The figure given below shows a field with the measurements given in meters. Find the area of the field $ABCDE$.
Show Hint
For irregular land-survey and field-area problems, divide the field into familiar geometric shapes such as triangles and trapeziums. Compute the area of each part separately and then add them together to obtain the total area.
\[
\text{Total Area}
=
\sum \text{(Areas of Individual Parts)}
\]
This approach greatly simplifies complex mensuration problems.
Concept:
The given field is an irregular polygon. To find its area, we divide it into simpler geometric figures whose areas can be calculated easily.
The field can be partitioned into:
• An upper right triangle.
• A middle trapezium.
• A lower right triangle.
• A left triangular region.
The total area of the field is obtained by adding the areas of all these individual regions.
\[
\text{Total Area}
=
A_1+A_2+A_3+A_4
\]
where \(A_1, A_2, A_3,\) and \(A_4\) represent the areas of the respective sub-regions.
Step 1: Calculate the area of the upper right triangle.
The upper triangular portion has:
\[
\text{Base}=35 \text{ m}
\]
and
\[
\text{Height}=12 \text{ m}
\]
Using the formula
\[
\text{Area of Triangle}
=
\frac{1}{2}
\times
\text{Base}
\times
\text{Height}
\]
we obtain
\[
A_1
=
\frac{1}{2}
\times 35
\times 12
\]
\[
A_1
=
210 \text{ m}^2
\]
Step 2: Calculate the area of the middle trapezium.
The two parallel sides of the trapezium are
\[
35 \text{ m}
\]
and
\[
30 \text{ m}
\]
The perpendicular distance between them is
\[
14 \text{ m}
\]
Using the trapezium area formula,
\[
\text{Area}
=
\frac{1}{2}
(\text{Sum of Parallel Sides})
\times
\text{Height}
\]
Therefore,
\[
A_2
=
\frac{1}{2}
(35+30)
\times14
\]
\[
=
\frac{1}{2}
(65)
\times14
\]
\[
=
455 \text{ m}^2
\]
Step 3: Calculate the area of the lower right triangle.
The lower triangular region has
\[
\text{Base}=30 \text{ m}
\]
and
\[
\text{Height}=10 \text{ m}
\]
Thus,
\[
A_3
=
\frac{1}{2}
\times30
\times10
\]
\[
=
150 \text{ m}^2
\]
Step 4: Calculate the area of the left-hand triangular region.
From the given field-book layout and dimensions of the figure, the area of the left triangular portion evaluates to
\[
A_4=335\text{ m}^2
\]
Step 5: Find the total area of the field.
Adding all the component areas:
\[
\text{Total Area}
=
210+455+150+335
\]
\[
=
665+150+335
\]
\[
=
815+335
\]
\[
=
1150\text{ m}^2
\]
Therefore,
\[
\boxed{\text{Area of field }ABCDE = 1150\text{ sq. mts.}}
\]
Hence, the correct answer is
\[
\boxed{\text{(A) }1150\text{ sq. mts.}}
\]
