Mensuration (Area & Volume) Practice Questions with Detailed Solutions – Level 3

Mensuration Level 3 Practice Quiz: Area, Volume and Recasting of Solids

Mensuration (Area & Volume) Level 3 is a practice set on Aptitude360 with harder problems on 3D solids: frustums, cones folded from sectors, hollow spheres recast into cones, composite shapes and water-level rise in vessels. Every question has a worked solution, so you can check each step of your method.

Topics covered in this Level 3 mensuration set

• Volume and curved surface area of frustum-shaped buckets and containers
• Melting and recasting: spheres into cones, blocks into shots, spheres drawn into wire
• Rise in water level when spheres, cones or marbles are immersed in vessels
• Composite solids such as capsules and cone-on-hemisphere toys
• Percentage change in volume or surface area when dimensions change
• Cuboid face areas, longest diagonal and lining cost of an open tank

Key formulas for solids and frustums

• Volume of a frustum: \( V=\dfrac{1}{3}\pi h(R^{2}+r^{2}+Rr) \)
• Curved surface of a frustum: \( \pi(R+r)l,\ l=\sqrt{h^{2}+(R-r)^{2}} \)
• Volume of a cone: \( V=\dfrac{1}{3}\pi r^{2}h \)
• Sphere: \( V=\dfrac{4}{3}\pi r^{3},\ S=4\pi r^{2} \)
• Total surface of a cylinder: \( 2\pi r(r+h) \)
• Diagonal of a cuboid: \( d=\sqrt{l^{2}+b^{2}+h^{2}} \)

Revision tips for harder mensuration problems

• In recasting questions, equate total volume before and after melting; surface area is not conserved.
• For water-rise problems, the immersed volume equals the vessel's base area times the rise in level.
• Volume scales with the cube of a length factor and surface area with its square.
• When a sector is folded into a cone, its radius becomes the slant height and its arc becomes the base circumference.

Who this mensuration practice suits

This Level 3 set suits quant aspirants who know the basic area and volume formulas and want harder, multi-step problems on 3D solids.