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Geometry

Basics

Circle:

Area = π r2

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 r2 h

the area of a circle is π r2. We know that the volume of any prism is (area of base)*h so the volume of a cylinder is π r2 * 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 r2)

 

Sphere Volume:           4/3 π r3

Sphere Curved SA:    4π r2

 

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 r2) * 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

A2 = B2 + C2     where A is the hypotenuse

32 + 42 = x2

9 + 16 = x2

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.

 

   

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