Q13The curved surface areas of two cones are equal. The base radius of one is $5\text{ cm}$ and its slant height is $12\text{ cm}$. Calculate the …
• WAEC 2019 · Paper 2 · Q2The slant height of a cone is $18.7\text{ cm}$ and the diameter is $24\text{ cm}$. Calculate, correct to three significant figures, the curved …
• WAEC 2018 · Paper 2 · Q9A donkey is tied with a rope to a post which is $15\text{ m}$ from a fence. If the length of the rope between the donkey and the post is $17\text{ m}$, …
• WAEC 2018 · Paper 2 · Q2In a right-angled triangle, $\sin X = \frac35$. Evaluate, leaving the answer as a fraction, $5(\cos X)^2 - 3$.
• WAEC 2018 · Paper 2 · Q8A container in the form of a cone resting on its vertex is full when 4.158 litres of water is poured into it. If the radius of its base is $21\text{ cm}$, …
• WAEC 2016 · Paper 2 · Q3In the diagram, $\angle RTS = 28^\circ$, $\angle VRM = 46^\circ$ and $MQ$ is a tangent to the circle $VRSTU$ at the point $R$. Find $\angle VUS$.
• WAEC 2016 · Paper 2 · Q9Solve: $3\log_{10}2 - 2\log_{10}3 = 1 + \log_{10}\left(\frac1x\right)$.
• WAEC 2014 · Paper 2 · Q5The diagram shows a right pyramid with a rectangular base $WXYZ$ and vertex $O$. If $|WX| = 8\text{ cm}$, $|ZW| = 6\text{ cm}$ and $|OX| = 13\text{ cm}$, …
• WAEC 2014 · Paper 2 · Q7A spherical tank of diameter $3\text{ m}$ is filled with water from a pipe of radius $30\text{ cm}$ at $0.2\text{ m}$ per second. Calculate, …
• WAEC 2013 · Paper 2 · Q12An open rectangular tank is made from a steel plate of area $1440\text{ m}^2$. Its length is twice its width. If the depth of the tank is $4\text{ m}$ …
• WAEC 2013 · Paper 2 · Q2The diagram shows a triangular prism with $|QR| = |MN| = |OP| = 10\text{ cm}$ and $|NR| = |QM| = 8\text{ cm}$. If $\angle RON = 90^\circ$ and …
• WAEC 2012 · Paper 2 · Q11In the diagram, $ABCD$ is a rectangular garden $(3n - 1)\text{ m}$ long and $(2n + 1)\text{ m}$ wide. A wire mesh $135\text{ m}$ long is used …
• WAEC 2012 · Paper 2 · Q11A sector of a circle with radius $20\text{ cm}$ has an area of $396\text{ cm}^2$. Calculate, correct to 1 decimal place, the: $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2012 · Paper 2 · Q13The internal diameter of a spherical bowl, half full of water, is $20\text{ cm}$. The content is poured into an empty cylindrical vessel with …
• WAEC 2011 · Paper 2 · Q10The total surface areas of two spheres are in the ratio $9 : 49$. If the radius of the smaller sphere is $12\text{ cm}$, find, correct to the …
• WAEC 2011 · Paper 2 · Q3A sector of a circle with radius $21\text{ cm}$ has an area of $280\text{ cm}^2$. Calculate, correct to 1 decimal place, the perimeter of the …
• WAEC 2010 · Paper 2 · Q4The diagram shows a cone with slant height $10.5\text{ cm}$. If the curved surface area of the cone is $115.5\text{ cm}^2$, calculate, correct …
• WAEC 2010 · Paper 2 · Q13The second, fourth and sixth terms of an Arithmetic Progression (A.P.) are $x - 1$, $x + 1$ and $7$ respectively. Find the: (i) common difference; …
• WAEC 2010 · Paper 2 · Q5The diagonals of a rhombus are $14\text{ cm}$ and $9\text{ cm}$. Calculate, correct to the nearest centimetre, the perimeter of the rhombus.
• WAEC 2010 · Paper 2 · Q9$OABCD$ is a right pyramid with a rectangular base $ABCD$. Its vertical height is $OG$. If $|AB| = 6\text{ cm}$, $|BC| = 8\text{ cm}$ and each …
• WAEC 2009 · Paper 2 · Q5In the diagram, $ABCDEF$ is a triangular prism. $\angle ABC = \angle DEF = 90^\circ$, $|AB| = 24$ cm, $|BC| = 7$ cm and $|CD| = 40$ cm.
