Surface Areas and Volumes: Practice Worksheet

Surface area, volume, capacity, and displacement across distinct combined-solid models.

18 problems·35–45 min·★★★★☆

Print worksheet (PDF) →

Problem 1
surface-area·★★☆☆☆
Two cubes of edge 7 cm are joined face-to-face. Find the surface area of the resulting cuboid.
▶Answer
$490$ cm$^2$
▶Step-by-step solution
The new cuboid is $14\times7\times7$ cm.
Its surface area is $2(lw+lh+wh)=490$ cm$^2$.
Report an issue →
Problem 2
surface-area·★★★☆☆
A hollow cylindrical cup of radius 5 cm and height 12 cm has a hollow hemispherical bowl of the same radius fixed at one end. Find its inner surface area.
▶Answer
$170\pi$ cm$^2$
▶Step-by-step solution
Add only the inner curved surfaces.
$2\pi rh+2\pi r^2=170\pi$ cm$^2$.
Report an issue →
Problem 3
surface-area·★★★★☆
A cone of radius 6 cm and height 8 cm is joined to a hemisphere of the same radius. Find the exposed surface area.
▶Answer
$132\pi$ cm$^2$
▶Step-by-step solution
The cone's slant height is $\sqrt{6^2+8^2}=10$ cm.
Add $\pi rl+2\pi r^2=132\pi$ cm$^2$.
Report an issue →
Problem 4
surface-area·★★★★★
A hemisphere of the greatest possible radius is mounted on a cube of edge 12 cm. Find the exposed surface area.
▶Answer
$864+36\pi$ cm$^2$
▶Step-by-step solution
The greatest radius is 6 cm.
Subtract the covered base circle from the cube and add the hemisphere's curved area, leaving a net $+\pi r^2$.
Report an issue →
Problem 5
surface-area·★★☆☆☆
A hemispherical depression of greatest possible radius is cut into one face of a cube of edge 12 cm. Find the remaining surface area.
▶Answer
$864+36\pi$ cm$^2$
▶Step-by-step solution
Remove the circular opening and add the curved surface of the depression.
The net change is $-\pi r^2+2\pi r^2=+\pi r^2$.
Report an issue →
Problem 6
surface-area·★★★☆☆
A capsule has radius 3 cm and total length 14 cm, including two hemispherical ends. Find its surface area.
▶Answer
$84\pi$ cm$^2$
▶Step-by-step solution
The cylindrical length is $14-2(3)=8$ cm.
Add $2\pi rh+4\pi r^2=84\pi$ cm$^2$.
Report an issue →
Problem 7
surface-area·★★★★☆
An open-bottom tent consists of a cylinder of radius 12 m and height 9 m topped by a cone of height 16 m. Find the canvas area.
▶Answer
$456\pi$ m$^2$
▶Step-by-step solution
Cone slant height is 20 m.
Canvas area $=2\pi rh+\pi rl=456\pi$ m$^2$.
Report an issue →
Problem 8
surface-area·★★★★★
A cone of radius 3 cm and depth 4 cm is hollowed from one end of a cylinder with the same radius and height 4 cm. Find the total exposed surface area.
▶Answer
$48\pi$ cm$^2$
▶Step-by-step solution
Add the cylinder's curved surface, the untouched circular base, and the cavity's curved surface.
The cone slant height is 5 cm, giving $48\pi$ cm$^2$.
Report an issue →
Problem 9
surface-area·★★☆☆☆
A hemisphere is scooped from each flat end of a cylinder of radius 4 cm and height 13 cm. Find the exposed surface area.
▶Answer
$168\pi$ cm$^2$
▶Step-by-step solution
The exposed parts are the cylinder's curved surface and two hemispherical cavities.
$2\pi rh+4\pi r^2=168\pi$ cm$^2$.
Report an issue →
Problem 10
volume·★★★☆☆
A cone of radius 9 cm and height 10 cm stands on a hemisphere of the same radius. Find its volume.
▶Answer
$\frac{2268}3\pi$ cm$^3$
▶Step-by-step solution
Add cone and hemisphere volumes.
$\frac13\pi r^2h+\frac23\pi r^3=\frac{2268}3\pi$ cm$^3$.
Report an issue →
Problem 11
volume·★★★★☆
A cylinder of radius 9 cm and length 5 cm has a cone of height 9 cm attached at each end. Find the total volume.
▶Answer
$\frac{2673}3\pi$ cm$^3$
▶Step-by-step solution
Add one cylinder and two cone volumes.
$\pi r^2h+2(\frac13\pi r^2H)=\frac{2673}3\pi$ cm$^3$.
Report an issue →
Problem 12
volume·★★★★★
A capsule has a cylindrical middle of radius 6 cm and length 11 cm, plus two hemispherical ends. Find its volume.
▶Answer
$\frac{2052}3\pi$ cm$^3$
▶Step-by-step solution
The hemispheres form one sphere.
$\pi r^2h+\frac43\pi r^3=\frac{2052}3\pi$ cm$^3$.
Report an issue →
Problem 13
volume·★★☆☆☆
A 20 cm by 10 cm by 3 cm wooden block has 6 conical depressions, each radius 1 cm and depth 3 cm. Find the remaining volume.
▶Answer
$600-6\pi$ cm$^3$
▶Step-by-step solution
Block volume is 600 cm$^3$ and each cone has volume $\pi$ cm$^3$.
Subtract the volume of all depressions.
Report an issue →
Problem 14
volume·★★★☆☆
Small spheres of radius 2 cm are dropped into a full inverted cone of radius 6 cm and height 16 cm. If their combined volume equals the displaced liquid, how many spheres are needed?
▶Answer
$18$
▶Step-by-step solution
Equate the cone volume to the total sphere volume.
$n(4\pi(2)^3/3)=\pi(6)^2(16)/3$, so $n=18$.
Report an issue →
Problem 15
volume·★★★★☆
A stepped solid consists of cylinders $(r,h)=(4,8)$ cm and $(2,5)$ cm. Its material density is 3 g/cm$^3$. Find its mass.
▶Answer
$444\pi$ g
▶Step-by-step solution
Total volume is $\pi(4^28+2^25)=148\pi$ cm$^3$.
Multiply by density.
Report an issue →
Problem 16
volume·★★★★★
A cylinder of radius 6 cm and height 15 cm is full of water. A solid cone of height 8 cm joined to a hemisphere, both radius 3 cm, is immersed completely. Find the water remaining.
▶Answer
$498\pi$ cm$^3$
▶Step-by-step solution
Cylinder volume is $540\pi$ cm$^3$.
The immersed solid displaces $42\pi$ cm$^3$; subtract.
Report an issue →
Problem 17
volume·★★☆☆☆
A vessel consists of a spherical body of radius 6 cm and a cylindrical neck of radius 2 cm and height 5 cm. Find its capacity.
▶Answer
$\frac{924}3\pi$ cm$^3$
▶Step-by-step solution
Add the sphere and neck volumes.
$4\pi r^3/3+\pi R^2h=924\pi/3$ cm$^3$.
Report an issue →
Problem 18
volume·★★★☆☆
A cone of radius 9 cm and height 12 cm is joined to a hemisphere and placed snugly inside a cylinder of radius 9 cm and height 21 cm. Find the volume inside the cylinder but outside the solid.
▶Answer
$891\pi$ cm$^3$
▶Step-by-step solution
Subtract the cone-plus-hemisphere volume from the circumscribing cylinder.
The difference is $(1701-810)\pi=891\pi$ cm$^3$.
Report an issue →