# Properties of a Trapezoid

A trapezoid is a quadrilateral with one pair of parallel sides. A parallelogram can also be called a trapezoid since it has two parallel sides. The pair of parallel sides are called bases, while the non-parallel sides are called lateral sides. An isosceles trapezoid is a trapezoid that has lateral sides of the same length. Depending on whether the trapezoid is isosceles or not, it can have different properties.

Here, we will look at the fundamental properties of “normal” trapezoids and isosceles trapezoids. In addition, we will look at its most important formulas along with some examples.

##### GEOMETRY

Relevant for

Learning about the fundamental properties of a trapezoid.

See properties

##### GEOMETRY

Relevant for

Learning about the fundamental properties of a trapezoid.

See properties

## Fundamental properties of trapezoids

The fundamental properties of the trapezoid are:

• A pair of opposite sides are parallel.
• Two pairs of adjacent angles add up to 180 degrees.
• The two non-parallel sides are unequal.
• The diagonals intersect.
• The line joining the midpoints of the non-parallel sides is parallel to the parallel sides and is equal to half the sum of both parallel sides.
• A trapezoid can be inscribed in a circle.
• The sum of the four exterior angles is 4 right angles.
• The sum of the four interior angles is 4 right angles.

The fundamental properties of the isosceles trapezoid are:

• Two adjacent angles add up to 180 degrees.
• When the two non-parallel sides are equal and two pairs of adjacent angles are equal, it is called an isosceles trapezoid.
• An isosceles trapezoid can be inscribed in a circle.
• The diagonals of an isosceles trapezoid are equal.
• The diagonals of an isosceles trapezoid form a pair of congruent triangles with equal sides as the base.
• The two non-parallel sides are equal.
• The line joining the midpoints of the non-parallel sides is parallel to the parallel sides and is equal to half the sum of those sides.
• The sum of the four exterior angles is 4 right angles.
• The sum of the four interior angles is 4 right angles.
• By rotating the isosceles trapezoid with respect to the shorter parallel side, we get concave cones at both ends.
• By rotating the isosceles trapezoid with respect to the longest parallel side, we get convex cones at both ends.

## Important formulas of trapezoids

The following formulas are the most important for solving problems related to trapezoids.

Area of a trapezoid: The area of a trapezoid can be calculated using the length of its bases (parallel sides) and the length of its height, which is the perpendicular distance between the parallel sides. The formula for the area of a trapezoid is:

where,

• $latex b_{1}$ is the length of one of the parallel sides of the trapezoid
• $latex b_{2}$ is the length of the other parallel side of the trapezoid
• $latex h$ is the length the height of the trapezoid

Perimeter of the trapezoid: The perimeter of the trapezoid is calculated by adding the lengths of all the sides. Therefore, we have the formula:

where,

• $latex a, ~b, ~c, ~d$ represent the lengths of the four sides of the trapezoid

## Examples of trapezoids problems

### EXAMPLE 1

What is the area of a trapezoid that has parallel sides of length 14 m and 16 m and a height of 10 m?

Solution: We have the following information:

• Base 1, $latex b_{1} =14$ m
• Base 2, $latex b_{2}=16$ m
• Height, $latex h=10$ m

Therefore, replacing these values in the area formula, we have:

$latex A=\frac{(b_{1}+b_{2})h}{2}$

$latex =\frac{(14+16)10}{2}$

$latex =\frac{(30)10}{2}$

$latex =\frac{300}{2}$

$latex A=150$

The area of the trapezoid is 150 m².

### EXAMPLE 2

What is the perimeter of a trapezoid that has sides with lengths of 9 m, 15 m, 13 m, and 12 m?

Solution: We have the following information:

• Side 1, $latex a=9$ m
• Side 2, $latex b=15$ m
• Side 3, $latex c=13$ m
• Side 4, $latex d=12$ m

Thus, substituting these values in the perimeter formula, we have:

$latex p=a+b+c+d$

$latex =9+15+13+12$

$latex p=49$

The perimeter of the trapezoid is 49 m.  