• WAEC 2009 · Paper 2 · Q12Three positive numbers are in arithmetic progression (A.P.). The sum of the squares of the three numbers is 155, while the sum of the numbers …
• WAEC 2009 · Paper 2 · Q4A sector of angle $135^\circ$ is cut from a thin circular metal sheet of radius 40 cm. The sector is then folded, with its straight edges coinciding, …
• WAEC 2008 · Paper 2 · Q3A bucket is $12\text{ cm}$ in diameter at the bottom, $20\text{ cm}$ in diameter at the open end and $16\text{ cm}$ deep. If the bucket is filled …
• WAEC 2008 · Paper 2 · Q5A rectangular field is $l$ metres long and $b$ metres wide. Its perimeter is $280$ metres. If the length is two and a half times its breadth, …

Objective (76)

• WAEC 2025 · Paper 1 · Q23A cylindrical container, closed at both ends, has radius $7\text{ cm}$ and height $5\text{ cm}$. Find its total surface area. $\left[\pi = \frac{22}{7}\right]$
• WAEC 2025 · Paper 1 · Q24Find the total surface area of a solid circular cone with base radius $3\text{ cm}$ and slant height $4\text{ cm}$. $\left[\pi = \frac{22}{7}\right]$
• WAEC 2025 · Paper 1 · Q25A water tank of height $\frac12\text{ m}$ has a square base of side $1\frac12\text{ m}$. It is filled from a tanker holding 1500 litres. How …
• WAEC 2025 · Paper 1 · Q26A sphere has volume $k\text{ cm}^3$ and surface area $k\text{ cm}^2$. Calculate its diameter.
• WAEC 2025 · Paper 1 · Q27A cone is cut to leave a frustum with radii $12\text{ cm}$ and $6\text{ cm}$ and slant height $12\text{ cm}$. Calculate the slant height of the …
• WAEC 2025 · Paper 1 · Q29A $120^\circ$ sector of a circle of radius $21\text{ cm}$ is bent to form a cone. What is the base radius of the cone?
• WAEC 2025 · Paper 1 · Q30A cylindrical container, closed at both ends, has radius $7\text{ cm}$ and height $5\text{ cm}$. What is its volume? $\left[\pi = \frac{22}{7}\right]$
• NECO 2024 · Paper 1 · Q42The base radius and height of a cone are $4\text{ cm}$ and $6\text{ cm}$ respectively. Calculate its volume, correct to the nearest whole number.
• NECO 2024 · Paper 1 · Q44Calculate the surface area of a sphere with diameter $10.4\text{ cm}$, correct to 3 significant figures. [Take $\pi = 3.142$]
• NECO 2024 · Paper 1 · Q46The volume of a spherical ball is $114\text{ cm}^3$. Find its radius to the nearest whole number.
• WAEC 2024 · Paper 1 · Q14A cylindrical metallic barrel of height $2.5\text{ m}$ and radius $0.245\text{ m}$ is closed at one end. Find, correct to one decimal place, …
• WAEC 2024 · Paper 1 · Q26The length and breadth of a cuboid are $15\text{ cm}$ and $8\text{ cm}$ respectively. If the volume of the cuboid is $1{,}560\text{ cm}^3$, calculate …
• WAEC 2024 · Paper 1 · Q32A cone and a cylinder are of equal volume. The base radius of the cone is twice the radius of the cylinder. What is the ratio of the height of …
• NECO 2023 · Paper 1 · Q48If 1.109 litres of water is poured into a cylindrical container of base radius $4.2\text{ cm}$, find the level of water, correct to two significant …
• WAEC 2023 · Paper 1 · Q16The radius and height of a solid cylinder are $8\text{ cm}$ and $14\text{ cm}$ respectively. Find, correct to two decimal places, the total surface …
• WAEC 2023 · Paper 1 · Q18The radius of a sphere is $3\text{ cm}$. Find, in terms of $\pi$, its volume.
• WAEC 2023 · Paper 1 · Q23An empty cylindrical tank is $140\text{ cm}$ in diameter. If 200 litres of water are poured into the tank, calculate, correct to the nearest …
• NECO 2022 · Paper 1 · Q37Find the total surface area of the cuboid in the figure.
• NECO 2022 · Paper 1 · Q41Find the curved surface area of a cone of height 4 cm and base radius 3 cm, correct to 2 decimal places.
• NECO 2022 · Paper 1 · Q48A plastic spherical ball of radius 6 cm was melted and reshaped into a cube. Find the length of the cube, correct to the nearest centimetre.
• WAEC 2022 · Paper 1 · Q24A cylinder, open at one end, has a radius of $3.5\text{ cm}$ and height $8\text{ cm}$. Calculate the total surface area. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2022 · Paper 1 · Q40Find, correct to two decimal places, the volume of a sphere whose radius is $3\text{ cm}$. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2022 · Paper 1 · Q42Find the volume of a cone of radius $3.5\text{ cm}$ and vertical height $12\text{ cm}$. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2021 · Paper 1 · Q26A solid brass cube is melted and recast as a solid cone of height $h$ and base radius $r$. If the height of the cube is $h$, find $r$ in terms …
• WAEC 2021 · Paper 1 · Q37A cone has a base radius of $8\text{ cm}$ and height $11\text{ cm}$. Calculate, correct to two decimal places, the curved surface area. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2020 · Paper 1 · Q18A solid cuboid has a length of $7\text{ cm}$, a width of $5\text{ cm}$ and a height of $4\text{ cm}$. Calculate its total surface area.
