Area of Composite Shapes - Steps, Examples & Questions - Free Printable

Let's solve the problem step by step.

Problem:

We are given a composite shape made up of two rectangles and a triangle that is cut out from one of them. We need to find the area of the shaded (pink) region.

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Step 1: Understand the Shape

The figure consists of:

- A large rectangle on the left:
- Width = 7 m
- Height = 6 m

- A smaller rectangle on the right:
- Width = 4 m
- Height = 7 m

But wait — there’s a white triangle removed from the center, so we must subtract its area.

Also, notice:
- The total width of the entire shape is 11 m, which is $7 + 4 = 11$, so the two rectangles are side by side.
- The height of the left rectangle is 6 m, and the right one is 7 m, but they overlap in height?
- Actually, the top of the left rectangle is only 6 m high, while the right one is 7 m high. But the top edge of the whole shape appears to be aligned at 7 m, so maybe the left rectangle extends upward?

Wait — let's look carefully.

From the diagram:

- The left part has a height of 6 m, and is 7 m wide.
- The right part is 4 m wide and 7 m tall.
- The total width is $7 + 4 = 11$ m.
- The height of the right rectangle is 7 m, and the left rectangle is only 6 m high, but there's a 2 m gap above it? Wait — no.

Actually, looking at the dashed line:

- There's a horizontal dashed line at the top of the white triangle, which is 2 m above the base of the left rectangle.
- The total height of the left rectangle is 6 m, so the dashed line is at 2 m from the top of the left rectangle? No — let's recheck.

Wait: the vertical arrow on the left says 6 m, from bottom to top of the left rectangle. Then, above it, there's a 2 m label pointing from the top of the left rectangle to the top of the right rectangle. So the right rectangle is taller — total height 6 + 2 = 8 m? But it says 7 m on the right side.

Wait — contradiction?

Let’s read the labels again:

- Left rectangle: width = 7 m, height = 6 m
- Right rectangle: width = 4 m, height = 7 m
- Total width: 11 m → 7 + 4 = 11 → OK
- On the right side, the total height is labeled as 7 m
- On the left side, the height is labeled 6 m
- Above the left rectangle, there's a 2 m label going from the top of the left rectangle to the top of the right rectangle → so the right rectangle extends 2 m higher than the left one.

So the bottom of the right rectangle is aligned with the bottom of the left one? But then the right one would be 6 + 2 = 8 m tall, but it says 7 m.

Wait — inconsistency.

Wait, actually, the vertical arrow on the far right says 7 m, meaning the total height of the right rectangle is 7 m.

But the left rectangle is 6 m tall, and the dashed line is 2 m above the top of the left rectangle? That can't be if the right rectangle is only 7 m tall.

Wait — perhaps the dashed line is not at the top of the left rectangle.

Let’s analyze the triangle.

The white triangle is inside the pink area, and:

- Its base is along the bottom, and spans 5 m (as labeled).
- Its height is 6 m, from the base to the apex (which is aligned with the dashed line).

The dashed line is horizontal and located 2 m below the top of the right rectangle?

Wait — let’s interpret all labels:

From the image:

- The total width is 11 m: 7 m (left) + 4 m (right)
- The left rectangle: 7 m wide, 6 m tall
- The right rectangle: 4 m wide, 7 m tall
- The triangle is cut out, with:
- Base = 5 m (on the bottom)
- Height = 6 m (from base to tip)
- The tip of the triangle reaches up to the dashed line, which is 2 m below the top of the right rectangle

Wait — the right rectangle is 7 m tall, and the dashed line is 2 m below the top, so the dashed line is at 7 - 2 = 5 m from the bottom?

But the triangle height is 6 m, so that can’t be.

Wait — no, the height of the triangle is labeled 6 m, from the bottom to the tip.

So the tip is at 6 m above the base.

Now, the right rectangle is 7 m tall, so the triangle tip is 6 m up, and the right rectangle goes to 7 m, so it's 1 m below the top of the right rectangle.

But the dashed line is 2 m from the top of the left rectangle, and the left rectangle is 6 m tall, so the dashed line is at 6 + 2 = 8 m? But that exceeds the right rectangle’s height.

This suggests misalignment.

Wait — perhaps the dashed line is horizontal across both rectangles, and is 2 m below the top of the right rectangle, and also 2 m above the top of the left rectangle?

But the left rectangle is 6 m tall, so if the dashed line is 2 m above it, then it would be at 8 m, but the right rectangle is only 7 m tall — impossible.

So something’s wrong.

Let me reinterpret based on standard geometry problems like this.

Perhaps the entire shape is composed of:

- A large rectangle of size 11 m × 7 m (since total width 11 m, height 7 m), but a triangle is cut out from it.

But the left part shows a 7 m wide section, and the right is 4 m wide.

Wait — the bottom of the shape is 11 m long.

The left rectangle is 7 m wide and 6 m tall.

The right rectangle is 4 m wide and 7 m tall.

So the bottom edges are aligned, and the tops are not.

But the white triangle is drawn overlapping both, with:

- Base of 5 m on the bottom (likely centered or offset)
- Height of 6 m

And the dashed line is at 2 m below the top of the right rectangle, which is 7 m tall → so dashed line is at 5 m from bottom.

