Area of Parallelograms, Triangles, and Trapezoids
Whereas a linear measurement (line) is a one-dimensional measurement, area is a two-dimensional
measurement. It refers to how many square units of surface an object has. Below is a square that is 3
units long on a side (linear measurement), but contains 9 square units (area).
3 units
The length of 1 side is written as 3 units.
The area of the square is 9 units2.
3 units
Notice the difference in labeling. The linear measurement has a
label of units while the area measurement has a label of units2.
This is because the area of a square is found by multiplying a square’s
width by its height; a bit like multiplying the number of columns by the number of rows.
3 units x 3 units = 3 x 3 x units x units
or 9 x units2 or
9 units2
One can think of the exponent 2 in the units2 as saying it’s a two-dimensional measurement (area).
A square is just a special form of a rectangle, so the area of a rectangle can be found the same way.
The area of this rectangle is 4 units x 3 units or 12 units2.
The bottom side of the rectangle is called the base (b).
3 units
The side of the rectangle is called the height (h). The height
of an object is always measured at right angles (900) to the base.
Since a rectangle has 4 right angles the side of the rectangle is
its height (also called altitude). The formula for the area of a
rectangle can then be written as b x h = Area or bh = A.
4 units
Rectangles and squares are just special cases of a quadrilateral (4-sided figure) called a parallelogram.
A parallelogram can be cut to form a rectangle (see below), so the formula for the area of a
parallelogram is also bh = A.
height
height
base
base
Triangle cut from here, and moved to here.
Area of Parallelograms, Triangles, and Trapezoids
The diagonal of any parallelogram divides the parallelogram into two congruent triangles.
height
height
base
base
Therefore the area of a triangle is one-half the area of the parallelogram it came from.
The formula is
bh x ½ = Area or bh = A
2
A trapezoid can be turned into a rectangle by cutting it as follows.
base 2
This triangle rotated
up to fill the gap here.
midline
This triangle rotated
up to fill the gap here.
base 1
Notice that this forms a rectangle whose height is the same as the trapezoid it came from, and whose
base is the same length as the midline of the trapezoid. The length of the midline is just the average of
base 1 and base 2 which is found by adding the two bases together and dividing by 2. Since the area of
the rectangle, formed by cutting the trapezoid, can be found by multiplying the length of the midline by
the height of the trapezoid, the following formula can be derived.
Area = midline x height or
(base 1 + base 2) x height = Area or (b1 + b2)h
2
2
Another proof for the area of a trapezoid can be seen below.
Cut along this line and rotate the little
base 2
triangle about the midpoint to form a larger
triangle. The base of this new triangle is
equal to base 1 + base 2. Since the are of a
triangle is equal to bh/2, the area of the
height
midpoint
triangle, cut from the trapezoid, would be
base 1
(base 2)
(b1 + b2)h = Area
2
Area of Parallelograms, Triangles, and Trapezoids
Determine the areas of the following triangles, parallelograms and trapezoids.
1.
2.
6 cm
6 cm
5 cm
12 cm
11 cm
3.
4.
26 m
8m
24 m
15 m
20 m
14 m
5.
6.
14 in
10 in
30 m
18 m
24 in
12 m
7. Determine the area of the non-shaded portion below. (It’s two-dimensional)
20 m
7m
trapezoid
rectangle
parallelogram.
12 m
15 m
9m
30 m
6m