• WAEC 2020 · Paper 1 · Q28The dimensions of a rectangular base of a right pyramid are $9\text{ cm}$ by $5\text{ cm}$. If the volume of the pyramid is $105\text{ cm}^3$, …
• WAEC 2020 · Paper 1 · Q38The diameter of a sphere is $12\text{ cm}$. Calculate, correct to the nearest $\text{cm}^3$, the volume of the sphere. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• WAEC 2019 · Paper 1 · Q21The total surface area of a solid cylinder is $165\text{ cm}^2$. If the base diameter is $7\text{ cm}$, calculate its height. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• JAMB 2018 · UTME · Q36The volume of a hemispherical bowl is $718\frac23\text{ cm}^3$. Find its radius. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• JAMB 2017 · UTME · Q38A right triangular prism is $8\text{ cm}$ long. Each triangular end is a right-angled triangle with hypotenuse $5\text{ cm}$ and one of the other …
• JAMB 2017 · UTME (set 2) · Q4The curved surface area of a cylinder $5\text{ cm}$ high is $110\text{ cm}^2$. Find the radius of its base. $\left[\pi = \frac{22}{7}\right]$
• JAMB 2017 · UTME (set 2) · Q40A cylindrical tank has a capacity of $3080\text{ m}^3$. What is the depth of the tank if the diameter of its base is $14\text{ m}$? $\left[\pi = \frac{22}{7}\right]$
• JAMB 2016 · UTME · Q15Find the radius of a sphere whose surface area is $154\text{ cm}^2$. $\left[\text{Take }\pi = \frac{22}{7}\right]$
• JAMB 2016 · UTME · Q23A cylindrical pipe $50\text{ m}$ long with radius $7\text{ m}$ has one end open. What is the total surface area of the pipe?
• JAMB 2015 · UTME · Q11A solid cylinder of radius $3\text{ cm}$ has a total surface area of $36\pi\text{ cm}^2$. Find its height.
• JAMB 2014 · UTME · Q30A cylindrical tank has a capacity of $6160\text{ m}^3$. What is the depth of the tank if the radius of its base is $28\text{ m}$? $\left[\pi = \frac{22}{7}\right]$
• JAMB 2012 · UTME · Q30Calculate the volume of a cuboid of length $0.76\text{ cm}$, breadth $2.6\text{ cm}$ and height $0.82\text{ cm}$.
• JAMB 2011 · UTME · Q30A solid metal cube of side $3\text{ cm}$ is placed in a rectangular tank of dimensions $3\text{ cm}$, $4\text{ cm}$ and $5\text{ cm}$. What volume …
• JAMB 2010 · UTME · Q33A cylindrical pipe $50\text{ m}$ long with radius $7\text{ m}$ has one end open. What is the total surface area of the pipe?
• JAMB 2004 · UME · Q38What is the rate of change of the volume $V$ of a hemisphere with respect to its radius $r$ when $r = 2$?
• JAMB 2002 · UME · Q16A solid hemisphere has radius 7 cm. Find its total surface area. $\left[\pi = \frac{22}{7}\right]$
• JAMB 2002 · UME · Q9A bucket is 12 cm in diameter at the top, 8 cm in diameter at the bottom and 4 cm deep. Calculate its volume.
• JAMB 2001 · UME · Q23A cylindrical tank has a capacity of $3080\text{ m}^3$. What is the depth of the tank if the diameter of its base is 14 m? $\left[\pi = \frac{22}{7}\right]$
• JAMB 2001 · UME · Q24A sector of a circle of radius 7.2 cm which subtends an angle of $300^\circ$ at the centre is used to form a cone. What is the radius of the …
• JAMB 2001 · UME · Q36Find the rate of change of the volume $V$ of a sphere with respect to its radius $r$ when $r = 1$.
• JAMB 2000 · UME · Q24A frustum of a pyramid with a square base has its upper and lower sections as squares of sides 2 m and 5 m respectively, and the distance between …
• JAMB 1999 · UME · Q24A frustum has top radius 3 cm, bottom radius 6 cm and vertical height 4 cm. Find its slant height $l$.
• JAMB 1999 · UME · Q38Find the volume of the solid generated when the area enclosed by $y = 0$, $y = 2x$ and $x = 3$ is rotated about the $x$-axis.
