Parallelogram, triangle and trapezium by decomposing

What is it about?

You already know the area of a rectangle: length times width. Parallelograms, triangles and trapeziums can be turned into rectangles by cutting, moving and completing – the area does not change.

What matters is the base $g$ and the matching height $h$: the perpendicular distance of the opposite corner (or side) from the base. If its foot lies outside, extend the base.

• Parallelogram: cut off a triangle along the height and slide it to the other side: a rectangle $g$ long and $h$ wide, so $A = g \cdot h$. If it is steeply slanted, one cut is not enough – then it is cut into several strips.
• Triangle: a copy rotated by $180^\circ$ about the midpoint $M$ of a side completes it to a parallelogram with the same base and height. The triangle is half of it, so $A = g \cdot h : 2$.
• Trapezium with parallel sides $a$ and $c$: the rotated copy gives a parallelogram with base $a + c$, so $A = (a + c) \cdot h : 2$. Alternatively split it along a diagonal into two triangles: $a \cdot h : 2 + c \cdot h : 2$.

Example: $a = 7\,\text{cm}$, $c = 3\,\text{cm}$, $h = 4\,\text{cm}$ gives $A = (7 + 3) \cdot 4 : 2\,\text{cm}^2 = 20\,\text{cm}^2$.

One grid square is $1\,\text{cm}$ wide, so it is $1\,\text{cm}^2$; “Count the squares” checks every result.

Try it

1. Press “Cut and move” for this parallelogram. Which piece moves where? Why does it fit exactly?
2. Then count the squares. Does the number match $g \cdot h$?
3. Cut up a steeply slanted parallelogram. Where is the foot of the height? Into how many pieces is it cut?
4. Double the triangle and rearrange the parallelogram into a rectangle. Count the squares: which belong to the triangle, which to the copy – and what about the cut ones?
5. Press “Slide the top” for the triangle or drag the top $C$ yourself. What changes about the shape, what about the area?
6. Compare both ways for the trapezium: rotated copy and two triangles. Do they give the same result?
7. Set c = 0 cm and c = a for the trapezium. Which shapes appear, and what becomes of the formula?

Tasks

Task 1: Parallelogram

Load task Find the area of the parallelogram. Then cut it up to check.

Task 2: Triangle

Load task Calculate the area of the triangle.

Task 3: An obtuse triangle

Load task The corners are $A(0, 0)$, $B(4, 0)$ and $C(7, 3)$. Mia calculates $7 \cdot 3 : 2 = 10.5$. What went wrong?

Task 4: Trapezium two ways

Load task Calculate the area with the trapezium formula and by splitting it into two triangles.

Task 5: Same size, different shape

Find parallelograms with a whole-number base and height (in cm) and an area of $24\,\text{cm}^2$ (base at most $10\,\text{cm}$, height at most $8\,\text{cm}$). How many different parallelograms are there if you may also change the offset?

Task 6: Working backwards

A trapezium has the parallel sides $a = 8\,\text{cm}$ and $c = 4\,\text{cm}$ and an area of $30\,\text{cm}^2$. How high is it?

Ideas for the classroom

• Use: grade 6, area of parallelogram, triangle and trapezium derived by cutting and completing. Good as an introduction on the whiteboard: let the class guess first, then cut.
• Order: parallelogram → rectangle, then triangle → parallelogram, finally the trapezium in two ways. With “Show the area right away” off, the result only appears after cutting or counting.
• Shear: “Slide the top” shows that only base and height matter. This leads to the question why the slanted side appears in no formula.
• Model: all corners lie on grid points, the base is horizontal. The height is drawn from the top left corner; for the trapezium $h$ is the distance between the two parallel sides.
• Common stumbling blocks:
  • The slanted side is taken as the height – the height is perpendicular to the base.
  • For triangles “$: 2$” is forgotten; the doubled triangle is a reminder.
  • For obtuse triangles or steeply slanted parallelograms the foot of the height lies outside – the base must be extended, but its length stays $g$.
  • For the trapezium, $a$ and $c$ are multiplied instead of added.