Every area formula on this page traces back to the rectangle. Seeing the chain makes them far harder to forget, and it explains the one detail students most often get wrong: height always means the perpendicular height, never the slanted side.
The formulas
| Shape | Formula | Where it comes from |
|---|---|---|
| Rectangle | length x width | counting unit squares in rows |
| Parallelogram | base x height | cut a triangle off one end and slide it to the other, making a rectangle |
| Triangle | half of base x height | two identical triangles make a parallelogram |
| Trapezoid | half of the sum of the two parallel sides, times the height | two identical trapezoids make a parallelogram |
| Circle | pi times radius squared | cut into thin wedges and rearrange into a near rectangle |
Perpendicular height
In every formula above, the height is measured at a right angle to the base. In a slanted parallelogram the sloping side is longer than the height, and using it inflates the area. Diagrams usually mark the true height with a dashed line and a small square.
A parallelogram with base 10 and slant side 6
the perpendicular height is 5 correct: 10 x 5 = 50 square units wrong: 10 x 6 = 60 square units using the slant side overstates the area
Worked examples
Triangle with base 12 and height 7
area = (12 x 7) / 2 = 42 square units
Trapezoid with parallel sides 8 and 14, height 6
area = (8 + 14) / 2 x 6 = 11 x 6 = 66 square units
Circle with radius 5
area = pi x 5 x 5 = 25 pi about 78.5 square units
Radius, not diameter
The circle formula uses the radius. Given a diameter of 10, the radius is 5, and squaring the diameter instead makes the answer four times too large. Halve before you square.
Common mistakes
Using a slanted side as the height
The commonest error in this whole topic. Height is perpendicular to the base.
Forgetting to halve the triangle
A triangle is half of the parallelogram with the same base and height.
Check yourself
Triangle, base 9, height 4.
(9 x 4) / 2 = 18 square units.
Parallelogram, base 15, perpendicular height 8.
120 square units.
Trapezoid, parallel sides 5 and 11, height 4.
(5 + 11) / 2 x 4 = 32 square units.
Circle with a diameter of 14. Find the area.
The radius is 7, so the area is 49 pi, about 153.9 square units.
See also
Sources
- Standard area formulas and their derivations as taught in United States middle grades geometry. Examples on this page are our own.