• JAMB 1998 · UME · Q30A cylindrical drum of diameter 56 cm contains 123.2 litres of oil when full. Find the height of the drum in centimetres. $\left[\pi = \frac{22}{7}\right]$
• JAMB 1998 · UME · Q6A market woman sells oil in cylindrical tins 10 cm deep and 6 cm in diameter at ₦15.00 each. If she bought a full cylindrical jug 18 cm deep …
• JAMB 1997 · UME · Q30A prism has a uniform cross-section in the shape of a trapezium with parallel sides 8 cm and 10 cm, 5 cm apart. If the prism is 11 cm long, find …
• JAMB 1997 · UME · Q31A cone with a sector angle of $45^\circ$ is cut out of a circle of radius $r$ cm. Find the base radius of the cone.
• JAMB 1995 · UME · Q33The base diameter of a cylinder is 14 cm and its height is 12 cm. Calculate the total surface area if the cylinder has both a base and a top. …
• JAMB 1994 · UME · Q29An open rectangular box is made of wood 2 cm thick. If the internal dimensions of the box are 50 cm long, 36 cm wide and 20 cm deep, what is …
• JAMB 1994 · UME · Q33In a frustum of a cone (an upturned cup shape), the top diameter is twice the bottom diameter. If the height of the frustum is $h$ cm, find the …
• JAMB 1993 · UME · Q36Find the radius of a sphere whose surface area is $154\text{ cm}^2$. $\left[\pi = \frac{22}{7}\right]$
• JAMB 1991 · UME · Q45Find the total surface area of a solid cylinder whose base radius is 4 cm and height is 5 cm.
• JAMB 1991 · UME · Q46Find the volume of the figure: a cone of height $x$ on top of a cylinder of height $y$, both of radius $a$.
• JAMB 1990 · UME · Q42A cylindrical metal pipe is 3 cm thick. If the internal radius of the pipe is 10 cm, find the volume of metal used in making 3 m of the pipe.
• JAMB 1990 · UME · Q43The heights of two circular cylinders are in the ratio $2 : 3$ and their base radii in the ratio $9 : 8$. What is the ratio of their volumes?
• JAMB 1989 · UME · Q44A cylindrical metal pipe 1 m long has an outer diameter of 7.2 cm and an inner diameter of 2.8 cm. Find the volume of metal used for the pipe.
• JAMB 1989 · UME · Q45$OXYZW$ is a pyramid with a square base such that $OX = OY = OZ = OW = 5$ cm and $XY = XW = YZ = WZ = 6$ cm. Find the height $OT$.
• JAMB 1989 · UME · Q9If the surface area of a sphere is increased by $44\%$, find the percentage increase in its diameter.
• JAMB 1988 · UME · Q44A metal pipe 10 cm long has an external diameter of 12 cm and a thickness of 1 cm. Find the volume of the metal used in making the pipe.
• JAMB 1988 · UME · Q45A solid consists of a hemisphere surmounted by a right circular cone, both of radius 3.0 cm; the cone has height 6.0 cm. Find the volume of the …
• JAMB 1987 · UME · Q41A room is 12 m long, 9 m wide and 8 m high. Find the cosine of the angle which a diagonal of the room makes with the floor.
• JAMB 1987 · UME · Q43The base of a pyramid is a square of side 8 cm. If its vertex is directly above the centre, find the height, given that each slant edge is $4\sqrt3$ …
• JAMB 1986 · UME · Q36A solid prism has the trapezium $PQRS$ as its uniform cross-section, with parallel vertical sides $PQ = 6$ m and $SR = 11$ m, $QR = 12$ m perpendicular …
• JAMB 1986 · UME · Q41Find the total surface area of a solid cone of radius $2\sqrt3$ cm and slant height $4\sqrt3$ cm.
• JAMB 1986 · UME · Q44An open rectangular box externally measures $4\text{ m} \times 3\text{ m} \times 4\text{ m}$ (length × width × height). Find the total cost of …
• JAMB 1985 · UME · Q31The ratio of the lengths of two similar rectangular blocks is $2 : 3$. If the volume of the larger block is $351\text{ cm}^3$, then the volume …
• JAMB 1985 · UME · Q39A solid sphere of radius 4 cm has a mass of 64 kg. What will be the mass of a shell of the same metal whose internal and external radii are 2 …
• JAMB 1985 · UME · Q41$GHIJKLMN$ is a cube of side $a$, and $HN$ is a diagonal joining opposite corners. Find the length of $HN$.
• JAMB 1984 · UME · Q27A cone is formed by bending a sector of a circle with an angle of $210^\circ$. Find the radius of the base of the cone if the diameter of the …
• JAMB 1983 · UME · Q20$PQRS$ is a desk of dimensions $2\text{ m} \times 0.8\text{ m}$ which is inclined at $30^\circ$ to the horizontal ($PQ = 2\text{ m}$ is horizontal …