A field is in the shape of a trapezium whose parallel sides are 25m and 10 m. The non-parallel sides are 14m and 13m. Find the area of the field.

Explanation

We draw a triangle inside the trapezium and by using Heron’s formula, we can calculate the area of a triangle, and then find the height of the triangle which will also be the height of the trapezium hence, we will calculate the area of the trapezium.
Heron's formula for the area of a triangle, Area `= sqrt(s(s - a)(s - b)(s - c))`
Where `a, b` and `c` are the sides of the triangle, and`s =` Semi-perimeter `=` Half the Perimeter of the triangle `= (a + b + c)/2`
Given `ABCD` is a field. Draw `CG ⊥ AB` from `C` on `AB`, and `CF` parallel to `DA`.
`DC = AF = 10 m, DA = CF = 13 m` (opposite sides of parallelogram)
So, `FB = 25 - 10 = 15 m`
In` triangle CFB, a = 15 m, b = 14 m, c = 13 m.`
Semi Perimeter`(s) = (a + b + c)/2`
`s = (15 + 14 + 13)/2`
`s = 42/2`
`s = 21 m`
By using Heron’s formula,
Area of` triangle CFB = sqrt(s(s - a)(s - b)(s - c))`
`= sqrt(21(21 - 15)(21 - 14)(21 - 13))`
`= sqrt(21 xx 6 xx 7 xx 8)`
`= 84 m^2`
Also,
Area of `triangleCFB = 1/2 xx` base `xx` height
`84 = 1/2 xx BF xx CG`
`84 = 1/2 xx 15 xx CG`
`CG = (84 xx 2)/15`
`CG = 11.2 m `
Area of trapezium `ABCD = 1/2 xx` sum of parallel sides `xx` distance between them
`= 1/2 xx (AB + DC) xx CG`
`= 1/2 xx (25 + 10) xx 11.2`
`= 196 m^2`
Hence the area of the field is `196 m^2`.