Mathematics K–6 · Years 5–6

Cut and rearrange: parallelograms and triangles from rectangles

Measurement and space

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

A parallelogram cut along its height and rearranged becomes a rectangle of the same area, and a rectangle cut along a diagonal gives two triangles of half its area.

What you need

1 cm grid paper
scissors, a ruler and a set square
card parallelograms: base 6 cm and height 4 cm, one with its top edge shifted 1 cm along from the base and one shifted 5 cm
card rectangles 6 cm by 4 cm and 10 cm by 5 cm

How to do it

Count the area of the 6 cm by 4 cm rectangle on the grid. Cut it along a diagonal and lay the two triangles on each other to check they match.
Draw a line from one corner of a parallelogram at right angles to the base (its height). Cut along it and slide the triangle to the other end. Describe the shape you now have and find its area on the grid.
Do the same with the more slanted parallelogram of the same base and height. Compare the two areas.
Draw a triangle on the grid, draw the rectangle that just encloses it with one side along the triangle's base, and cut the leftover pieces to show they fill the triangle.
Write the area of each shape as a calculation from its base and height.

What you should see

The 6 by 4 rectangle has area 24 square centimetres and each triangle from the diagonal cut has 12. Both parallelograms with base 6 cm and height 4 cm become a 6 by 4 rectangle after the cut and slide, so each has area 24 regardless of slant. The enclosing rectangle around a triangle of base 10 cm and height 5 cm has area 50 and its leftover pieces exactly fill the triangle, which therefore has area 25. Each area comes out as base times height for a parallelogram and half of base times height for a triangle.

What changes

This activity lists no variables to change, measure and keep the same.

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A more slanted parallelogram has more area because its sides are longer.
The area of a triangle is base times height.

Safety card

Low riskLearners carry it out

Hazards

scissors

Controls

rounded-tip scissors; cut on the desk

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package is not needed; ordinary classroom supervision under the school's usual risk management.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K-10 Syllabus (2022), NSW Education Standards AuthorityMA3-2DS-03 MAO-WM-01
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities