Kite and Trapezoid
A kite is a quadrilateral with two pairs of equal adjacent sides, and a trapezoid is a quadrilateral with at least one pair of parallel sides, called the bases. In brief: the area of a kite is half the product of its diagonals, A = (1/2)d1d2, and the area of a trapezoid is the average of the bases times the height, A = (1/2)(a + c)h. For a trapezoid with bases 5 and 9 and height 4, the area is 28.
These two quadrilaterals are not parallelograms (in general), and each has its own formulas and properties. The isosceles trapezoid and the right trapezoid are the important special cases. This guide explains the properties, the area and perimeter formulas, the midsegment of a trapezoid and a table comparing the shapes, with worked problems.
What are the properties of a kite?
A kite ABCD has AB = AD and CB = CD. The vertices A and C, where the equal sides meet, lie on the axis of symmetry AC.
- The diagonals are perpendicular.
- The diagonal AC (the axis of symmetry) bisects the other diagonal BD and bisects the angles at A and C.
- One pair of opposite angles (at B and D) is equal. The angles at A and C are generally different.
- The area is A = (1/2) · AC · BD. This is true for any quadrilateral with perpendicular diagonals.
- A kite has one axis of symmetry. A rhombus is a kite with all sides equal, and then it has two axes.
Example 1: kite area and perimeter
A kite has diagonals 10 and 6. The short diagonal crosses the axis of symmetry and splits the long diagonal into the parts 3 and 7. Half of the short diagonal is 3. The two short sides are √(32 + 32) = √18 ≈ 4.24 and the two long sides are √(32 + 72) = √58 ≈ 7.62. The perimeter is 2 · 4.24 + 2 · 7.62 ≈ 23.72 and the area is (1/2) · 10 · 6 = 30.
Example 2: angles
A kite has angles 100° at A and 60° at C. The two equal angles at B and D are (360° − 160°)/2 = 100° each.
What are the properties of a trapezoid?
The parallel sides are the bases, a and c, and the other two sides are the legs. The height h is the distance between the bases. The angles next to each leg are supplementary, since the bases are parallel (co-interior angles).
- Area: A = (1/2)(a + c)h.
- Isosceles trapezoid: equal legs, equal base angles, equal diagonals, and one axis of symmetry through the midpoints of the bases.
- Right trapezoid: one leg is perpendicular to the bases, so two angles are 90°, and this leg equals the height.
For an isosceles trapezoid with bases a > c and leg l, the height is h = √(l2 − ((a − c)/2)2). Dropping perpendiculars from the ends of the shorter base creates two right triangles, each with a base (a − c)/2.
Example 3: isosceles trapezoid
Bases 14 and 6, legs 5. The offset is (14 − 6)/2 = 4, so h = √(25 − 16) = 3. Area: (1/2)(14 + 6) · 3 = 30. Perimeter: 14 + 6 + 5 + 5 = 30.
Example 4: right trapezoid
Bases 8 and 5, and the perpendicular leg 4. The slanted leg is √(42 + 32) = 5. Area = (1/2)(13) · 4 = 26. Perimeter: 8 + 5 + 4 + 5 = 22.
What is the midsegment of a trapezoid?
The midsegment joins the midpoints of the legs. It is parallel to the bases and its length is the average of the bases: m = (a + c)/2. Therefore the area is A = m · h. For bases 5 and 9, m = 7. The segment through the intersection of the diagonals parallel to the bases has the harmonic-mean length 2ac/(a + c) = 90/14 ≈ 6.43. The ratios in a trapezoid also follow from similar triangles: the triangles formed by the diagonals and the bases are similar, with the ratio a : c.
Worked problems
Example 5: finding a base
A trapezoid has area 48, height 6 and one base 7. Then (1/2)(7 + c) · 6 = 48, so 7 + c = 16 and c = 9.
Example 6: an isosceles trapezoid from angles
An isosceles trapezoid has a base angle of 60° and legs of 8. The height is 8 sin 60° = 4√3 ≈ 6.93 and the offset is 8 cos 60° = 4. If the shorter base is 10, the longer base is 10 + 2 · 4 = 18, and the area is (1/2)(28)(6.93) ≈ 97.0.
Example 7: coordinates
A(0, 0), B(8, 0), C(6, 4), D(2, 4). AB ∥ DC (both horizontal). The legs: AD = √(4 + 16) = √20 and BC = √(4 + 16) = √20, so it is an isosceles trapezoid with bases 8 and 4 and height 4. Area = (1/2)(12)(4) = 24.
Kite and trapezoid compared
| Feature | Kite | Trapezoid | Isosceles trapezoid |
|---|---|---|---|
| Parallel sides | none (in general) | at least one pair | one pair |
| Equal sides | two pairs of adjacent | none (in general) | the legs |
| Diagonals | perpendicular | no special property | equal |
| Axes of symmetry | 1 | 0 (in general) | 1 |
| Area | d1d2/2 | (a + c)h/2 | (a + c)h/2 |
| Cyclic (vertices on a circle) | only if right kite | no | yes |
Where do these shapes come from?
The word trapezoid comes from the Greek trapeza meaning table. In Euclid’s Elements, a trapezium was any quadrilateral without parallel sides, and this meaning survives in British English, where the American “trapezoid” is a “trapezium”, and vice versa. The area formula for a trapezoid was known in Babylonian mathematics. The shape called a kite is named after the toy.
Where are they used today?
Trapezoids appear in the cross-sections of canals, embankments and dams, in roof gables and in the shape of bridge piers. Surveyors find the area of an irregular plot by dividing it into trapezoids, which is also the idea of the trapezoidal rule for numerical integration. Kite shapes are used in design and for flying toys, and the kite structure appears in the layout of some architectural windows and in the shape of road signs.
Which mistakes are common?
Using the slanted leg as the height
In a trapezoid, the height is perpendicular to the bases. Only in a right trapezoid does a leg equal the height.
Averaging the wrong sides
The area uses the average of the two parallel sides, not of all four sides.
Mixing up kite and rhombus
A rhombus has four equal sides. A kite needs only two pairs of adjacent equal sides.
Forgetting which diagonal is bisected
In a kite the axis of symmetry bisects the other diagonal, but it is not generally bisected by it.