What is a Trapezium

A quadrilateral in which one pair of opposite sides are parallel is called as a trapezium. The parallel sides are called as the bases of the trapezium. The non-parallel sides are simply called the sides of the trapezium.

Properties of Trapezium:

• One of the two opposite sides are parallel in trapezium.
• The other opposite sides need not be parallel.

The basic properties of trapezium are different from the parallelogram.

The total space inside the boundary of the trapezium is called as the area of the trapezium. Area is measured in terms of square unit.

In the figure shown below, ABCD is a trapezium in which `barAB` and `barDC` are parallel. The diagonal `barBD` divides the trapezium into two triangles namely triangle ABD and triangle BCD. Draw DF perpendicular to AB. Again draw BE perpendicular to `barDC` . Now, E is in the extension of `barDC` . Here FBED isa rectangle.

Area of the trapezium ABCD = Area of the ABD +Area of the triangle BCD

Area ofthe triangle ABD = 1/2 x `barAB` x `barDF`

Area of the triangle BCD = 1/2 x `barDC` x `barBE`

Since FBED is a rectangle, opposite sides `barDF` = `barEB` = h(say)

Therefore, area of the triangle ABD + Area of the triangle BCD = 1/2 x `barAB` x h+ 1/2 x `barDC` x h

= 1/2 h(`barAB` + `barDC` )

Area of the trapezium, ABCD = 1/2 x height (Sum of the parallel sides)

Trapezoid is one of the types of quadrilaterals. It has four sides in which two sides are parallel to each other. The perpendicular distance between the two parallel sides is called as the altitude of a trapezoid. It is also called as the height of the trapezoid.

The parallel sides of a trapezoid are called as the base of the trapezoid. The total space occupied by the trapezoid is called as the area of the trapezoid.

Altitude of a Trapezoid Formula

The formula for the altitude of a trapezoid can be easily derived from the formula to calculate the area of the trapezoid.

Area of a trapezoid(A) = `1/2` x h x (a + b) square units

h is the Altitude

a and b are the lengths of the parallel sides

Altitude of the trapezoid (h) = `(2A) / (a + b)` units

Construct a trapezium ABCD with the following measurements and calculate its area. `barAB` is parallel to `barDC` , `barAB` = 8 cm, `barBC` = 5 cm, `barAC` = 6 cm and `barCD` = 4.5 cm.

Solution:

Given:

`barAB` is parallel to `barDC`

`barAC` = 8 cm `barBC` = 5 cm

`barAC` = 6 cm `barCD` = 4.5 cm

Steps to construct a Trapezium:

Step 1: Draw a rough figure and mark the given measures

Step 2: Draw a line segment AB = 8 cm

Step 3: With A and B as centers and with 6 cm, 5 cm as radii respectively draw two arcs. Let them cut at C.

Step 4: draw `barAC` and `barBC`

Step 5: Draw `barCX` parallel to `barBA`

Step 6: With C as center and with 4.5cm as radius draw an arc cutting `barCX` at D.

Step 7: Draw `barAD`

Example 1: Find the altitude of trapezoid whose area 28.5 cm² and sides a = 2.5 cm and b = 7 cm.

Solution:

Area (A) = 28.5 cm²

a = 2.5 cm

b = 7 cm

Altitude of the trapezoid (h) = `(2A) / (a + b)` units

= `(2 X 28.5) / (2.5 + 7)`

= `57 / 9.5`

Altitude of trapezoid (h) = 6 cm

Example 2: Find the altitude of trapezoid whose area 40.25 cm² and sides a = 3.5 cm and b = 7 cm.

Solution:

Area (A) = 40.25 cm²

a =3.5 cm

b = 7 cm

Altitude of the trapezoid (h) = `(2A) / (a + b)` units

= `(2 X 40.25) / (3.5 + 7)`

= `80.5 / 10.5`

Altitude of trapezoid (h) = 7 cm

Example 3: Find the altitude of trapezoid whose area 54 cm² and sides a = 4.5 cm and b = 9 cm.

Solution:

Area (A) = 54 cm²

a = 4.5 cm

b = 9 cm

Altitude of the trapezoid (h) = `(2A) / (a + b)` units

= `(2 X 54) / (4.5 + 9)`

= `108 / 13.5`

Altitude of trapezoid (h) = 8 cm