A triangle and a parallelogram with the same perpendicular height have equal areas when the triangle’s base is twice as long. The triangle uses half of base × height, while the parallelogram uses the whole product.
MOE’s primary mathematics syllabus includes finding the areas of triangles and parallelograms. The examples below show how to decide whether the base or the height must change in an equal-area question.
- Identify the perpendicular height for each chosen base.
- Write the two area expressions.
- Use the shared measurement to find the missing one.
Worked example | Same height, different bases
A parallelogram has a base of 8 cm and a perpendicular height of 6 cm. A triangle also has a perpendicular height of 6 cm and has the same area. Find the triangle’s base.
The parallelogram covers 8 × 6 = 48 cm². To get a triangle of area 48 cm² at height 6 cm, the base must be 48 × 2 ÷ 6 = 16 cm.
Check by substitution. Half of 16 × 6 is 48. If the triangle also had an 8 cm base, its area would be 24 cm², half the parallelogram’s area.
What if they have the same base instead?
Keep the parallelogram at base 8 cm and height 6 cm. Give the triangle the same 8 cm base. To match the area, solve ½ × 8 × height = 48. The required height is 12 cm.
The extra factor of two can go into the base or into the height, depending on what stays fixed. If both measurements change, calculate their product and compare the areas directly.
Try a height outside the shape
A slanted parallelogram has a base of 9 cm. Its perpendicular height is 4 cm, shown by a line outside its outline. A triangle has the same area and the same perpendicular height. Its base is 18 cm because 9 × 4 = 36 and ½ × 18 × 4 = 36.
A parent can ask the pupil to point to the right angle between the height and the base or its extension. The height remains usable even if that right angle falls outside the outline.
Frequently asked questions
Do I use the sloping side as the height?
Use the distance measured perpendicular to the chosen base.
Do equal areas mean equal perimeters?
The area calculation gives the enclosed surface. Check the actual boundary lengths for perimeter.
Official information
MOE’s 2021 Primary Mathematics syllabus, October 2025 update, Measurement and Geometry. Checked 9 November 2026.