1. Math 10
GEOMETRY

2. 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.

3. What is Geometry? History Geometry is the study of shapes.
They studied Geometry in Ancient Mesopotamia & Ancient Egypt.
Geometry is important in the art and construction fields.

4. Geometry in Real Life
title

5. Shapes Vocabulary Review
equilateral, isosceles, right
Know the different types of triangles
4th
rhombus, square, rectangle,
parallelogram, trapezoid
Know the different types of quadrilaterals

6. Identify, describe, and classify solid geometric figures.
5th

7. Quadrilaterals and Triangles
6th

8. Quadrilateral: A four-sided polygon
rhombus
Square
quad
rectangle
Parallelogram

9. Square: A rectangle with 4 congruent sides

10. Parallelogram: A quadrilateral whose opposite sides are parallel and congruent.
Rhombus: A parallelogram whose four sides are congruent and whose opposite angles are congruent.
parallelogram

11. Rectangle: A parallelogram with 4 right angles

12. Trapezoid: A quadrilateral with only two parallel sides

13. Triangle: A three-sided polygon

14. Equilateral triangle: A triangle with three congruent sides

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

16. Scalene triangle: A triangle with no congruent sides

17. Activity PICK A PARTNER! Go outside the classroom.
Now it’s your turn to find these shapes in the real world.
PICK A PARTNER!
Go outside the classroom.
Gather any 5 materials or collect pictures that has
distinctive shapes.
3) Present it in the class and identify what shape it is.
stop

18. Shapes in Real Life

19. A quadrilateral whose opposite sides are parallel and congruent
Parallelogram:
A quadrilateral whose opposite sides are parallel and congruent
Rhombus: A parallelogram whose four sides are congruent and whose opposite angles are congruent
parallelogram

20. A quadrilateral whose opposite sides are parallel and congruent
rectangle

21. Equilateral triangle: A triangle with three congruent sides

22. triangle Can you Identify all trapezoid These shapes? rectangle
parallelogram

25. 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 PACKAGING
OF THE PRODUCT
* ECONOMICAL RATE OF THE CONTAINER
* DISPOSAL OF THE CONTAINER

26. 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:

30. Volumes Of Solids 8m 5m
7cm
5 cm
14cm
6cm
4cm
4cm
3cm
10cm

31. is the amount of space occupied by any 3-dimensional object.
Volume
is the amount of space occupied by any 3-dimensional object.
1cm
1cm
1cm
Volume = base area x height
= 1cm2 x 1cm
= 1cm3

32. 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
We will now see how to calculate the volume of various shapes.

33. Volumes Of Cuboids Look at the cuboid below: 4cm 3cm 10cm
We must first calculate the area of the base of the cuboid:
The base is a rectangle measuring 10cm by 3cm:
3cm
10cm

34. 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

35. 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.

36. 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.

37. 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.

38. What Goes In The Box ? Calculate the volumes of the cuboids below: (1)
14cm
5 cm
7cm
(2)
3.4cm
490cm3
39.3cm3
(3)
8.9 m
2.7m
3.2m
76.9 m3

39. The Volume Of A Cylinder
Consider the cylinder below:
It has a height of 6cm .
4cm
6cm
What is the size of the radius ?
2cm
Volume = cross section x height
What shape is the cross section?
Circle
Calculate the area of the circle:
A =  r 2
A = 3.14 x 2 x 2
A = cm2
The formula for the volume of a cylinder is:
V =  r 2 h
r = radius h = height.
Calculate the volume:
V =  r 2 x h
V = x 6
V = cm3

40. A beverage can has the following dimensions. What is its volume
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 × 64 A = A = 201 cm2   V = Ah   V = (201 cm2) (18 cm)   V = 3618 cm3     The volume of the beverage can is 3618 cm3.

41. The Volume Of A Triangular Prism
Consider the triangular prism below:
5cm
8cm
Volume = Cross Section x Height
What shape is the cross section ?
Triangle.
Calculate the area of the triangle:
A = ½ x base x height
A = 0.5 x 5 x 5
A = 12.5cm2
Calculate the volume:
Volume = Cross Section x Length
The formula for the volume of a triangular prism is :
V = ½ b h l
b= base h = height l = length
V = 12.5 x 8
V = 100 cm3

42. A chocolate bar is sold in the following box
A chocolate bar is sold in the following box. Calculate the space inside the box.   Solution   V = Ah V = (1600 mm2) (200 mm) V = mm3   The space inside the box is mm3.

43. What Goes In The Box ? Calculate the volume of the shapes below: (2)
(1)
16cm
14cm
(3)
6cm
12cm 8m
2813.4cm3
30m3
288cm3

44. Volume Of A Cone Consider the cylinder and cone shown below: D
The diameter (D) of the top of the cone and the cylinder are equal. H
The height (H) of the cone and the cylinder are equal.
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 ?
This shows that the cylinder has three times the volume of a cone with the same height and radius.
3 times.

45. 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: h r
r = radius h = height

46. Calculate the volume of the cones below:
(2) 9m 6m
(1)

47. More Complex Shapes Calculate the volume of the shape below: 20m 23m
Volume = Cross sectional area x length.
V = 256 x 23 A1 A2
V = 2888m3
Calculate the cross sectional area:
Area = A1 + A2
Area = (12 x 16) + ( ½ x (20 –12) x 16)
Area =
Area = 256m2

48. For the solids below identify the cross sectional area required for calculating the volume:
(2)
(1)
Right Angled Triangle.
Circle
(4)
(3) A2 A1
Rectangle & Semi Circle.
Pentagon

49. Example
Calculate the volume of the shape below:
12cm
18cm
10cm A2 A1
Calculate the volume.
Volume = cross sectional area x Length
V = x 18
V = cm3
Calculate the cross sectional area:
Area = A1 + A2
Area = (12 x 10) + ( ½ x  x 6 x 6 )
Area =
Area = cm2

50. What Goes In The Box? 11m (1) 4466m3 14m 22m (2) 18m 17cm 19156.2cm3

51. Class Work!

56. Summary Of Volume Formulah
V =  r 2 h l b h
V = l b h b l h
V = ½ b h l h r

58. HOMEWORK : Answer Check Your Understanding # 6-8 on pages 22.
Study the vocabulary of different polygons for the next lesson.