The area of the shaded green polygon is equal to 8 square units.
The area of the shaded red polygon is equal to 9 square units.
In Mathematics and Geometry, the area of a triangle can be calculated by using the following mathematical equation (formula):
Area of triangle = 1/2 × b × h
Where:
- b represent the base area.
- h represent the height.
Note: When the green polygon is dissected, three triangles would be formed.
By substituting the given side lengths into the formula for the area of a triangle, we have the following;
Area of triangle = (1/2 × 1 × 1) + (1/2 × 3 × 1) + (1/2 × 3 × 4)
Area of triangle = 0.5 + 1.5 + 6
Area of triangle = 8 square units.
In order to determine the area of the shaded red polygon, we would have to apply Pick's theorem;
Area of polygon = (b/2 + i) - 1
where:
- i is the number of lattice points inside a polygon.
- b is the number of lattice points on the perimeter of a polygon.
By substituting the number of lattice points, we have the following;
Area of polygon = (b/2 + i) - 1
Area of red polygon = (4/2 + 8) - 1
Area of red polygon = 10 - 1
Area of red polygon = 9 square units.