Math Area Formulas: Finding the Area of a Parallelogram or Trapezoid
A parallelogram is a quadrilateral with opposite sides parallel. In a parallelogram, opposite sides are always of equal length.
The formula for the area of a parallelogram, where area is A, the base (which can be any of the sides) is b, and the height (measured perpendicular to the base) is h, is
A = bh
A rectangle is a parallelogram with four right angles. In a rectangle, the base and height are equal to the length and width. In a parallelogram with non-right angles, it is impossible to calculate the area based on the lengths of the sides alone. The farther the angles are from 90 degrees, the more “squashed” the parallelogram will appear, and the smaller the area will be.
Area of a Parallelogram Example Problem 1
A parallelogram is 3 yards long. The height, measured perpendicular to this side, is 1.5 yards. What is the area?
How to solve:
Since they are perpendicular, the side measurement and the height can be used in the formula.
A = (3)(1.5)
A = 4.5 square yards
Area of a Parallelogram Example Problem 2
A parallelogram has adjacent sides with lengths 25 cm and 8 cm. You use a protractor to measure one of the angles of the parallelogram, and find that it is 90 degrees. What is the area?
How to solve:
If one angle of a parallelogram is 90 degrees, then it is a rectangle. Therefore one side can be used as the base and the other side can be used as the height.
A = (25)(8)
A = 200 square cm
Area of a Parallelogram Example Problem 3
A parallelogram has adjacent sides with lengths 5 inches and 6 inches. What is the area?
How to solve:
This problem cannot be solved with the information given. Either the height (perpendicular to one of the sides) or one interior angle (requiring the use of trigonometry) must be known as well. Watch out for trick questions like this!
Area of a Trapezoid Formula
A trapezoid is a quadrilateral with one pair of parallel sides, with the other pair of sides not parallel. A right trapezoid has exactly two right angles. Three values must be known to find the area of a trapezoid: the lengths of both parallel sides and the height (a perpendicular line segment between the parallel sides). In a right trapezoid, the shorter of the non-parallel sides is equal to the height. The formula for area of a trapezoid, where area is A, one parallel side is b, the other parallel side is c, and height is h, is
A = ½_h_(b + c)
Area of a Trapezoid Example Problem 1
A trapezoid has two parallel sides of lengths 6 inches and 10 inches. The height is 12 inches. What is the area?
How to solve:
A = ½(12)(6 + 10)
A = ½(12)(8)
A = ½(96)
A = 48 square inches
Area of a Trapezoid Example Problem 2
A right trapezoid has two parallel sides of lengths 4 meters and 6 meters. The non-parallel sides have lengths 7 meters and 9 meters. What is the area?
How to solve:
Since this is a right trapezoid, the shorter of the non-parallel sides is equal to the height.
A = ½(7)(4 + 6)
A = ½(7)(10)
A = ½(70)
A = 35 square meters
