Find the area of each of the eight shapes in the composite figure.
(Hint: The hexagon can broken into two trapezoids.)
Triangle 24 square units
To find the area of each shape in the composite figure, we first need to determine the dimensions of each shape.
Let's break down the composite figure into its individual shapes:
1. The triangle: Area = 24 square units
2. The square: The square has four equal sides so we can call one side x. The area of a square is side squared, so Area = x^2
3. The hexagon: The hexagon can be broken down into two trapezoids. Let's call the two bases of the trapezoids a and b, and the height h. The area of a trapezoid is (a + b) x h / 2
Now, let's solve for the dimensions of each shape:
1. Triangle:
Area = 1/2 * base * height
24 = 1/2 * base * height
base * height = 48
2. Square:
Area = side^2
Since all sides of a square are equal, we can set x = base of the triangle
x = sqrt(48)
x ≈ 6.93
3. Hexagon:
Since the hexagon can be divided into two trapezoids, each trapezoid will have a base = x and height = x/2
Let's calculate the area of one trapezoid:
Area = (a + b) x h / 2
Area = (x + x) x (x/2) / 2
Area = (2x) x (x/2) / 2
Area = 2x^2 / 4
Area = x^2 / 2
Since we already calculated x for the square:
Area = (6.93)^2 / 2
Area ≈ 24.08 square units
Therefore, the areas of the individual shapes in the composite figure are:
- Triangle: 24 square units
- Square: 48 square units
- Hexagon (each trapezoid): 24.08 square units
Total area of the composite figure: 24 + 48 + 2(24.08) = 120.16 square units.