Trapezium and kite

A trapezium and a kite each break the parallelogram rules in a different way. A trapezium has just one pair of parallel sides instead of two. A kite has adjacent equal sides instead of opposite ones. Both shapes appear in real life , bridges, sails, paper kites, and traffic signs.

Concept

Trapezium.

A trapezium is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the parallel sides (or bases), and the other two are the non-parallel sides (or legs).

Properties:

• The angles on the same side of one of the parallel sides are supplementary (because they are co-interior angles between two parallel lines).
• The sum of all four angles is $360^\circ$ as usual.

A right trapezium has two right angles (the two adjacent ones at a leg). An isosceles trapezium has its non-parallel sides equal; this also makes its base angles equal and its diagonals equal in length.

Area formula: Area of trapezium $= \dfrac{1}{2}(a + b)h$, where $a, b$ are the lengths of the parallel sides and $h$ is the perpendicular distance between them.

Kite.

A kite is a quadrilateral with two pairs of adjacent sides equal. In kite $ABCD$, $AB = AD$ and $CB = CD$.

Properties:

• One diagonal is the axis of symmetry of the kite.
• The diagonals are perpendicular to each other.
• The longer diagonal bisects the shorter one.
• One pair of opposite angles ($\angle B$ and $\angle D$ here) are equal.

Area formula: Area of kite $= \dfrac{1}{2} d_1 d_2$, where $d_1, d_2$ are the diagonals. (The same formula works for a rhombus, since a rhombus is a special kite.)

The big family tree.

Adding shapes:

              Quadrilateral
              /     |      \
        Trapezium  Kite   Parallelogram
                          /      \
                  Rectangle    Rhombus
                          \      /
                           Square

A square is in fact a special case of every named quadrilateral above: it is a parallelogram, a rectangle, a rhombus, and (depending on how strictly "exactly" is read) sometimes a trapezium too.

Worked examples

Example 1. A trapezium has parallel sides $12$ cm and $8$ cm. The distance between them is $5$ cm. Find its area.

• Area $= \frac{1}{2}(12 + 8) \cdot 5 = \frac{1}{2} \cdot 20 \cdot 5 = 50$ cm$^2$.

Example 2. In trapezium $ABCD$, $AB \parallel CD$. If $\angle A = 70^\circ$, find $\angle D$.

• Co-interior angles: $\angle A + \angle D = 180^\circ$.
• $\angle D = 180 - 70 = 110^\circ$.

Example 3. A kite has diagonals $24$ cm and $10$ cm. Find its area.

• Area $= \frac{1}{2} \cdot 24 \cdot 10 = 120$ cm$^2$.

Example 4. In kite $WXYZ$, $WX = WZ = 7$ cm, $YX = YZ = 12$ cm. If diagonals meet at $O$ and $WO = 4$ cm, find $XO$.

• $\triangle WXO$ is right-angled at $O$.
• $XO = \sqrt{WX^2 - WO^2} = \sqrt{49 - 16} = \sqrt{33}$ cm.

Try it yourself

1. The parallel sides of a trapezium are $11$ cm and $7$ cm and the height is $6$ cm. Find the area.
2. In a trapezium $ABCD$ with $AB \parallel CD$, $\angle B = 75^\circ$. Find $\angle C$.
3. The diagonals of a kite are $18$ cm and $14$ cm. Find the area.
4. Sketch a kite. Identify which diagonal is the axis of symmetry.
5. In an isosceles trapezium, $\angle A = 60^\circ$. Find all the other angles.
6. Can a parallelogram be a kite? If yes, what is it called?
7. Find the height of a trapezium with parallel sides $15$ cm and $11$ cm and area $78$ cm$^2$.
8. Is every rhombus a kite? Justify.

Activity / Insight

Make a paper kite. Cut a rhombus or kite shape out of light paper, attach a string at the line of symmetry, add a tail, and try to fly it on a windy day. The kite's symmetry is what keeps it stable. A "kite" shape that isn't symmetric will tumble. This is a wonderful real-world demonstration of how a geometric property keeps a physical object behaving nicely.