# Parallelogram and Trapezoid Areas — Rearranging Rectangles
## Plain English first
Both of these shapes are rectangles in disguise. The trick in each case is to cut a piece off one end and move it to the other — and suddenly you have a rectangle whose area is obvious.
**Parallelogram:** Slice the slanted end off vertically. Slide that triangle to the other side. You get a perfect rectangle with the same base b and height h. Area = bh.
**Trapezoid:** Two trapezoids placed together (one flipped) form a rectangle whose width is b₁+b₂ and height is h. One trapezoid is half of that. Area = ½h(b₁+b₂).
---
## Standard math notation
```
Parallelogram:
A = b × h
where b = base, h = perpendicular height (not the slant side)
Trapezoid:
A = ½ × h × (b₁ + b₂)
where b₁ = one parallel side (base)
b₂ = other parallel side (top)
h = perpendicular height between them
Think of it as: average of the two parallel sides × height
A = ((b₁ + b₂) / 2) × h
```
---
## Verbose Python with descriptive names
```python
def compute_parallelogram_area(
length_of_base,
perpendicular_height_not_slant_side
):
"""
Slice the slanted end off a parallelogram and reattach it to the other end.
You get a rectangle: base × height. Same area as the parallelogram.
Note: height is perpendicular to the base, not the length of the slanted side.
"""
area = length_of_base * perpendicular_height_not_slant_side
return area
def compute_trapezoid_area(
length_of_first_parallel_side,
length_of_second_parallel_side,
perpendicular_height_between_sides
):
"""
Flip a trapezoid and stick it to another copy — you get a rectangle.
That rectangle has width (b1 + b2) and height h.
One trapezoid is half that rectangle.
Area = ½ × h × (b1 + b2)
Equivalently: average of the two parallel sides × height.
"""
average_of_parallel_sides = (length_of_first_parallel_side + length_of_second_parallel_side) / 2
area = average_of_parallel_sides * perpendicular_height_between_sides
return area
print(compute_parallelogram_area(
length_of_base=8,
perpendicular_height_not_slant_side=5
)) # 40
print(compute_trapezoid_area(
length_of_first_parallel_side=4,
length_of_second_parallel_side=10,
perpendicular_height_between_sides=6
)) # 42.0
```
---
## Sora 2 video prompt (parallelogram)
```
5-second animation. A parallelogram with grid fill. A vertical cut slices
off the right triangle at one end. That triangle slides across to fill the
left gap — forming a perfect rectangle. Label: 'same area, now obvious'.
Formula A = bh appears. Clean white background, warm earth tones.
```
## Sora 2 video prompt (trapezoid)
```
5-second animation, split screen or sequential. A trapezoid appears. A
second identical trapezoid flips and attaches — forming a rectangle with
width b₁+b₂ and height h. The full rectangle = h(b₁+b₂). One trapezoid
glows: ½ × h × (b₁+b₂). Formula assembles. Warm earth tones, clean.
```
---
## Builds on
- [Rectangle Area — Counting the Grid](/articles/66820765-0ec9-4197-8cfa-55bb81c2bf6d)
- [Triangle Area — Half a Rectangle](/articles/6efb3a21-cfdd-4358-bc66-87ad756cb042)
## See also
- [Pythagorean Theorem — Squares on Sides](/articles/4d64ff4e-28db-4c2b-8415-8751fd5adecf)
- [Math Foundations — Visual Table of Contents](/articles/d404884f-54fc-4289-b3f1-baaad2bec6b2)