__Geometry__

**Basics**

Circle:

Area = π r^{2}

Circumpherence =
2 π r

__Areas of common shapes__

Rectangle: b*h

Triangle:
^{1}/_{2} b*h

*note that the area of a right angled triangle is half that of
the smallest rectangle that can be drawn around it, however this formula works
for ALL TRIANGLES!*

**Trapeziums, parallelograms and the rhombus!**

**Trapezium **(any shape with at least
one pair of parallel sides): ^{1}/_{2}
(a+b) h

*the formula basically says "take the average length of the
two parallel sides and then multiply it by the perpendicular height"*

**Parallelogram** (a __special
trapezium__ that has not only one pair of parallel sides, but two!)

Of course, it is still a Trapezium because it has *"at least
one pair of parallel sides" *so we can still use* *
^{1}/_{2} (a+b) h *
*

In addition, for parallelograms you can also use
^{1}/_{2} (diag a)*(diag b) and
base*h *NB- you can take any side as the base so long as
you take "h" as the height perpendicular (at a right angle) to this base.*

**Rhombus** (a __special parallelogram__
that has not only *"two pairs of parallel sides"*, but both sets are
exactly the same length)

A Rhombus is a special parallelogram and a very special
trapezium

Of course, it is still a Trapezium because it has *"at least
one pair of parallel sides" *so we can still use* *
^{1}/_{2} (a+b) h *
*

Of course, it is still a Parallelogram because it has *"two
pairs of parallel sides" *so we can still use* *
^{1}/_{2} (diag a)*(diag b) and
base*h

In addition, for rhombuses you can also use
^{1}/_{2} (diag a)^{2}
since (diag a)*(diag b) is the same as (diag a)*(diag a), which is the same as
(diag a)^{2} if (diag a) = (diag b). This is true in a rhombus but NOT
in a parallelogram and certainly not in a trapezium!

**Arc lengths**

An arc is a part of the Circumpherence of a circle (see circle
diag above and below)

The formula for the Circumpherence of a circle is
2 π r so
the length of say, half the Circumpherence is obviously
1/2 (2 π r ) and the length of an arc which forms one quarter of a
circle is 1/4 (2 π r )!
So, if we look at the example above, the portion of the circle that the arc
covers is 65 parts out of a possible 360, hence the formula is
^{65}/_{360} (2 π r ).

The general formula for finding an arc length is
^{Angle}/_{360} (2 π r )

**Curved surface areas and volumes ***NB) You
need to calculate volumes from lengths or rearrange equations to find lengths
from volumes*

Prism volume (any shape where the end will look the same no
matter how far up you slice it): Area of
base * height

Cylinder Volume: π r^{2}
h

*the area of a circle is *
π r^{2}.
We know that the volume of any prism is (area of base)*h so the volume of a
cylinder is π r^{2} * h

Cylinder Curved surface area: Circumpherence of base circle *
height = (2 π r) h

*remember, the total SA includes (curved SA) +
(the top and bottom flat circles) so is *
(2 π
r) h + 2(π r^{2})

Sphere Volume: ^{ 4}/_{3}
π r^{3}

Sphere Curved SA: 4π r^{2}

Pyramid (any shape on the ground with all its edges coming to a point)
Volume: ^{1}/_{3} floor area * height

Cone (a special pyramid with a circle as the base shape) Volume:
^{1}/_{3} floor area * height = ^{1}/_{3}
(π r^{2})
* height

Cone Curved SA:
π r L *L is the "slant height"
(see diag)*

*Cones*

**frustrum**

A **right cone ** is one where the edges all meet directly above the
centre of the base circle

A **frustrum **is a cone with the top part cut off making a 'lampshade'
shape

**Trigonometry**

__
SOHCAHTOA__

This is an
acronym for...

__A
Triangle__

__
__

__Using
the equations__

__
__

1)
To find lengths…

2)
To find angles…

*Remember the brackets*

__Pythagoras' theorem__

**A**^{2} = B^{2} + C^{2
}* where A is the
hypotenuse*

3^{2 }+ 4^{2} = x^{2}

9 + 16 = x^{2}

x = √(9+16)

x = √25

x = 5

*It is unusual to get a whole number answer to a Pythagoras question so you
should be familiar with the famous "3-4-5" triangle*

Also remember about multiples of this such as the "6-8-10" and the "30-40-50"
triangles.