Activity:
Discovering Area Formulas of Quadrilaterals by Using Composite
Figures
Format:
Small group or Large Group
Objectives:
Participants will investigate the area formulas for a triangle,
parallelogram, and trapezoid.
Related 2009 SOL(s):
8.11 The student will solve practical area and perimeter problems
involving composite plane figures.
Materials:
Scissors
Tape
Pencil or pen
Nine cards made of one-inch grid paper printed on cardstock cut into
3”x5” (see appendix for template) per participant or nine 3x5 grid index
cards
One-inch square tiles (if desired)
Document camera and projector (if desired)
Time Required:
Directions:
60 minutes
1. Model the cutting of the cards for the participants. Use a document
camera to show how to cut the cards or hold up the cards as you cut
them.
2. Basic Assumption: area of a rectangle is base times height. From
where does this premise come? One interpretation of multiplication
is an array interpretation. We can divide our card into one-inch
squares and count the number of one-inch square to determine its
area. Since the card is 3 inches by five inches, we can create an
array by laying out three rows of five tiles. The result is 15 tiles,
which is the product of 3 and 5.
3. Have the participants label the base and height of a rectangle on one
of their cards. Does it matter which side is labeled the base and
which side is labeled the height? No. Base and height are
perpendicular to one another. The orientation of our card does not
matter.
4. In the middle of the card, have the participants write Area = base *
height and underneath that write A = bh . We will compare all other
figures with this card.
5. See instructions for cutting the cards with the diagrams on page 3.
6. As you cut these various triangles and quadrilaterals, review with
the participants the similarities and differences between triangles
and quadrilaterals, the properties of quadrilaterals, and vocabulary
words such as base, height/altitude, diagonals, congruent, obtuse,
right, acute, scalene, isosceles, etc. Have the participants label base
and height and write the formula on their cards.
1
7. Right Triangle and Non-right Triangle: A =
1
bh
2
8. Parallelogram: A = bh
For our example, we will use a non-rectangular parallelogram. Once
we have the formula for the area a parallelogram, we can find the
area of other quadrilaterals. Which other quadrilaterals area also
parallelograms? Rectangle (which we already knew and were using
as our comparison), rhombus, and square.
1
9. Trapezoid: A = ( b1 + b2 ) h
2
As an extension, you can discuss the median of a trapezoid.
1
10. For an extension, show the Rhombus: A = d1d2 .
2
Will this formula work for other quadrilaterals? Only the square.
11. Discussion – how would you use this with students?
Closing and
Debriefing:
Possible questions to ask:
• What did you learn from this session?
• How would you apply this to your classroom?
• What is still unclear?
• Comments and/or concerns?
Reflection for Presenter: (Please reflect on and complete the questions below immediately
after delivering the session)
What specific examples of learning did you note?
What specific errors and/or misconceptions still need to be corrected?
Summarize the workshop evaluations.
2
Instructions/Diagrams for Cutting the Cards
Area of a Rectangle A = bh
3 by 5 Index card
We will compare all other areas to this rectangle.
Area = base * height
h
A = bh
b
1
bh
2
Example 1: Right Triangle
Area of a Triangle A =
h
Area = 1 base * height
2
A=
1
bh
2
b
Take a card and draw in one diagonal using a
straightedge. Cut along the diagonal to form 2 congruent
triangles. We have cut the area in half, but the base and
height remain the same length. Since it takes two
congruent triangle to form the rectangle, the area of the
triangle is half the area of the rectangle; therefore, the
1
area of the triangle is bh .
2
Example 2: A Non-Right Triangle
Take a card and mark a point anywhere along one side.
Then use a straightedge to draw two lines connecting the
vertices on the opposite side to that point. Cut along
h
those lines to form three triangles. Is the area of the two
smaller triangles the same as the larger triangle? Yes.
1
Area = base * height
We can tape together the two smaller triangles to form a
2
triangle congruent to the larger triangle. Is the base and
1
A = bh
height of the larger triangle the same as the original
2
b
rectangle? Yes. We have cut the area in half, but the
1
base and height remain the same length, so the area of the triangle is bh .
