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.
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