Triangles and parallelograms on the same base
Draw a parallelogram ABCD and a triangle ABE on the same base AB, with their tops on a second parallel line. Slide the top along the line and change the base and the gap, and see which areas change and which do not.
Core concept
Parallelograms on the same base and between the same parallels are equal in area, and a triangle on the same base and between the same parallels is half of such a parallelogram
Topic: Triangles and quadrilaterals
Two parallel lines are always the same distance apart. Take a base AB on the lower line. Any parallelogram ABCD with its top side CD on the upper line has area: How much flat space a shape covers, measured in square units like cm². All key words base × height, where the height is the gap between the lines. Slide CD along the upper line and the parallelogram leans more or less, but its base and height do not change, so neither does its area. A triangle ABE with its top E on the upper line has area ½ × base × height: half of the parallelogram, wherever E is. These theorems are used in construction too: to draw a triangle equal in area to a quadrilateral, a corner is slid along a line parallel to a diagonal, which keeps the area the same.
What you will learn
- State the theorem for parallelograms on the same base and between the same parallels.
- Explain why a triangle on the same base and between the same parallels is half the parallelogram.
- Use base × height and ½ × base × height to find these areas.
Before you start
- Know the areas of a parallelogram and a triangle, and what parallel lines are.
Materials
- Squared paper, a ruler and a pencil
Equipment
- A set square to draw the parallel lines
Procedure
- Record the 6 cm base with the parallels 4 cm apart.
- Slide the top 4 cm to the right and record; slide it 4 cm to the left and record.
- Make the base 8 cm and the gap 5 cm, and record.
How this simulation is simplified
- The top of the parallelogram and the top of the triangle stay on the upper parallel line.