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Maths: Areas & Volumes

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Graeme Brooke

on 19 November 2013

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Transcript of Maths: Areas & Volumes

Multiply the length ( ) of the rectangle by its width ( ) and you have the Area of the Rectangle
Triangles can be
thought of as having
a base ( )
and a height ( )
Areas
Areas
Areas
of rectangles, triangles, parallelograms,
rhombuses, kites, trapeziums and circles... then... surface areas

The Area of a Triangle
Rectangles are Easy
What we're going to do now
is look at AREA measurements (sometimes known as AREAL measurements)
Parallelograms
Remember this?
These are LINEAR measurements
The metre was first defined as being 1 x 10 millionth of the distance from the north pole to the equator
A = l w
l
w
b
h
The area of a triangle
is equal to half it's base
times it's height
A = ½(b h)
= ½bh
A rhombus is a special parallelogram where all four sides are equal in length.
You can think of it as a square that has been "pushed over"
The Rhombus
A kite is a quadrilateral with two pairs of equal, adjacent sides and one pair of equal angles
Kites
A Parallelogram is a quadrilateral with two pairs of parallel sides
The AREA of a shape is the amount of flat surface enclosed by that shape
of Prisms & Other 3 Dimensional Solids
A = lw
l
w
The Area of a Parallelogram
The area of a parallelogram is twice the area of each triangle, so the area of a parallelogram is given by the formula A = b x h, where b is the base and h is the height of the parallelogram
These are LINEAR measurements
10 000 km =
10 000 000 m
b
h
A = bh
The Area of a Kite
The area of a kite can be determined by dividing the kite into two congruent triangles and using the formula for the area of a triangle
b
h
The area of a circle is given by the formula



where r is the radius of a circle
and has an approximate value of 3.14
The Area of a Circle
2
r
Pi
Pi is a letter in the Greek alphabet that looks like this
In mathematics, pi is a constant that is the ratio of a circle's circumference to its diameter, C/D,
and is approximately 3.14
A = r
The Annulus
The word annulus is the Latin word for ring.
An annulus is the shape formed between
two circles with a common centre
(called "concentric circles")
The area of an annulus is simply
the difference between
the areas of the two circles
Areas
To find the area of a rhombus, just use the formula for the area of a parallelogram
Trapeziums
A trapezium is a quadrilateral with one pair of parallel sides
The Area of a Trapezium
To determine the area of a trapezium, draw two lines to create two triangles and one rectangle
Then it's easy
Volumes
We looked at LINEAR measurements...
...we looked at AREA measurements...
...now we're going to explore VOLUME measurements
Prisms
"Prisms" are solid shapes with identical opposite ends joined by straight edges. They are three dimensional
objects that can be cut into identical ‘slices’, called cross-sections. Prisms are named according to the shape of their cross-section
Rectangular
Prism
Triangular
Prism
Hexagonal
Prism
The Volume of a Prism
The volume of any prism is given by the formula V = A x H, where A is the cross-sectional area of a prism and H is the height of a prism
& Volumes
A = bh
kite
Rectangle
Rectangle
x
Triangle
x
Parallelogram
circle
p.255
Maths Quest 8, Chapter 10, p.285
A = b h
Parallelogram
x
A = ½bh + ½bh
kite
p.252
i)
ii)
p.253
p.256
i)
ii)
i)
ii)
p.254
p.256
The kite they're making here looks to be about 80cm long and they suggest that the littler cross-member should be a quarter shorter than the long bit. Whilst you're watching this, in your head, work out the area of this kite.
Graeme Brooke, October 2013
This Prezi is available to look at again, at
http://goo.gl/5Z1fSU
In the "real" world, kites can be all sorts of different shapes, but in the mathematical world, kites have a specific shape...
The Area of a Sector
The Area of an Ellipse
Heron's Formula
The VOLUME of a 3 dimensional figure is the amount of space it takes up
Prisms
Cones
Pyramids
Spheres
Compound objects
Yr. 10 p.208
Yr. 10 p.210
& Volumes
Nets
A "net" is a 2 dimensional representation of something that can be folded up to make a 3 dimensional object.
The net of a triangular prism might look like this...
Surface area is the total area of all the sides in a 3 dimensional object.

To find the surface area of a prism, you have to add up the areas of all the sides of the prism.

These sides are all 2-dimensional shapes, and we already know how to calculate their areas
The Surface Area of a Prism
Cylinders
Top
Base
Curved Surface
The Net for a Cylinder might look like this...
This V8 engine block has 8 cylinders
The Total Surface Area of a Cylinder is sum of the areas of its individual parts:
Top:


Base:


Curved
surface:
3 Dimensional Objects
Calculate the Total Surface Area of Theses Cylinders
Year 10
Year 8
8, 9, 10
8, 9, 10
Calculate the areas of these compound shapes based on parts of circles
3 Dimensional Objects
Areas
Note: In this particular net, this is a 3:4:5 triangle.
They're special.
10
Find the volumes of each of the following prisms:
The Volume of a Cylinder
The volume of any cylinder is given by the formula V = A x H, where A is the cross-sectional area of the cylinder and H is the height of the cylinder
Find the volumes of each of the following cylinders:
In the case of a cylinder,
so...
This engine block has 4 cylinders
A car’s cylinders each have a radius of 4 cm and a height of 7.4 cm.

a) Calculate the volume of 1 cylinder. Use pi = 3.1416.
b) If the car has 4 cylinders and 1 litre = 1000 cm3, find the total volume (called the engine’s displacement) in litres.
= 371.96 cubic cm
= 371.96 cubic cm
= 371.96 cubic cm
371.96 * 4 = 1487.84 cubic cm
= 1.5 litre 4 cylinder car
= 371.96 cubic cm
Volume
This wave was 110 feet high - and one foot = 12 inches and one inch = 25.4mm and the wave was 30m from front to back and if we assumed it was a triangular prism, what would the volume of the wave be if it was 250m wide?
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