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Area of the lake is sum of the three triangles shown in figure.

Find the diagonals of the polygon using cosine law.

Let the third side of the first triangle be . \ \

Cosine law: .

Here and .

.

Find third side third triangle.

Interial angle of  third triangle is  .

Cosine law: . \ \

Here and .

.

Find the area of the three triangles.

Apply Herons area formula: , where .

Find the area of the first triangle.

and .

Here and .

.

.

.

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Substitute corresponding values in the area formula.

.

Find the area of the second triangle.

and .

Here and .

.

.

.

.

.

Find the area of the third triangle.

and .

Here and .

.

.

.

.

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Area of the lake is

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. \ \

Area of the lake is . \ \

Area of the lake is . \ \