Cuboid, cube, and cylinder

The three simplest three-dimensional shapes are the cuboid, the cube, and the cylinder. The first two are rectangular boxes; the third has a circular cross-section. All three have the property that you can lay them flat and look at the surface as a sum of polygons (cuboid, cube) or polygons plus circles (cylinder).

Cuboid

A cuboid has six rectangular faces, joined to make a box. It has three independent dimensions: length $\ell$, breadth $b$, and height $h$.

• Volume: $V = \ell b h$. Just the product of the three side-lengths.
• Total surface area: the cuboid has 3 pairs of opposite faces, each pair identical. Their areas are $\ell b, bh, h\ell$. Summing twice: $S = 2(\ell b + bh + h\ell).$
• Lateral (curved) surface area: the area of the four sides excluding the top and bottom. This is the perimeter of the base ($2\ell + 2b$) times the height $h$: $L = (2\ell + 2b) h = 2h(\ell + b).$
• Diagonal: the longest distance inside a cuboid, between two opposite corners, is $\sqrt{\ell^2 + b^2 + h^2}$.

Cube

A cube is the special case of a cuboid with $\ell = b = h = a$. Plugging in:

• Volume: $V = a^3$.
• Total surface area: $S = 6 a^2$ (six identical square faces).
• Lateral surface area: $L = 4 a^2$ (four sides, excluding top and bottom).
• Diagonal: $d = a\sqrt 3$.

Cylinder (right circular)

A cylinder has a circular cross-section and a flat top and bottom. The two important measurements are the radius $r$ and the height $h$.

Imagine peeling the curved surface and unrolling it: it becomes a rectangle of height $h$ and width $2\pi r$ (the circumference of the base). The area of this rectangle is the curved surface area (CSA):

$\text{CSA} = 2\pi r h.$

The two flat circular ends each have area $\pi r^2$. So the total surface area (TSA) is

$\text{TSA} = 2\pi r h + 2 \cdot \pi r^2 = 2\pi r(h + r).$

The volume of a cylinder is the area of the base times the height:

$V = \pi r^2 h.$

A hollow cylinder (a pipe) of inner radius $r_1$ and outer radius $r_2$, height $h$:

• Volume of material: $\pi h (r_2^2 - r_1^2)$.
• Total surface area (outer + inner + two annular ends): $2\pi h(r_1 + r_2) + 2\pi(r_2^2 - r_1^2)$.

Worked examples

Example 1. A cuboid has sides $5, 4, 3$ cm. Find its volume and surface area.

$V = 5 \cdot 4 \cdot 3 = 60$ cm$^3$.

$S = 2(20 + 12 + 15) = 2 \cdot 47 = 94$ cm$^2$.

Example 2. A cube of side $7$ cm. Find $V$ and $S$.

$V = 343$ cm$^3$. $S = 6 \cdot 49 = 294$ cm$^2$.

Example 3. A cylinder of radius $7$ and height $20$. Find CSA, TSA, and volume.

CSA $= 2 \cdot 22/7 \cdot 7 \cdot 20 = 880$ cm$^2$.

TSA $= 880 + 2 \cdot 22/7 \cdot 49 = 880 + 308 = 1188$ cm$^2$.

$V = 22/7 \cdot 49 \cdot 20 = 22 \cdot 7 \cdot 20 = 3080$ cm$^3$.

Example 4. A cube of side $4$ cm is melted and recast into smaller cubes of side $1$ cm. How many smaller cubes are formed?

$V_{\text{large}} = 64$. $V_{\text{small}} = 1$. Number $= 64$.

Example 5. A cylindrical tank of diameter $14$ m and depth $5$ m. How many litres can it hold? ($1$ m$^3 = 1000$ L.)

$r = 7$, $V = 22/7 \cdot 49 \cdot 5 = 770$ m$^3 = 770\,000$ L.

Example 6. A hollow cylindrical pipe is $21$ cm long, with outer diameter $4$ cm and inner diameter $3$ cm. Find the volume of the material.

$r_2 = 2$, $r_1 = 1.5$. $V = \pi h (r_2^2 - r_1^2) = 22/7 \cdot 21 \cdot (4 - 2.25) = 22 \cdot 3 \cdot 1.75 = 115.5$ cm$^3$.

Example 7. A rectangular room is $10$ m by $6$ m by $4$ m. Find the cost of whitewashing the four walls and ceiling at Rs. $8$ per m$^2$.

Lateral area + ceiling = $2 \cdot 4 \cdot (10 + 6) + 10 \cdot 6 = 128 + 60 = 188$ m$^2$. Cost $= 188 \cdot 8 =$ Rs. $1504$.

Try it yourself

1. Find the volume and surface area of a cuboid of sides $8, 6, 5$ cm.
2. Find the volume and surface area of a cube of side $10$ cm.
3. Find the diagonal of a cuboid of sides $3, 4, 12$ cm.
4. A cylinder has radius $7$ cm and height $10$ cm. Find the curved surface area and volume.
5. A cylindrical can is open at the top, with radius $7$ cm and height $14$ cm. Find the total surface area (excluding the top).
6. The volume of a cube is $216$ cm$^3$. Find its surface area.
7. A cylinder has volume $1540$ cm$^3$ and radius $7$ cm. Find the height.
8. A cuboid of dimensions $40 \times 30 \times 20$ cm is to be wrapped. Find the wrapping paper needed (surface area).
9. A solid cube of side $12$ cm is melted to form $216$ smaller equal cubes. Find the side of a small cube.
10. A hollow cylinder has outer radius $5$ cm, inner radius $4$ cm, and height $14$ cm. Find the volume of the material.
11. Two cubes of side $7$ cm are joined end to end. Find the surface area of the resulting cuboid.
12. A rectangular tank holds $1080$ litres of water. If its base is $1.5$ m by $1$ m, find the height of the water.

Pitfalls / Insight

(1) Volume has units of length cubed (cm$^3$, m$^3$, etc.). Surface area has units of length squared (cm$^2$, m$^2$, etc.). Verify your final unit.

(2) When two cubes are joined to make a cuboid, the surface area is not twice the cube's surface area , two faces are now internal and lost. The new surface area is $2 \cdot 6 a^2 - 2 \cdot a^2 = 10 a^2$.

(3) "Open at the top" cylinder: subtract one $\pi r^2$ from the total surface area.

(4) Convert units before computing volumes, especially when mixing m and cm. $1$ m$^3 = 10^6$ cm$^3$, not $10^3$.