But the triangle’s height is 6 m, so tip is at 6 m.

So tip is at 6 m, dashed line is at 5 m — so tip is above dashed line? But dashed line is shown as the top of the triangle.

Wait — the dashed line is horizontal, and the triangle's apex touches it.

So the height of the triangle is 6 m, and it ends at the dashed line.

So the dashed line is 6 m above the base.

But the right rectangle is 7 m tall, so the dashed line is at 6 m from bottom → 1 m below the top.

And the left rectangle is 6 m tall, so the dashed line is at the top of the left rectangle.

Ah! That makes sense.

So:

- The left rectangle is 7 m wide, 6 m tall.
- The right rectangle is 4 m wide, 7 m tall.
- They are placed side by side, sharing a common bottom edge.
- The top of the left rectangle is at 6 m.
- The top of the right rectangle is at 7 m.
- The dashed line is at 6 m, which is the top of the left rectangle and 1 m below the top of the right rectangle.
- The white triangle has:
- Base = 5 m (on the bottom)
- Height = 6 m, reaching up to the dashed line at 6 m.

So the triangle is entirely within the bottom 6 m of the shape.

Now, the shaded area is the union of the two rectangles minus the area of the triangle.

But wait — is the triangle overlapping both rectangles?

Yes — because the base is 5 m, and the total width is 11 m.

Where is the base of the triangle?

It’s shown starting at some point on the bottom.

Looking at the diagram:

- From the left, the triangle base is 5 m wide.
- The left rectangle is 7 m wide, so if the triangle starts at the left edge, it would extend 5 m into the left rectangle.
- The right rectangle starts at 7 m, so the triangle does not reach into it unless the base extends beyond 7 m.

But the triangle base is only 5 m, so if it starts at x=0, it ends at x=5, so it's entirely within the left rectangle.

But then why is the right rectangle shown?

Wait — the right rectangle is 4 m wide, starting at x=7, ending at x=11.

The triangle base is 5 m, and if it starts at x=0, it ends at x=5 — so it doesn’t reach the right rectangle.

But the dashed line is horizontal across both rectangles, suggesting the triangle might extend into the right side.

Wait — the apex of the triangle is at the dashed line, and it’s shown as a right triangle, with one leg vertical, one horizontal.

Looking at the shape: the triangle has a vertical leg of 6 m, and a horizontal base of 5 m.

But it’s drawn such that the right angle is at the bottom-left corner of the triangle.

So it’s a right triangle, with legs 5 m and 6 m.

But where is it placed?

The base is on the bottom, 5 m long.

The vertical leg goes up 6 m to the dashed line.

Now, the dashed line is at 6 m height, and runs across both rectangles.

So the triangle is underneath the dashed line.

But the left rectangle is 7 m wide, so if the triangle starts at the left edge, it occupies x=0 to x=5, y=0 to y=6.

Then, the right rectangle starts at x=7, so there's a gap between x=5 and x=7 — 2 m wide.

But the right rectangle is 4 m wide, so it goes from x=7 to x=11.

So the triangle is only in the left part.

But then the right rectangle is fully shaded, and the left rectangle has a triangle cut out.

But the left rectangle is 7 m wide, and the triangle is 5 m wide, so it's not filling the whole width.

So the shaded area is:

- Area of left rectangle (7 m × 6 m) minus area of triangle
- Plus area of right rectangle (4 m × 7 m)

But is that correct?

Wait — the total height of the left rectangle is 6 m, and the triangle goes up to 6 m, so yes.

But the right rectangle is 7 m tall, so it includes the area from y=0 to y=7.

But the dashed line is at y=6, so from y=6 to y=7, the right rectangle has an extra strip.

But the triangle doesn’t go there.

So yes, the shaded area is:

$$
\text{Area} = (\text{Left rect}) + (\text{Right rect}) - (\text{Triangle})
$$

Let’s compute.

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Step 2: Compute Areas

#### Left Rectangle:
- Width = 7 m
- Height = 6 m
- Area = $7 \times 6 = 42 \text{ m}^2$

#### Right Rectangle:
- Width = 4 m
- Height = 7 m
- Area = $4 \times 7 = 28 \text{ m}^2$

#### Total rectangle area before subtraction:
$42 + 28 = 70 \text{ m}^2$

#### White Triangle:
- It’s a right triangle with:
- Base = 5 m
- Height = 6 m
- Area = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 6 = 15 \text{ m}^2$

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Step 3: Subtract the Triangle

$$
\text{Shaded Area} = 70 - 15 = 55 \text{ m}^2
$$

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✔ Final Answer:

$$
\boxed{55} \text{ m}^2
$$

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Explanation Summary:

- The shape consists of two rectangles:
- Left: 7 m × 6 m = 42 m²
- Right: 4 m × 7 m = 28 m²
- Total area: 70 m²
- A right triangle (5 m base, 6 m height) is cut out from the left rectangle, area = 15 m²
- Shaded area = 70 − 15 = 55 m²

✔ Therefore, the area of the shaded region is 55 square meters.