1. MATHS –CLASS IX
Topic :
AREA OF
Parallelogram

2. Area Of Parallelograms
Definition:
A parallelogram is a quadrilateral
with opposite sides parallel.

3. Review of Characteristics of Parallelograms
Opposite sides are parallel.
Consecutive angles are
supplementary. (SSI angles)
Opposite sides are congruent.
Opposite angles are congruent.

4. Review of Characteristics of Parallelograms
Opposite sides are parallel.
Consecutive angles are
supplementary. (SSI angels)
Opposite sides are congruent.
Opposite angles are congruent.
Diagonals bisect each other.

5. Areas of Parallelograms
Find the area by counting squares. 8
Full blocks = 4
Half blocks =
Total Area =
10 su.

6. Areas of Parallelograms
Find the area by counting squares.
Full blocks = 12
Half blocks = 6
Total Area =
15 su.

7. Areas of Parallelograms
Find the area by counting squares. 20
Full blocks = 8
Half blocks =
Total Area =
24 su.

8. There is!!! Don’t you agree? ? ? There must be ? ? an easier way! ? ?
It is tough to compute
all the partial blocks. ? ?
There must be
an easier way! ? ? ? ?
There is!!! ? ?
I like rectangles much better!
They are real easy!
Don’t you agree?

9. Let’s cut off a corner and start to make a rectangle.
Doesn’t a parallelogram look
like a rectangle with it’s side kicked in?
Let’s cut off a corner and
start to make a rectangle.

10. Wow! We have part of a rectangle.
Now watch what else we have.

11. Let’s move the triangle to the other side.
The two triangles are congruent.
Let’s move the triangle to the other side.

12. What has happened?
We have now created
a rectangle
with the same area
as the parallelogram.

13. We have now created
a rectangle
with the same area
as the parallelogram.
Therefore,
the formula for the area
of a parallelogram
is the same
as that of a rectangle.

14. Parallelogram h b

15. Parallelogram b2 h2
Note that there is another base and another height.
Sometimes you must use the other height.

16. Triangle Now, we can use the parallelogram formula to derive
No, we are not going
to add up squares again.
Now, we can use the
parallelogram formula
to derive
the area of a triangle.

17. Triangle Let’s construct a line through
the vertex parallel to the base.

18. Triangle We have just created a parallelogram.
Let’s construct another line through
the right vertex parallel to another side.
We have just created a parallelogram.

19. Triangle SAS, SSS, ASA, or AAS We can do this for any triangle.
Note that both the triangles are congruent by…
SAS, SSS, ASA, or AAS

20. SAS Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Opposite sides of a parallelogram are congruent.
SAS

21. SSS Opposite sides of a parallelogram are congruent.
Reflexive property.
SSS

22. ASA AIA: Alternate Interior Angles are congruent.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
ASA

23. AAS AIA: Alternate Interior Angles are congruent.
Opposite angles of a parallelogram are congruent.
Opposite sides of a parallelogram are congruent.
AAS

24. Triangle
All triangles are just
Half a parallelogram.
Therefore…

25. Triangle Note where the height is located. It is the height of both
the parallelogram
and triangle. h

26. Sample Problems A= 121.6875 A= 121.7 Find the area. 8.25 14.75 14.75

27. Finding Areas of Parallelogram12 16 20 12 16

28. Find the area: fractions

29. Find the area: fractions
Convert to decimals.

30. Find the area: mixed modes
Convert to decimals.

31. Find the area: mixed modes
Convert to decimals.
A =
A = 59.8

32. Find Area of Triangle 5 3 4 3 4

33. Find the base associated with the corresponding height.
Triangle ABC has three altitudes or heights.
They are …
Each side is a corresponding base.
Find the base associated with
the corresponding height.

34. h h If the area = 96 and …. Find the values of AD and FC.
This is a backwards problem.
You always start
with the formula. h

35. h h If the area = 96 and …. Find the values of AD and FC.
This is a backwards problem.
You always start
with the formula. h

36. Backward Problems 2 2 h 25 h 125 25 If the area is 125 sf,
and the base is 25, find the height. h 2 2 25
Divide by 25 h
125 25

37. 182 h 30 30 h If the area is 182 sf, and the base is 30,
find the height. h 30
182 30 h

38. Compound Complex Multiple Stage Problems
With these problems, you…
1 Plan solution with equations or written strategy
as if you have all the information needed.
2. Write the equation, leaving empty parentheses
to insert the needed values.
3. Go find the needed values in sidebar stages,
substituting back into the original strategy or equation.
4. When all values are found,
complete the original strategy or equation.

39. Find Area of Triangle 13 5 12 12 5
5, 12, 13 triangle

40. Find Area of Triangle 8 8
600 8 4 8

41. Find Area of Triangle 8 15 15
600
triangle

42. Find Area of Triangle 11 5 b 5 b 11 5

43. Find Area of Parallelogram8 h 16 16
Need to find value of height.

44. Find Area of Triangle 8 15 15 8
triangle

45. Find the area of pentagon.
Add line and label figure.
600 10 5 10 10 10

46. They all have the same area. Why?
Which triangle has
the largest area?
They all have
the same area.
Why?

47. They each have
the same base: AB
They each have
the same height.

48. h h is 13 cm. M is midpoint of Find the area of triangles ABM and ACM.
h is 13 for both triangles.

49. M is midpoint of
h is 13 cm. h

50. Summary 1. The area of a parallelogram is…
2. The area of a triangle is half the area of parallelogram.
3. A triangle has three heights or altitudes.
4. A triangle has three bases (sides)
to correspond with each height.