The surface area and the volume of a box

Geometry, grade 7 Lesson 16 of 18Learning

The surface area and the volume of a box

The surface area

$\displaystyle a$ – the length
$\displaystyle b$ – the width
$\displaystyle c$ – the height
$\displaystyle S$ – the surface area

The surface area of a box is the sum of the areas of its six faces – the area of its net. It is how much paper it takes to wrap the box without overlaps. The net has three pairs of congruent rectangles:

The base and the lid: each of area $\displaystyle ab$ .
The front and back faces: $\displaystyle ac$ .
The two side faces: $\displaystyle bc$ .

Add the three different areas, and multiply by 2:

\displaystyle S

\displaystyle =

\displaystyle 2(ab+bc+ac)

A cube has six faces that are equal squares, each of area $\displaystyle a^2$ :

\displaystyle S

\displaystyle =

\displaystyle 6a^2

Finding the surface area

A present is in a box 20 cm long, 15 cm wide and 10 cm high.
How much wrapping paper does it take, at least?

The base and the lid: $\displaystyle {20\cdot15}={300}$ cm² each.
The front and back faces: $\displaystyle {20\cdot10}={200}$ cm² each.
The sides: $\displaystyle {15\cdot10}={150}$ cm² each.
The surface area: $\displaystyle {2\cdot(300+200+150)}={1300}$ cm².

It takes at least 1300 cm² of paper – in fact a little more, for the folds.

Exercise 1

Find the surface area, in cm².

A box 6 cm long, 4 cm wide and 3 cm high

/ – 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

A cube whose edge is 7 cm

/ – 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

The volume

\displaystyle V

– the volume

The volume of a box is how much room it takes up: how many unit cubes – cubes of 1 cm by 1 cm by 1 cm – fill it. A unit cube’s volume is one cm³. Fill a box 4 cm long, 3 cm wide and 2 cm high:

One layer covers the base: 3 rows of 4 cubes, $\displaystyle {4\cdot3}={12}$ cubes – like the base’s area, 12 cm².
The height is 2 cm, so there are two layers: $\displaystyle {12\cdot2}={24}$ cubes.
The volume is 24 cm³.

So it is in every box: the volume is the base’s area times the height – length times width times height:

\displaystyle V

\displaystyle =

\displaystyle a\cdot b\cdot c

In a cube the length, the width and the height are equal, so the volume is $\displaystyle a\cdot a\cdot a$ , written $\displaystyle a^{3}$ for short:

\displaystyle V

\displaystyle =

\displaystyle a^3

Units of volume

cm³ – the volume of a cube whose edge is 1 cm.
A litre – the volume of a cube whose edge is 10 cm: $\displaystyle {10\cdot10\cdot10}={1000}$ cm³.
m³ – the volume of a cube whose edge is 1 m: 1000 litres.

All the measurements must be in the same unit: if one is in metres and the rest in centimetres, convert first.

Finding the volume

An aquarium is 50 cm long, 30 cm wide and 40 cm high.
How many litres of water does it hold?

The volume: $\displaystyle {50\cdot30\cdot40}={60{,}000}$ cm³.
A litre is 1000 cm³: $\displaystyle {60{,}000:1000}={60}$ litres.

Exercise 2

Find:

the volume of a box 5 cm long, 4 cm wide and 3 cm high, in cm³

/ – 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

the volume of a cube whose edge is 6 cm, in cm³

/ – 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

how many litres a storage box 40 cm long, 25 cm wide and 20 cm high holds

/ – 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

A missing edge from the volume

Knowing the volume and the two dimensions of the base, the height is found by dividing.

Example. A box’s volume is 60 cm³. It is 4 cm long and 3 cm wide. How high is it?

One layer of cubes holds $\displaystyle {4\cdot3}={12}$ cubes.
There are 60 cubes, so there are $\displaystyle {60:12}={5}$ layers.
The height is 5 cm: $\displaystyle {\frac{60}{4\cdot3}}={5}$ .

Exercise 3

An aquarium holds 72 litres.
It is 60 cm long and 30 cm wide.
How high is it, in cm?

/ – 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 room is 4 m long, 3 m wide and 2.5 m high.
Its four walls and its ceiling are painted, but not its floor.
Leave out the door and the window.

How many m² are painted?

/ – 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

A tin of paint is enough for 10 m². How many tins must be bought? Explain.

Exercise 5

A cube’s volume is 125 cm³.

How long is its edge? Explain how you found it.

/ – 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

What is its surface area, in cm²?

/ – 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 6

One cube’s edge is 2 cm, and a second cube’s edge is 4 cm – double.
How many times larger is the second cube’s surface area?
And how many times larger is its volume?
Work them out and explain.

Surface area

/ – 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

Volume

/ – 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: geometry, grade 7

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Angles

The sum of angles – the ray $\displaystyle BD$ runs inside $\displaystyle \angle ABC$
$\displaystyle \angle ABC$ $\displaystyle =$ $\displaystyle \angle ABD+\angle DBC$
The difference of angles
$\displaystyle \angle DBC$ $\displaystyle =$ $\displaystyle \angle ABC-\angle ABD$
The bisector $\displaystyle BD$ cuts $\displaystyle \angle ABC$ into two equal angles
$\displaystyle \angle ABD=\angle DBC=\frac{\angle ABC}{2}$
Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
The angles around a point
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$

The angles of a triangle and a quadrilateral

The angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
The two acute angles of a right triangle
$\displaystyle \alpha+\beta=90^{\circ}$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$
The angles of a quadrilateral
$\displaystyle \angle A+\angle B+\angle C+\angle D=360^{\circ}$

The perimeter and area of a rectangle and a square

The area of a rectangle – $\displaystyle a$ , $\displaystyle b$ two sides that meet at a vertex
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The perimeter of a rectangle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\cdot(a+b)$
The area of a square – $\displaystyle a$ the side
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^2$
The perimeter of a square
$\displaystyle P$ $\displaystyle =$ $\displaystyle 4\cdot a$

The areas of polygons

The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot b}{2}$ $\displaystyle \angle C=90^{\circ}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$

The box and the cube

The surface area of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle 2(ab+bc+ac)$
The surface area of a cube – $\displaystyle a$ the edge
$\displaystyle S$ $\displaystyle =$ $\displaystyle 6a^2$
The volume of a box
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\cdot b\cdot c$
The volume of a cube
$\displaystyle V$ $\displaystyle =$ $\displaystyle a^3$