Mathematics 7–10 · Year 7
Parallelogram and triangle area by cutting and rearranging
Measurement and space
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The idea
A parallelogram cut along a perpendicular height and rearranged becomes a rectangle with the same base and height, so its area is base × height whatever its slant, and any triangle is half of the parallelogram made from two copies of it.
What you need
- 1 cm grid paper, 2 sheets per learner
- 30 cm ruler with millimetre markings and a set square
- 180 degree protractor
- Round-ended scissors and a glue stick
- Two coloured pencils (one for bases, one for perpendicular heights)
How to do it
- On grid paper draw parallelogram P1 with base 8.0 cm, perpendicular height 5.0 cm and slanting sides at 60 degrees to the base (the top edge starts 2.9 cm to the right of the bottom-left corner). Draw parallelogram P2 with the same base and height and sides at 45 degrees (top edge shifted 5.0 cm).
- Count the squares inside each shape, pairing part squares into whole squares, and record both counts.
- Measure the slanting sides of each parallelogram and compute both perimeters.
- Cut out P1, draw its perpendicular height from one top corner, cut along it and slide the right-angled triangle to the other end. Measure the rectangle formed, compute its area and glue it in.
- Draw an acute, a right-angled and an obtuse triangle, each with base 8.0 cm and perpendicular height 5.0 cm. Fold the paper and cut so that each triangle comes out twice, rotate one copy 180 degrees and join the two along a side to form a parallelogram; record its area and halve it.
- Write the two rules in words and symbols (A = bh for a parallelogram, A = ½bh for a triangle) and check them against every count.
What you should see
Both parallelograms have area 8.0 × 5.0 = 40 cm², and square counts that pair part squares come out close to 40, differing only where part squares are paired by eye. Their perimeters differ: the 60 degree sides are 5.77 cm long (measured about 5.8 cm), giving a perimeter of 27.5 cm, and the 45 degree sides are 7.07 cm (about 7.1 cm), giving 30.1 cm, so equal areas do not mean equal perimeters. The rearranged P1 is an 8.0 cm by 5.0 cm rectangle. Each of the three triangles has area 20 cm², half of the parallelogram built from two copies, even though the obtuse one looks larger. With ±1 mm on each ruler reading the computed area can be out by about ±1.3 cm² (3 per cent). The learner knows it worked when the count, the rearranged rectangle and the formula agree within about 2 cm² for every shape, a margin that allows for the ±1.3 cm² of ruler reading and for part squares paired by eye.
What changes
- What you change
- slant of the sides (60 or 45 degrees) with base and perpendicular height held fixed
- What you measure
- area and perimeter
- What you keep the same
- base 8.0 cm
- perpendicular height 5.0 cm
- the same rule for counting part squares
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The area of a parallelogram is its base times its slanting side.
- Shapes with the same area have the same perimeter.
- An obtuse triangle has a larger area than an acute one with the same base and height.
- The height of a triangle must be one of its sides.
Safety card
Hazards
- scissors
Controls
- round-ended scissors, cutting on the desk
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
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), Stage 4 Area; page read 2026-09-22MA4-ARE-C-01
- Australian Curriculum v9AC9M7M01
Sources
The pages the author read to write this activity.