2
3
Area of a Non-Rectangular Parallelogram A = bh
Area = base * height
h
A = bh
b
Take a card and mark a point anywhere along one side. Then use a straightedge to draw a line
connecting one vertex on the opposite side to that point. Cut along that line to form a triangle and
a trapezoid. Tape the triangle to the opposite side to form a parallelogram. Is the area the same as
the original rectangle? Yes. Is the base and height of the parallelogram the same as the original
rectangle? Yes, so the area of the parallelogram is bh .
1
(b1 + b2 ) h
2
Example 1: Creating an isosceles trapezoid
Area of a Trapezoid A =
b1
Area = 1 *the sum of the bases * height
2
1
A = ( b1 + b2 ) h
2
h
b2
Take a card and mark a point anywhere along one side. Then use a straightedge to draw a line
connecting one vertex on the opposite side to that point. Cut along that line to form a triangle and
a trapezoid. Flip the triangle and tape it to the opposite side of a trapezoid to form a larger
trapezoid. Is the area the same as the original rectangle? Yes. Is the height of the trapezoid the
same as the original rectangle? Yes. What about the bases (the one pair of parallel sides)?
Neither is equal to the original base, but the average of them is equal to the original base
1
1
(b1 + b2 ) = b . Then the area of the trapezoid is (b1 + b2 ) h .
2
2
4
Example 2:
b1
b1
h
b2
b2
h
Area = 1 *the sum of the bases * height
2
1
A = ( b1 + b2 ) h
2
Take a card and create a trapezoid by cutting off two triangles. Label the bases and the height on
the card. Trace the trapezoid onto another card to create a congruent trapezoid. Tape the two
trapezoids together to create a parallelogram. The area of the parallelogram is ( b1 + b2 ) h . Since
the parallelogram is formed from 2 congruent trapezoids, the area of the trapezoid is half that of
1
the parallelogram or ( b1 + b2 ) h .
2
Extension: The median of a trapezoid is the segment parallel to the bases whose endpoints are
the midpoints of the non-parallel sides. Fold the trapezoid in half, lining up the parallel sides.
This fold creates the median of the trapezoid. Compare it
b1
with the base of your original rectangle and notice that
they are the same length. The median is the average of
the two bases (parallel sides) in the trapezoid. So the
median = the average of the bases
area of a trapezoid can also be expressed as the
1
median = ( b1 + b2 )
product of the median and the base:
2
1
A = ( b1 + b2 ) h = ( median ) ( h ) .
2
b2
5
1
d1d2
2
A rhombus is a parallelogram and one can find its area by using the parallelogram area formula,
A = bh . Here is another way to find the area of a rhombus using its diagonals.
Extension: Area of a Rhombus: A =
Take a card and draw both diagonals using a
straightedge. Cut along the diagonals to form 2 pairs of
congruent, isosceles triangles. How do you know the
triangles are isosceles? The diagonals of a rectangle are
equal in length and bisects each other. Tape the bases of
two of the congruent isosceles triangles together to form
a rhombus. (Note: you can make two rhombi from the
index card.)
Now that we have a rhombus, let’s draw in its diagonals. The
diagonals in a rhombus are perpendicular to each other. Why?
They are the base and height of the two isosceles triangles that
we taped together. If we add together the areas of the two
congruent triangles we will have the area of the rhombus. Since
the triangles are congruent their areas are equal. The area of
1
1 ⎛1 ⎞
one triangle is bh = d1 ⎜ d2 ⎟ . Doubling the area of the
2
2 ⎝2 ⎠
triangle gives us the area of the rhombus; therefore, the area of
1 ⎛1 ⎞ 1
the rhombus is A = 2i d1 ⎜ d2 ⎟ = d1d2 .
2 ⎝2 ⎠ 2
d2
d2
d1
d1
6
Appendix
7