What it shows
A triangle cut from one end of a parallelogram and slid to the other end makes a rectangle of the same base and height, so the area of a parallelogram is base times height. Cutting the parallelogram along a diagonal gives two equal triangles, each with area half the base times the height.
The science
Area does not change when a shape is cut and the pieces are moved without overlapping. A parallelogram with base b and perpendicular height h can be cut along the height from one corner; the triangle removed fits exactly on the other side to make a b by h rectangle, so the area is b times h. The slanted side length is not used, which is the common mistake. A diagonal cuts any parallelogram into two congruent triangles, so a triangle of base b and height h has area half of b times h.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Two sheets of coloured card
- A ruler, set square and pencil
- Scissors
- A board with two card rails to slide the pieces
- Glue
Safety
For this project: An adult must help with the scissors for thick card.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Draw a parallelogram with base 12 cm, slanted side 8 cm and perpendicular height 6 cm, and cut it from card.
Draw the height from the top left corner straight down to the base and cut along it to remove a right triangle.
Glue two thin card rails on the board, 6 cm apart, so the pieces slide between them.
Slide the triangle across to the right end and fit it against the slanted edge; a 12 cm by 6 cm rectangle appears.
Measure and record: rectangle area 12 times 6 = 72 square cm, so the parallelogram area is 72 square cm.
Cut a second parallelogram of the same size along its diagonal into two triangles and lay one on the other to show they match.
Write the triangle area, half of 12 times 6 = 36 square cm, on each piece.
Mount both demonstrations side by side with the formulas written in large letters.
What you should see
The pieces slide into a 12 by 6 rectangle, giving area 72 square cm for the parallelogram and 36 square cm for each triangle, with the slanted side playing no part.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why is the slanted side not used in the area formula?
Only the perpendicular height matters; the slanted side can be any length for the same base and height.
2What is the area of a parallelogram with base 10 cm and height 4 cm?
40 square cm.
3How does the triangle formula follow from the parallelogram?
A diagonal splits the parallelogram into two equal triangles, so each has half its area.
4Does cutting and sliding change the area?
No. Moving pieces without overlap keeps the total area the same.
Take it further
Cut a trapezium into two triangles and show that its area is half the sum of the parallel sides times the height.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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