Understanding Area Formulas and Shapes

Area Formulas
Area Formulas
Rectangle
 
 
Rectangle
What is the area formula?
 
 
Rectangle
What is the area formula?
bh
 
 
Rectangle
What is the area formula?
bh
What other shape has 4 right angles?
 
 
Rectangle
What is the area formula?
bh
What other shape has 4 right angles?
Square!
 
 
Rectangle
What is the area formula?
bh
What other shape has 4 right angles?
Square!
Can we use the same
area formula?
 
 
Rectangle
What is the area formula?
bh
What other shape has 4 right angles?
Square!
Can we use the same
area formula?
Yes
 
 
Practice!
Practice!
Rectangle
Square
10m
17m
14cm
Answers
Answers
Rectangle
Square
10m
17m
14cm
196 cm
2
170 m
2
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
Triangle
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
 
Triangle
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
 
2 Triangles
Triangle
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
 
2 Triangles
So then what happens to
the formula?
Triangle
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
 
2 Triangles
So then what happens to
the formula?
 
 
Triangle
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
 
2 Triangles
So then what happens to
the formula?
 
 
bh
Triangle
So then what happens if we
cut a rectangle in half?
    What shape is made?
 
 
 
 
2 Triangles
So then what happens to
the formula?
 
 
bh
2
Practice!
Practice!
Triangle
5 ft
14 ft
 
Answers
Answers
35 ft
2
Triangle
5 ft
14 ft
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
What will the area formula
be now that it is a
rectangle?
Parallelogram
Let’s look at a
parallelogram.
 
 
 
 
What happens if we slice
off the slanted parts on the
ends?
 
 
 
What will the area formula
be now that it is a
rectangle?
bh
Parallelogram
Be careful though!  The
height has to be
perpendicular from the
base, just like the side of
a rectangle!
 
 
 
 
 
 
 
bh
 
Parallelogram
Be careful though!  The
height has to be
perpendicular from the
base, just like the side of
a rectangle!
 
 
 
 
 
 
 
bh
 
Parallelogram
Be careful though!  The
height has to be
perpendicular from the
base, just like the side of
a rectangle!
 
 
 
 
 
 
 
bh
 
Rhombus
The rhombus is just a
parallelogram with all
equal sides!  So it also
has bh for an area
formula.
 
 
 
 
 
 
bh
 
Practice!
Practice!
Parallelogram
Rhombus
3 in
9 in
4 cm
 
 
2.7 cm
Answers
Answers
10.8 cm
2
27 in
2
Parallelogram
Rhombus
3 in
9 in
4 cm
2.7 cm
Let’s try something new
with the parallelogram.
 
 
 
 
Let’s try something new
with the parallelogram.
 
 
 
 
Earlier, you saw that you
could use two trapezoids to
make a parallelogram.
Let’s try something new
with the parallelogram.
 
 
 
 
Earlier, you saw that you
could use two trapezoids to
make a parallelogram.
Let’s try to figure out the
formula since we now
know the area formula for a
parallelogram.
Trapezoid
 
 
 
 
Trapezoid
 
 
 
 
 
Trapezoid
 
 
 
 
 
So we see that
we are
dividing the
parallelogram
in half.  What
will that do to
the formula?
Trapezoid
 
 
 
 
 
So we see that
we are
dividing the
parallelogram
in half.  What
will that do to
the formula?
bh
Trapezoid
 
 
 
 
 
So we see that
we are
dividing the
parallelogram
in half.  What
will that do to
the formula?
bh
2
Trapezoid
 
 
 
 
 
But now there
is a problem.
What is wrong
with the base?
bh
2
Trapezoid
 
 
 
 
 
b
h
2
So we need to account
for the split base, by
calling the top base,
base 1
, and the bottom
base, 
base 2
.  By
adding them together,
we get the original base
from the parallelogram.
The heights are the
same, so no problem
there.
Trapezoid
 
 
 
 
 
(
b1
 + 
b2
)
h
2
So we need to account
for the split base, by
calling the top base,
base 1
, and the bottom
base, 
base 2
.  By
adding them together,
we get the original base
from the parallelogram.
The heights are the
same, so no problem
there.
base 2
base 1
base 1
base 2
Practice!
Practice!
Trapezoid
11 m
3 m
 
5 m
Answers
Answers
35 m
2
Trapezoid
11 m
3 m
5 m
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
So there is just one
more left!
 
 
 
 
 
 
 
So there is just one
more left!
 
 
 
 
 
 
Let’s go back to the
triangle.
A few weeks ago
you learned that by
reflecting a triangle,
you can make a kite.
 
Kite
So there is just one
more left!
 
 
 
 
 
 
Let’s go back to the
triangle.
A few weeks ago
you learned that by
reflecting a triangle,
you can make a kite.
 
 
Kite
Now we have to
determine the
formula.  What is
the area of a triangle
formula again?
 
