ShapesProperties
- Opposite sides are equal and parallel.
- Opposite angles are equal, and neighbouring angles add up to 180°.
- The diagonals cut each other in half.
- Rectangles, rhombuses and squares are all special parallelograms.
Why the formulas work
Cut a right-angled triangle off one end of a parallelogram and slide it to the other end. The shape becomes a rectangle with the same base and the same height, so the area is base times height. The slant side does not come into the area at all. It matters only for the perimeter.
Worked example
Take a parallelogram with base 12 cm, height 7 cm and slant side 9 cm.
- Area. Area = b × h. Here: 12 × 7 = 84 cm².
- Perimeter. Perimeter = 2(a + b). Here: 2 × (9 + 12) = 42 cm.
Finding a missing measurement
Height = area ÷ base: an area of 84 cm² on a base of 12 cm means a height of 7 cm. Any side can be the base, as long as you use the height at right angles to it. With sides of 12 cm and 9 cm, the height to the 9 cm side is 84 ÷ 9, about 9.33 cm. The longer base always has the shorter height.
Mistakes to avoid
- Multiplying the base by the slant side. Use the perpendicular height.
- A parallelogram and a rectangle with the same sides do not have the same area. The more it leans, the smaller the area.
Where you see it
The step sides of an escalator, a leaning stack of books seen from the side, some tiles and the shape of an eraser are parallelograms.
Try these
Work each one out, then tap to check.
A parallelogram has base 18 cm, height 5 cm and slant side 6.5 cm. Find its area and perimeter.
Area = 90 cm²; Perimeter = 49 cm.
A parallelogram has base 9 cm, height 8 cm and slant side 10 cm. Find its area and perimeter.
Area = 72 cm²; Perimeter = 38 cm.