3.4 Area and Perimeter of Trapezoids on the Coordinate Plane
Bell Work: Given trapezoid ABCD with AB  BC and DC  BC,
Find the perimeter of ABCD. Use the distance formula
if necessary.
Length AB = 7
Length CD = 2
A(0, 0)
D(8, -5)
length BC = 8
length AD =
8  0   5  0
2
2
B(0, -7)
C(8, -7)
 64  25  89
Perimeter = about 26.4
Trapezoids are quadrilaterals with exactly one pair of parallel sides.
The parallel sides are known as the bases of the trapezoid and the non-parallel sides are called the
legs of the trapezoid.
Plot the following points and then connect them in order.
A(-2, 2), B(-2, -3), C(3, -3), D( 1, 2)
A
1. What are the bases of the trapezoid?
AD and BC
A’
2. What are the legs of the trapezoid?
AB and DC
B
3. Confirm that this is a trapezoid by showing that the bases are
parallel and that the legs are not parallel.
Recall: Parallel lines have slopes that are the same.
slope AD = (2 – 2)/(1 - -2) = 0/3 = 0
D
D’
C
B’
C’
slope BC = (-3 - -3)/(3 - -2) = 0/5 = 0
slope AB = (-3 – 2)/(-2 - -2) = -5/0
slope CD = (2 - -3)/(1 – 3) = 5/-2 = -5/2
= undefined
AD and BC have equal slopes thus they are parallel
AB and CD do not have equal slopes thus they are not parallel.
3. Perimeter and area will be easier to find if we translate the trapezoid so that some of the vertices
contain coordinates which are zero.
a. Describe a transformation that will accomplish this. (up/down, right/left, how many units)
right two, down two
b. Do the transformation and label the images A’, B’, C’ and D’.
c. What are these ordered pairs.
A’(0, 0)
B’(0, -5),
C’(5, -5)
D’(3, 0).
4. Find the perimeter of the translated trapezoid. Use the distance formula if you need it.
Length A’B’ = 5
length B’C’ = 5
length A’D’ = 3
Length C’D’ =
3  52  0  52
 4  25  29
Perimeter = about 18.4
You may not know the formula for the area of a trapezoid.
Create the area formula for a trapezoid from the area formula for a parallelogram.
a. Take the parallelogram on the grid and divide it into two
trapezoids that are congruent.
Recall: Congruent means visually equal.
b. Call the shorter base b1 and the longer base b2.
Note: the base of the parallelogram = b1 + b2
c. Area of the parallelogram = bh = (b1 + b2)h____________________
2 ∙ Area of the trapezoid = Area of the parallelogram.
d. Area of the trapezoid = ½ area of the parallelogram
= ½ (b1 + b2)h
5. Back to trapezoid A’B’C’D’. Find the height of the trapezoid.
Height = 5 units
6. Use the area formula to find the area of a trapezoid.
area = ½ (3 + 5)(5) = 20 square units
Again this area was not difficult to find because the bases and the height were horizontal and vertical
segments. The problem would be much more complicated if the bases and height were not horizontal
and vertical segments.
Consider trapezoid WXYZ.
The length of the bases could be found using the distance formula,
but the height is the perpendicular distance between the two bases
and would either be drawn from W or from Z. This would again
require doing the point to line distance.
Recall: In section 3.3 we used a rectangle to surround the parallelogram
and reduced the problem to finding the areas of right triangles
and rectangles with horizontal and vertical bases and heights.
This made the mathematics of the problem easier to do.
Try that with trapezoid WXYZ.
1. What are the coordinates of WXYZ?
W(-1,0)
X(-1,-4)
Y(3, 2)
A
Z
D
Y
W
B
X
C
Z(1,3)
2. Sketch a rectangle that passes through points W, X, Y, Z so that each side of the rectangle passes
through one point of the trapezoid and all sides of the rectangle are either horizontal or
vertical. Label the vertices of the rectangle A, B, C, D.
a. What are the coordinates of each of these points?
A(-1,3)
B(-1,-4)
C(3, -4)
D(3, 3)
b. Calculate the area of trapezoid WXYZ by using the rectangle method.
Find the area of the outside rectangle, then subtract the areas of the right triangles leaving
the area of the trapezoid.
Area of rectangle ABCD = 4(7) = 28
Areas of the right triangles
½ (2)(3)= 3
½ (2)(1)= 1
Area of the trapezoid = 28 – 3 – 1 – 12 = 12 square units
½ (4)(6)=12