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Useful Formulae

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Useful Formulae Revision

Useful Formulae

This page contains useful formulae you may need in your exams. You should already be familiar with these equations from GCSE maths.

Area Calculations

Area is the amount of space that a 2D shape takes up.

Squares and Rectangles

$A=xy$

$x=$ length
$y=$ height

Triangles

$A=\dfrac{1}{2}bh$

$b=$ base
$h=$ height

Parallelograms

$A=bh$

$b=$ base
$h=$ perpendicular height

Trapezia

$A=\dfrac{1}{2}(a+b)h$

$a=$ width of the short side
$b=$ width of the long side
$h=$ height

Circles

You should know what the following words mean when talking about circles:

Radius – the distance between the entre of the circle and any point on the edge of the circle.
Diameter – the distance between two opposite edges of the circles, passing through the centre.
Circumference – the distance around the edge of the circle.

Area of a Circle

$A=\pi r^2$

$r=$ radius

Circumference

$C=2 \pi r = \pi d$

$r=$ radius
$d=$ diameter $=2r$

Volume Calculations

Volume is the amount of space a 3D shape takes up.

Cuboid

$V=xyz$

$x=$ width
$y=$ height
$z=$ length

Cylinders

$V=\pi r^2 l$

$r=$ radius
$l=$ length

Triangular Prism

$V=\dfrac{1}{2}bhl$

$b=$ base
$h=$ height
$l=$ length

Sphere

$V=\dfrac{4}{3} \pi r^3$

$r=$ radius

Example 1: Calculating the Area of an Inhibition Zone.

A biologist is performing an experiment on a new antibiotic drug. They measure a circular inhibition zone and find the diameter is $3.6 \text{ mm}$ .

Calculate the area of this inhibition zone.

[3 marks]

r=\dfrac{d}{2} = \dfrac{\textcolor{1c63ba}{3.6 \text{ mm}}}{2} = \textcolor{aa57ff}{\bold{1.8} \textbf{ mm}} \\ \bold{A=\pi r^2} = \pi \times (\textcolor{aa57ff}{1.8})^2 = \bold{10.2} \textbf{ mm}^2

Example 2: Calculating the Volume of Water Displaced

A wooden block with dimensions $\textcolor{f95d27}{10 \text{ cm}} \times \textcolor{f43364}{6 \text{ cm}} \times \textcolor{10a6f3}{3.5 \text{ cm}}$ is dropped into a container of water. Calculate the volume of water displaced by the wooden block.

[2 marks]

\text{volume of water displaced} \\ \begin{aligned} &= \text{volume of the block} \\ &= xyz \\ &= \textcolor{f95d27}{10} \times \textcolor{f43364}{6} \times \textcolor{10a6f3}{3.5} \\ &= \bold{210} \textbf{ cm}\bold{^3} \end{aligned}

Useful Formulae Example Questions

Question 1: Calculate the area of a square with sides of length $6 \text{ cm}$ .

[2 marks]

\begin{aligned} \bold{A} &= \bold{xy} \\ &= 6 \times 6 \\ &= \bold{36} \textbf{ cm}\bold{^2} \end{aligned}

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Question 2: Calculate the volume of a sphere with radius $40 \text{ mm}$ .

[2 marks]

\begin{aligned} \bold{V} &= \bold{\dfrac{4}{3} \pi r^3} \\ &= \dfrac{4}{3} \times \pi \times 40^3 \\ &= \bold{268 \, 083} \textbf{ mm}\bold{^3} \end{aligned}

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Question 3: Calculate the circumference of a circle with a diameter of $2 \text{ m}$ .

[2 marks]

\begin{aligned} \bold{C} &= \bold{\pi d} \\ &= \pi \times 2 \\ &= \bold{6.28} \textbf{ m} \end{aligned}

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