A Trapezoid is a 4-sided polygon (Quadrilateral) which has at least one pair of parallel sides. The sides of a Trapezoid that are parallel are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides. The perpendicular distance from one base to the other is called the altitude of a trapezoid. The line joining the midpoints of the two legs is called as the median or midline or midsegment of a trapezoid.

ABCD is a trapezoid with AB and CD as bases and AD and BC as legs. Perpendicular from B to CD is the altitude or height of the trapezoid. A line joining the midpoints of AD and BC is the median.

The median's length is the average of the two base lengths and is given by,
m = $\frac{b_1 + b_2}{2}$ where $b_1$ and $b_2$ are the lengths of the parallel bases.

## Isosceles Trapezoid

An Isosceles trapezoid is a trapezoid when the non parallel sides or legs are equal in length and both angles coming from a parallel side or the base angles are equal.

In the above figure, if AD = BC and $\angle$D = $\angle$C, then it is an Isosceles Trapezoid.

## Right Trapezoid

A Right trapezoid or a right-angled trapezoid is a trapezoid where the two adjacent angles are right angles.

## Area of Trapezoids

The area of a trapezoid is the product of the altitude and the average of the two base lengths times. It is given by the formula:

Area, A = ($\frac{b_1 + b_2}{2}$)h, where $b_1$, $b_2$ are the lengths of each base and h is the altitude (height).

The area of a trapezoid when the median is known, it is just the median times the height and the formula is given by
Area = mh, where m is the median and h is the altitude.

## Perimeter of Trapezoid

In Geometry, the Perimeter of any polygon is the total distance around the outside of a 2-dimensional shape. The perimeter of a trapezoid is the total distance around the outside, which can be found by adding together the length of each side.

Perimeter = a + b + c + d where a,b,c,d are the lengths of each side of a trapezoid.

## Trapezoid Properties

• By definition, in a trapezoid, at least one pair of opposite sides are parallel.
• The angles between pairs of parallel sides in a trapezoid are supplementary.
• Opposite sides of an isosceles trapezoid are equal.
• The angles on either side of the bases are the congruent.
• The diagonals of a trapezoid are congruent.

## Centroid of Trapezoid

The Centroid of a trapezoid geometrically lies on the median of the trapezoid which is between the two bases. If the lower left hand corner of the trapezoid is considered to be at the origin, then the coordinates of the Centroid is given by

$\overline{x}$ = $\frac{b}{2}$ + $\frac{[(2a + b)(c^2 - d^2)]}{6(b^2 - a^2)}$ and $\overline{y}$ = $\frac{b + 2a}{3(a + b)}$h where a, b are bases and c, d are legs.

## Trapezoid Theorems

1. A trapezoid is isosceles if and only if the base angles are congruent.
2. A trapezoid is isosceles if and only if the diagonals are congruent.
3. If a trapezoid is isosceles, the opposite angles are supplementary.

## Trapezoid Proofs

1. In the figure below, it is given that DE || AV and ΔDVA $\cong$ ΔEAV. Prove that DEVA is an isosceles trapezoid.

Proof:
StatementReason
1. DE ∥ AV.1. Given.
2. DAVE is a trapezoid.2. By the definition of trapezoid.
3. ΔDVA ≅ ΔEAV.3. Given.
4. DA ≅ EV.4. By CPCT congruent triangles.
5. DEVA is an isosceles trapezoid.5. By the definition of isosceles trapezoid.

2. Given in the figure below, trapezoid ABCD, AD || BC, BE and CF are altitudes drawn to AD, AE $\cong$ DF. Prove that trapezoid ABCD is isosceles.

Proof:
StatementReason
1. ABCD is a trapezium1. Given.
3. BE = CF3. Distance between same parallels.
4. ∠AEB = ∠DFC4. Both are altitudes, so 90° .
5. AE ≅ DF5.Given.
6. Δ AEB ≅ ΔDFC6. By SAS postulate.
7. AB = CD7. By CPCT.
8. ABCD is isosceles trapezoid.8. By 1 and 7 and the definition.

## Trapezoid Problems

### Solved Examples

Question 1: A trapezoid's two bases are 8 m and 6m, and it is 5m high. What is its Area?
Solution:

Base lengths, $b_1$ = 8 m and $b_2$ = 6m.
Altitude or height of trapezoid, h = 5m.

Area, A = ($\frac{b_1 + b_2}{2}$)h, where $b_1$, $b_2$ are the lengths of each base and h is the altitude (height).
Area of the given trapezoid, A = $\frac{8 + 6}{2}$ × 5  = 7  × 5  = 35 $m^2$.

Question 2: A trapezoid has side lengths of 5 cm, 11 cm, 7 cm and 9 cm, what is its Perimeter?
Solution:

Perimeter = a + b + c + d  where a, b, c, d are the lengths of each side of a trapezoid.
Perimeter of given trapezoid = 5 cm + 11 cm + 7 cm + 9 cm = 32 cm.

Question 3: The area of a trapezoid is 42 square inches and the bases are 8 inches and 6 inches. Find the height of the trapezoid.
Solution:

Given the base lengths, $b_1$ = 8 inches and $b_2$ = 6 inches.
Area, A = 42 square inches.

Area, A = ($\frac{b_1 + b_2}{2}$)h,  where $b_1$, $b_2$ are the lengths of each base and h is the height.

Thus Area = 42 = ($\frac{8+6}{2}$)h..

42 = ($\frac{14}{2}$)h.

42 = 7h.

h = $\frac{42}{7}$ = 6.

Height of the given trapezoid, h = 6 inches.