GEOMETRY

M at h 10

Students are expected to:

1) Determine and apply formulas for perimeter, area, surface area, and volume.

2) Demonstrate an understanding of the concepts of surface area and volume.

3) Determine the accuracy and precision of a measurement.

4) Explore properties of, and make and test conjectures about, two- and three-dimensional figures.

Geometry is the study of shapes.

History

They studied Geometry in Ancient Mesopotamia & Ancient Egypt.

Geometry is important in the art and construction fields.

What is Geometry?

Know the different types of triangles

equilateral, isosceles, right

rhombus, square, rectangle,parallelogram, trapezoid

Know the different types of quadrilaterals

Shapes Vocabulary Review

Identify, describe, and classify solid geometric figures.

Quadrilaterals and Triangles

Quadrilateral: A four-sided polygon

Square

rectangle

rhombus

Parallelogram

Square: A rectangle with 4 congruent sides

Parallelogram: A quadrilateral whose opposite sides are parallel and congruent.

Rhombus: A parallelogram whose four sides are congruent and whose opposite angles are congruent.

Rectangle: A parallelogram with 4 right angles

Trapezoid: A quadrilateral with only two parallel sides

Triangle: A three-sided polygon

Equilateral triangle: A triangle with three congruent sides

Isosceles triangle A triangle with two congruent sides and two congruent angle

Scalene triangle: A triangle with no congruent sides

Activity NOW IT’S YOUR TURN TO FIND THESE SHAPES IN THE REAL WORLD.

PICK A PARTNER!

1) Go outside the classroom.

2) Gather any 5 materials or collect pictures that has distinctive shapes.

3) Present it in the class and identify what shape it is.

stop

Shapes in Real Life

Parallelogram:

A quadrilateral whose opposite sides are parallel and congruent

Rhombus: A parallelogram whose four sides are congruent and whose opposite angles are congruent

A quadrilateral whose opposite sides are parallel and congruent

Equilateral triangle: A triangle with three congruent sides

trapezoid

parallelogram

triangle

rectangle

Can you Identify allThese shapes?

WHAT ARE THE FACTORS TO BE CONSIDERED IN CONTAINER DESIGN?

* NATURE OF THE PRODUCT

* VOLUME OF THE PRODUCT

* TRANSPORTATION OF THE PRODUCT

* SURFACE AREA OF THE PACKAGINGOF THE PRODUCT

* ECONOMICAL RATE OF THE CONTAINER

* DISPOSAL OF THE CONTAINER

WHAT IS VOLUME ?

The volume of a solid is the amount of space inside the solid.

Consider the cylinder below:

If we were to fill the cylinder with water the volume would be the amount of water the cylinder could hold:

VOLUMES OF SOLIDS

14cm5 cm

7cm

4cm

6cm

10cm

3cm

4cm

8m

5m

VOLUME

is the amount of space occupied by any 3-dimensional object.

1cm1cm

1cm

Volume = base area x height

= 1cm2 x 1cm

= 1cm3

MEASURING VOLUME

Volume is measured in cubic centimetres (also called centimetre cubed).

Here is a cubic centimetre

It is a cube which measures 1cm in all directions.1cm

1cm1cm

We will now see how to calculate the volume of various shapes.

VOLUMES OF CUBOIDSLook at the cuboid below:

10cm

3cm

4cm

We must first calculate the area of the base of the cuboid:

The base is a rectangle measuring 10cm by 3cm:

3cm

10cm

10cm

3cm

4cm

3cm

10cm

Area of a rectangle = length x breadth

Area = 10 x 3

Area = 30cm2

We now know we can place 30 centimetre squares on the base of the cuboid. But we can also place 30 cubic centimetres on the base:

10cm

3cm

4cm

We have now got to find how many layers of 1cm cubes we can place in the cuboid:

We can fit in 4 layers.

Volume = 30 x 4

Volume = 120cm3

That means that we can place 120 of our cubes measuring a centimetre in all directions inside our cuboid.

10cm

3cm

4cm

We have found that the volume of the cuboid is given by:

Volume = 10 x 3 x 4 = 120cm3

This gives us our formula for the volume of a cuboid:

Volume = Length x Breadth x Height

V=LBH for short.

THE CROSS SECTIONAL AREA

When we calculated the volume of the cuboid :

10cm

3cm

4cm

We found the area of the base : This is the Cross Sectional Area.

