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

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
• Triangle Area — Half a Rectangle

See also

• Pythagorean Theorem — Squares on Sides
• Math Foundations — Visual Table of Contents