Question

1. find the area of the shape below. break the larger shape into smaller pieces if needed.

8.4 m
8 m
4 m
11.4 m

Explanation:

Step1: Find the length of the left segment

The total base length is \(11.4\) m and the right segment is \(8.4\) m, so the left segment length \(l = 11.4 - 8.4 = 3\) m.

Step2: Calculate the area of the rectangle

The rectangle has length \(11.4\) m and height \(8\) m, but we can also consider the right rectangle (length \(8.4\) m, height \(8\) m) and the left part. Wait, alternatively, we can split the shape into a rectangle and a triangle. The rectangle part: length \(11.4\) m, height \(8\) m? No, wait, the left part has a triangle. Wait, actually, the shape can be seen as a large rectangle minus a triangle? Wait, no. Let's re - examine. The vertical side: the total height is \(8\) m, and there's a \(4\) m mark. Wait, maybe split into a rectangle (length \(11.4\) m, height \(8\) m) and a triangle? No, wait, the left - hand side: the horizontal segment is \(11.4 - 8.4=3\) m, and the vertical segment from the \(4\) m mark to the top is \(8 - 4 = 4\) m. So the triangle has base \(3\) m and height \(4\) m. The area of the rectangle (the main part) is \(11.4\times8\), but no, wait, actually, the shape can be considered as a rectangle with length \(11.4\) m and height \(8\) m minus the area of the triangle. Wait, no, let's do it step by step.

First, the area of the large rectangle (if we consider the outer rectangle) would be \(11.4\times8\), but there is a triangle missing? Wait, no, looking at the diagram, the left - hand side has a triangle. Let's split the shape into two parts: a rectangle with length \(8.4\) m and height \(8\) m, and a trapezoid or a rectangle plus a triangle. Wait, the horizontal length from the left to the \(8.4\) m mark is \(11.4 - 8.4 = 3\) m. The vertical length from the bottom to the \(4\) m mark is \(4\) m, and from \(4\) m to \(8\) m is \(4\) m. So the left part: we can consider a rectangle with length \(3\) m and height \(4\) m (the bottom part) and a triangle with base \(3\) m and height \(4\) m (the top part). Wait, no, the triangle has base \(3\) m and height \(4\) m (since \(8 - 4=4\)).

So the area of the rectangle (right part) is \(8.4\times8\). The area of the left part: the bottom is a rectangle with length \(3\) m and height \(4\) m, and the top is a triangle with base \(3\) m and height \(4\) m. The area of the bottom rectangle: \(3\times4 = 12\) \(m^{2}\). The area of the triangle: \(\frac{1}{2}\times3\times4=6\) \(m^{2}\). So the total area of the left part is \(12 + 6=18\) \(m^{2}\). The area of the right rectangle is \(8.4\times8 = 67.2\) \(m^{2}\). Then the total area is \(67.2+18 = 85.2\) \(m^{2}\).

Wait, another way: The shape can be considered as a trapezoid? No, let's use the method of adding a rectangle and a triangle. Wait, the total area can also be calculated as the area of the rectangle with length \(11.4\) m and height \(8\) m minus the area of the triangle? Wait, no, let's check the dimensions again.

Wait, the horizontal length: \(11.4\) m, vertical height: \(8\) m. The left - hand side has a triangle with base \(b = 11.4 - 8.4=3\) m and height \(h = 8 - 4 = 4\) m. Wait, no, the vertical segment from the \(4\) m mark to the top is \(8 - 4=4\) m, and the horizontal segment is \(3\) m. So the triangle is formed by these two.

Alternatively, the area of the shape is equal to the area of the rectangle with length \(11.4\) m and height \(8\) m minus the area of the triangle with base \(3\) m and height \(4\) m? Wait, no, that would be incorrect. Wait, let's draw the shape mentally: there is a right - angled rectangle on the right (length \(8.4\) m, height \(8\) m), and on the left, there is a shape that is a rectangle (length \(3\) m, height \(4\) m) and a triangle (base \(3\) m, height \(4\) m) on top of it. So the area of the left part is \(3\times4+\frac{1}{2}\times3\times4=12 + 6 = 18\). The area of the right part is \(8.4\times8=67.2\). Then total area is \(18 + 67.2 = 85.2\) \(m^{2}\).

Or, another approach: The shape can be divided into a rectangle (length \(11.4\) m, height \(4\) m) and a trapezoid (with bases \(8.4\) m and \(11.4\) m, height \(4\) m). Wait, the height of the trapezoid would be \(8 - 4 = 4\) m.

The area of the rectangle (bottom part): \(11.4\times4=45.6\) \(m^{2}\).

The area of the trapezoid (top part): \(\frac{(8.4 + 11.4)}{2}\times4=\frac{19.8}{2}\times4 = 9.9\times4 = 39.6\) \(m^{2}\).

Then total area: \(45.6+39.6 = 85.2\) \(m^{2}\).

Yes, this is a better way. Let's verify:

The formula for the area of a trapezoid is \(\frac{(a + b)}{2}\times h\), where \(a\) and \(b\) are the lengths of the two parallel sides, and \(h\) is the height between them.

Here, for the top trapezoid, \(a = 8.4\) m, \(b = 11.4\) m, \(h = 8 - 4=4\) m.

Area of trapezoid: \(\frac{(8.4 + 11.4)}{2}\times4=\frac{19.8}{2}\times4 = 9.9\times4 = 39.6\) \(m^{2}\).

Area of the bottom rectangle: length \(11.4\) m, height \(4\) m, so \(11.4\times4 = 45.6\) \(m^{2}\).

Total area: \(39.6+45.6 = 85.2\) \(m^{2}\).

Answer:

The area of the shape is \(\boldsymbol{85.2}\) square meters.