The Cross section is the shape that is repeated throughout the volume.We then calculated how many layers of cross section made up the volume.This gives us a formula for calculating other volumes:

Volume = Cross Sectional Area x Length.

What Goes In The Box ?

Calculate the volumes of the cuboids below:

(1)

14cm5 cm

7cm(2)

3.4cm

3.4cm

3.4cm

(3)

8.9 m

2.7m

3.2m

490cm3

39.3cm3

76.9 m3

THE VOLUME OF A CYLINDERConsider the cylinder below:

4cm

6cm

It has a height of 6cm .

What is the size of the radius ?2cm

Volume = cross section x heightWhat shape is the cross section?Circle

Calculate the area of the circle:A = r 2

A = 3.14 x 2 x 2A = 12.56 cm2

Calculate the volume:V = r 2 x hV = 12.56 x 6V = 75.36 cm3

The formula for the volume of a cylinder is:

V = r 2 h

r = radius h = height.

A beverage can has the following dimensions. What is its volume? Solution A = πr2 (Area of the Base)A = (3.14) (8)2

A = 3.14 × 64A = 200.96A = 201 cm2

V = Ah V = (201 cm2) (18 cm) V = 3618 cm3

The volume of the beverage can is 3618 cm3.

THE VOLUME OF A TRIANGULAR PRISMConsider the triangular prism below:

Volume = Cross Section x HeightWhat shape is the cross section ?Triangle.Calculate the area of the triangle:

5cm

8cm

5cmA = ½ x base x heightA = 0.5 x 5 x 5 A = 12.5cm2

Calculate the volume:Volume = Cross Section x Length

V = 12.5 x 8V = 100 cm3

The formula for the volume of a triangular prism is :

V = ½ b h l

b= base h = height l = length

A chocolate bar is sold in the following box. Calculate the space inside the box. Solution V = AhV = (1600 mm2) (200 mm)V = 320 000 mm3

The space inside the box is 320 000 mm3.

What Goes In The Box ?Calculate the volume of the shapes below:

(1)

16cm

14cm

(2)

3m

4m

5m

(3)

6cm12cm

8m

2813.4cm3

30m3

288cm3

VOLUME OF A CONEConsider the cylinder and cone shown below:

The diameter (D) of the top of the cone and the cylinder are equal.

D D

The height (H) of the cone and the cylinder are equal.

H H

If you filled the cone with water and emptied it into the cylinder, how many times would you have to fill the cone to completely fill the cylinder to the top ?

3 times. This shows that the cylinder has three times the volume of a cone with the same height and radius.

The formula for the volume of a cylinder is :

V = r 2 h

We have seen that the volume of a cylinder is three times more than that of a cone with the same diameter and height .

The formula for the volume of a cone is:

hr π3

1V 2

h

r

r = radius h = height

Calculate the volume of the cones below:

hr π3

1V 2

13m

18m(2)

9663.143

1V

9m

6m(1)

hr π3

1V 2

139914.33

1V

31102.14mV 3339.12mV

More Complex ShapesCalculate the volume of the shape below:

20m

23m

16m

12m

Calculate the cross sectional area:

A1A2

Area = A1 + A2Area = (12 x 16) + ( ½ x (20 –12) x 16)

Area = 192 + 64

Area = 256m2

Calculate the volume:

Volume = Cross sectional area x length.

V = 256 x 23

V = 2888m3

For the solids below identify the cross sectional area required for calculating the volume:

Circle

(2)

Right Angled Triangle.

(3)

Pentagon

(4)A2

A1

Rectangle & Semi Circle.

(1)

Calculate the volume of the shape below:

12cm 18cm

10cm

Calculate the cross sectional area:

A2

A1

Area = A1 + A2Area = (12 x 10) + ( ½ x x 6 x 6 )Area = 120 +56.52Area = 176.52cm2

Calculate the volume.

Volume = cross sectional area x LengthV = 176.52 x 18 V = 3177.36cm3

Example

What Goes In The Box?

18m

22m

14m

11m(1)

23cm 32cm

17cm

(2)

4466m3

19156.2cm3

Class Work!

Summary Of Volume Formula

lb

h

V = l b h

r

h

V = r 2 h

b

l

h

V = ½ b h l hr π3

1V 2

h

r

HOMEWORK :

– Answer Check Your Understanding # 6-8 on pages 22.– Study the vocabulary of different polygons for the

next lesson.