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

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?