The box and the cube: volume and surface area

The plane and space, grade 12 (3 units) Lesson 2 of 15Learning

The box and the cube: volume and surface area

The volume of a box

The volume of a solid is how much space it takes up – how much water, sand or air fits in it.

Volume is measured in cubic units: a cubic centimetre (cm³) is the volume of a cube whose edge is 1 cm, and a cubic metre (m³) that of a cube whose edge is 1 m.
A box with dimensions $\displaystyle a$ , $\displaystyle b$ and $\displaystyle c$ has $\displaystyle c$ layers, and each layer holds $\displaystyle a\cdot b$ unit cubes.

\displaystyle V

\displaystyle =

\displaystyle abc

In a cube all the edges are equal, $\displaystyle a$ :

\displaystyle V

\displaystyle =

\displaystyle a^3

The surface area

The surface area of a solid is the sum of the areas of all its faces: how much cardboard a package needs, how much paint the walls, how many tiles a pool.

A box has three pairs of equal faces – $\displaystyle ab$ , $\displaystyle ac$ and $\displaystyle bc$ :

\displaystyle A

\displaystyle =

\displaystyle 2(ab+ac+bc)

A cube has six equal square faces:

\displaystyle A

\displaystyle =

\displaystyle 6a^2

In a real question, only the faces that are really covered count: a pool has no ceiling, and a room’s floor is not painted.

Units of volume and area

Before calculating, write all the lengths in the same unit.

1 m³ = 1000 litres: a metre is 10 decimetres, so a cubic metre holds $\displaystyle {10\cdot10\cdot10}={1000}$ cubes of a decimetre – and each of them is a litre.
1 litre = 1000 cm³: a cube whose edge is 10 cm, $\displaystyle {10\cdot10\cdot10}={1000}$ . One cm³ is a millilitre (ml).
1 m² = 10,000 cm²: a square of 100 cm by 100 cm.

Example: a swimming pool

A swimming pool is a box: it is 25 m long, 12 m wide and 1.5 m deep.
A cubic metre of water costs 8 ₪.
How much does filling the pool cost, and how many m² of tiles cover its floor and walls?

The volume: $\displaystyle {V}={25\cdot12\cdot1.5}={450}$ m³, that is, 450,000 litres.
The cost: $\displaystyle {450\cdot8}={3600}$ ₪.
The tiles: the floor ( $\displaystyle 25\cdot12$ ), two long walls ( $\displaystyle 25\cdot1.5$ ) and two short walls ( $\displaystyle 12\cdot1.5$ ), with no ceiling – $\displaystyle {300+2\cdot37.5+2\cdot18}={411}$ m².

Exercise 1

An aquarium is a box: 80 cm by 40 cm, and 50 cm high.
How many litres of water fit in it?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A room’s floor is a rectangle of 4 m by 3.5 m, and the room is 2.6 m high.
The four walls and the ceiling are painted.
The room has a door of 0.9 m by 2 m and a window of 1.2 m by 1.2 m, which are not painted.
A litre of paint covers 10 m², and the room gets two coats.

a

How many m² are painted?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How many litres of paint do the two coats need?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The paint is sold in 4-litre tins. How many tins must be bought?

Exercise 3

A factory packs presents in cardboard boxes that are cubes. A box’s edge is 15 cm.

a

How many cm² of cardboard does one box have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How many m² of cardboard do 200 boxes need?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A water tank on a roof is a box of 2 m by 1.5 m, and 1 m high.
A family uses 450 litres of water a day.
How many days does a full tank last? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 12 (3 units)

The box and the cube

The volume of a box – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle V$ $\displaystyle =$ $\displaystyle abc$
The surface area of a box – three pairs of equal faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2(ab+ac+bc)$
The volume of a cube – $\displaystyle a$ the edge
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$
The surface area of a cube – six square faces
$\displaystyle A$ $\displaystyle =$ $\displaystyle 6a^2$

The prism

The volume of a prism – $\displaystyle B$ the base area, $\displaystyle h$ the height (the distance between the bases)
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$
The lateral area – $\displaystyle P$ the perimeter of the base
$\displaystyle L$ $\displaystyle =$ $\displaystyle P\cdot h$
The surface area – the two bases and the lateral surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2B+L$

The cylinder

The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2h$
The curved surface area – a rectangle as long as the base’s circumference
$\displaystyle L$ $\displaystyle =$ $\displaystyle 2\pi rh$
The surface area – the two bases and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle 2\pi r^2+2\pi rh$

The pyramid and the cone

The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$
In a square pyramid – $\displaystyle k$ the height of a side face, $\displaystyle h$ the pyramid’s height, $\displaystyle a$ the side of the base
$\displaystyle k^2$ $\displaystyle =$ $\displaystyle h^2+\left(\frac{a}{2}\right)^2$
The surface area of a square pyramid – the base and four triangles
$\displaystyle A$ $\displaystyle =$ $\displaystyle a^2+2ak$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2h}{3}$
The slant height $\displaystyle l$ – the hypotenuse of a triangle whose legs are the radius $\displaystyle r$ and the height $\displaystyle h$
$\displaystyle l^2$ $\displaystyle =$ $\displaystyle r^2+h^2$
The curved surface area of a cone
$\displaystyle L$ $\displaystyle =$ $\displaystyle \pi rl$
The surface area of a cone – the base and the curved surface
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2+\pi rl$

The sphere

The volume of a sphere – $\displaystyle r$ the radius
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4\pi r^3}{3}$
The surface area of a sphere
$\displaystyle A$ $\displaystyle =$ $\displaystyle 4\pi r^2$

Diagonals and distances

The diagonal of a box $\displaystyle d$ – $\displaystyle a$ , $\displaystyle b$ , $\displaystyle c$ its three dimensions
$\displaystyle d^2$ $\displaystyle =$ $\displaystyle a^2+b^2+c^2$
The diagonal of a cube – $\displaystyle a$ the edge
$\displaystyle d$ $\displaystyle =$ $\displaystyle a\sqrt{3}$
In a square pyramid – $\displaystyle s$ a side edge, $\displaystyle h$ the height, $\displaystyle a$ the side of the base
$\displaystyle s^2$ $\displaystyle =$ $\displaystyle h^2+\frac{a^2}{2}$

Trigonometry in a right-angled triangle

$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle, when both legs are known
$\displaystyle \alpha=\tan^{-1}\left(\frac{a}{b}\right)$

The shortest path on the surface of a box

A path from a vertex to the vertex farthest from it, across two faces: unfolded, the diagonal of a rectangle with sides $\displaystyle a+b$ and $\displaystyle c$
$\displaystyle p^2$ $\displaystyle =$ $\displaystyle (a+b)^2+c^2$