 
 
 
 
 
 
 
Kite
Now we have to
determine the
formula.  What is
the area of a triangle
formula again?
 
 
 
 
 
 
 
 
b
h
2
Kite
Now we have to
determine the
formula.  What is
the area of a triangle
formula again?
 
 
 
 
 
 
 
 
b
h
2
Fill in the blank.  A
kite is made up of
____ triangles.
Kite
Now we have to
determine the
formula.  What is
the area of a triangle
formula again?
 
 
 
 
 
 
 
 
b
h
2
Fill in the blank.  A
kite is made up of
____ triangles.
So it seems we
should multiply the
formula by 2.
Kite
 
 
 
 
 
 
 
 
b
h
2
*2 =
b
h
Kite
Now we have a different problem.  What is
the base and height of a kite?  The green
line is called the symmetry line, and the red
line is half the other diagonal.
 
 
 
 
 
 
 
 
b
h
2
*2 =
b
h
Kite
Let’s use kite
vocabulary instead to
create our formula.
 
 
 
 
 
 
 
 
Symmetry Line*
Half the Other Diagonal
Practice!
Practice!
Kite
 
2 ft
 
10 ft
Answers
Answers
20 ft
2
Kite
2 ft
10 ft
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
Summary so far...
Summary so far...
bh
 
 
bh
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
Symmetry Line
 * 
Half the Other Diagonal
Final Summary
Final Summary
Make sure all your formulas are written down!
Make sure all your formulas are written down!
b
h
 
 
b
h
2
 
 
 
 
(
b1
 + 
b2
)
h
2
 
 
 
 
Symmetry Line
 * 
Half the Other Diagonal
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Explore area formulas for rectangles and squares, learn how to calculate areas, discover shapes formed by cutting rectangles in half, and understand the transition of formulas when rectangles are transformed into triangles.


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  1. Area Formulas Area Formulas

  2. Rectangle Rectangle

  3. Rectangle Rectangle What is the area formula?

  4. Rectangle Rectangle What is the area formula? bh

  5. Rectangle Rectangle What is the area formula? What other shape has 4 right angles? bh

  6. Rectangle Rectangle What is the area formula? What other shape has 4 right angles? Square! bh

  7. Rectangle Rectangle What is the area formula? What other shape has 4 right angles? Square! bh Can we use the same area formula?

  8. Rectangle Rectangle What is the area formula? What other shape has 4 right angles? Square! bh Can we use the same area formula? Yes

  9. Practice! Practice! 17m 10m Rectangle Square 14cm

  10. Answers Answers 17m 10m Rectangle 170 m2 Square 196 cm2 14cm

  11. So then what happens if we cut a rectangle in half? What shape is made?

  12. Triangle Triangle So then what happens if we cut a rectangle in half? What shape is made?

  13. Triangle Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles

  14. Triangle Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula?

  15. Triangle Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles So then what happens to the formula?

  16. Triangle Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles bh So then what happens to the formula?

  17. Triangle Triangle So then what happens if we cut a rectangle in half? What shape is made? 2 Triangles bh 2 So then what happens to the formula?

  18. Practice! Practice! Triangle 14 ft 5 ft

  19. Answers Answers Triangle 14 ft 35 ft2 5 ft

  20. Summary so far... Summary so far... bh

  21. Summary so far... Summary so far... bh

  22. Summary so far... Summary so far... bh

  23. Summary so far... Summary so far... bh bh

  24. Summary so far... Summary so far... bh 2 bh

  25. Parallelogram Parallelogram Let s look at a parallelogram.

  26. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  27. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  28. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  29. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  30. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  31. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  32. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  33. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  34. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  35. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends?

  36. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends? What will the area formula be now that it is a rectangle?

  37. Parallelogram Parallelogram Let s look at a parallelogram. What happens if we slice off the slanted parts on the ends? What will the area formula be now that it is a rectangle? bh

  38. Parallelogram Parallelogram Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

  39. Parallelogram Parallelogram Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

  40. Parallelogram Parallelogram Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

  41. Rhombus Rhombus The rhombus is just a parallelogram with all equal sides! So it also has bh for an area formula. bh

  42. Practice! Practice! 9 in Parallelogram 3 in 2.7 cm Rhombus 4 cm

  43. Answers Answers 9 in Parallelogram 27 in2 3 in 2.7 cm Rhombus 10.8 cm2 4 cm

  44. Lets try something new with the parallelogram.

  45. Lets try something new with the parallelogram. Earlier, you saw that you could use two trapezoids to make a parallelogram.

  46. Lets try something new with the parallelogram. Earlier, you saw that you could use two trapezoids to make a parallelogram. Let s try to figure out the formula since we now know the area formula for a parallelogram.

  47. Trapezoid Trapezoid

  48. Trapezoid Trapezoid

  49. Trapezoid Trapezoid So we see that we are dividing the parallelogram in half. What will that do to the formula?

  50. Trapezoid Trapezoid So we see that we are dividing the parallelogram in half. What will that do to the formula? bh

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