Area of Trapezoids, Rhombuses, and Kites - Sixth Grade Mathematics

Special quadrilaterals have geometric formulas derived from decomposing their boundaries into congruent triangles and parallelograms.

Area of a Trapezoid

A trapezoid is a quadrilateral with at least one pair of parallel sides called bases (b1 and b2). The perpendicular distance between these parallel bases is the height (h).

Derivation of the Trapezoid Area Formula:
Take two identical copies of a trapezoid and rotate one upside down:

     Original Trapezoid:                Two Trapezoids Joined:
           Base 1 (b1)                         b1               b2
        +---------------+               +---------------+-----------------+
       /                \              /                \                /
    h /                  \ h         h /                  \              / h
     /                    \          /                    \            /
    +----------------------+         +----------------------+---------------+
           Base 2 (b2)                         b2               b1

The two trapezoids form one giant PARALLELOGRAM!
Base of the large parallelogram = (b1 + b2).
Height of the large parallelogram = h.
Area of large parallelogram = (b1 + b2) * h.
Since one trapezoid is half of the combined parallelogram:
Area of Trapezoid = (1 / 2) * (b1 + b2) * h   or   Area = ((b1 + b2) * h) / 2

Notice that (b1 + b2) / 2 is the average (mean) of the two parallel bases. Therefore, the area of a trapezoid is simply the average base length multiplied by the height!

Example: Trapezoid with b1 = 8 cm, b2 = 14 cm, and height h = 6 cm
         Average Base = (8 + 14) / 2 = 22 / 2 = 11 cm.
         Area = 11 * 6 = 66 cm^2.
         Formula Check: (1 / 2) * (8 + 14) * 6 = (1 / 2) * 22 * 6 = 11 * 6 = 66 cm^2.

Area of a Rhombus and Kite

A rhombus has four equal sides, and a kite has two pairs of equal adjacent sides. In both shapes, the diagonals (d1 and d2) intersect at exact 90-degree right angles.

Diagonals of Rhombuses and Kites:
          Rhombus                               Kite
             /\                                  /\
            / | \                              /  |  \
           /  |  \                            /   |   \
          +---+---+ d1                       +----+----+ d1
           \  |  /                            \   |   /
            \ | /                              \  |  /
             \/                                 \ | /
             d2                                  \|/
                                                  d2

In both shapes, the two perpendicular diagonals divide the interior into four right triangles. Enclosing either figure in a rectangle reveals that the figure occupies exactly half the rectangle area.

Formula for Area of Rhombus or Kite:
Area = (1 / 2) * d1 * d2   or   Area = (d1 * d2) / 2
where d1 and d2 are the full lengths of the two perpendicular diagonals.

Example 1: Rhombus with diagonals d1 = 12 cm and d2 = 16 cm
           Area = (1 / 2) * 12 * 16 = 6 * 16 = 96 cm^2.

Example 2: Kite with diagonals d1 = 9 inches and d2 = 20 inches
           Area = (1 / 2) * 9 * 20 = 9 * 10 = 90 in.^2.

A rhombus is also a parallelogram! If you are given its base side length and perpendicular height instead of its diagonals, you can also use Area = base * height. Both formulas yield the exact same area.

Exercises

Exercise 1. Calculate the area of each figure: (a) A trapezoid with bases 9 cm and 15 cm, and height 8 cm; (b) A trapezoid with bases 4.5 m and 7.5 m, and height 6 m; (c) A rhombus with diagonals 14 inches and 22 inches; (d) A kite with diagonals 8 cm and 25 cm.

Exercise 2. A trapezoid has an area of 120 square feet. The bases measure 14 feet and 16 feet. Find the perpendicular height of the trapezoid.

Exercise 3. A rhombus has an area of 72 square centimeters. If one of its diagonals measures 12 cm, what is the length of the other diagonal?

Exercise 4. A stained glass window pane is shaped like a kite with diagonals measuring 30 cm and 48 cm. (a) What is the area of the window pane in square centimeters? (b) If glass costs $0.05 per square centimeter, how much does the pane cost?

Solutions

Solution to Exercise 1. For (a), A = (1/2) * (9 + 15) * 8 = (1/2) * 24 * 8 = 12 * 8 = 96 cm^2. For (b), A = (1/2) * (4.5 + 7.5) * 6 = (1/2) * 12 * 6 = 6 * 6 = 36 m^2. For (c), A = (1/2) * 14 * 22 = 7 * 22 = 154 in.^2. For (d), A = (1/2) * 8 * 25 = 4 * 25 = 100 cm^2.

Solution to Exercise 2. Using A = (1/2) * (b1 + b2) * h: 120 = (1/2) * (14 + 16) * h. 120 = (1/2) * 30 * h = 15 * h. Divide by 15: h = 120 / 15 = 8 feet. The height is 8 feet.

Solution to Exercise 3. Using A = (1/2) * d1 * d2: 72 = (1/2) * 12 * d2. 72 = 6 * d2. Divide by 6: d2 = 72 / 6 = 12 cm. The other diagonal measures 12 cm (which means this particular rhombus also has equal diagonals, making it a square).

Solution to Exercise 4. For (a), Area = (1/2) * 30 * 48 = 15 * 48 = 720 cm^2. For (b), Cost = 720 cm^2 * $0.05/cm^2 = $36.00.