Area of Trapezium

Area of Trapezium

A trapezium is a quadrilateral having a pair of its opposite sides parallel. Area of trapezium is measured in terms of its two parallel sides and the height between them. Area of trapezium is given by the half of the product of the sum of parallel sides and height i.e. Area of trapezium = ½ (l₁ + l₂) × h, where l₁ and l₂ are two parallel sides of the trapezium and h is the height between them.

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Derivation of the Formula for Area of Trapezium

Let, ABCD in the given figure be a trapezium having AD//BC. AC is a diagonal. AE⊥BC drawn. Diagonal AC divides the trapezium ABCD into two triangles ΔABC and ΔACD. Since perpendicular distance between two parallel lines is same everywhere, the height of both the triangles is same as the height of two parallel lines.

Let, length of two parallel lines be l₁ and l₂ i.e. BC = l₁ AD = l₂ and height between two parallel sides = AE = h

Now,

Area of trapezium ABCD = area of triangle ABC + area of triangle ACD

                                        = ½ × BC × AE + ½ × AD × AE

                                        = ½ (BC + AD) × AE

                                        = ½ (l₁ + l₂) × h

∴ Area of trapezium = ½ (l₁ + l₂) × h, where l₁ and l₂ are two parallel sides and h is the height between them.

      

Workout Examples

Example 1: Calculate the area of given trapezium.

Solution: Here,

                                l₁ = 9cm

                                l₂ = 5cm

                                h = 4cm

                                Area of trapezium = ½ (l₁ + l₂) × h

                                                                   = ½ (9cm + 5cm) × 4cm

                                                                   = ½ × 14cm × 4cm

                                                                   = 28cm²

Example 2: In the given figure, ABCD is a trapezium in which AD//BC, ∠ABC = 90°, AB = 15cm and DC = 17cm. If the area of trapezium ABCD is 204cm², calculate the length of AD.

Solution: DE⊥BC drawn,

                 From the figure,                                   

                                   AB = DE = 15cm, DC = 17cm

                                   In right angle ΔDEC,

                                  

                                                

                                              

                                                                  Now,

                                               Area of trapezium ABCD = 204 cm²

                                i.e.          ½ (AD + BC) × AB = 204

                                or,          ½ (AD + BE + EC) × 15 = 204

                                or,          15/2 (AD + AD + 8) = 204

                                or,          2AD + 8 = 204 × 2/15

                                or,          2AD = 27.2 – 8

                                or,          2AD = 19.2

                                or,          AD = 19.2/2

                                or,          AD = 9.6cm

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