Question

cyu perimeter and area of composite shapes

1. determine the area and perimeter of the following:

a)

b)

Answer:

Part (a)

Step 1: Analyze the Composite Shape

The shape can be considered as a rectangle formed by combining the given segments. Let's find the length and width.

• The horizontal length: \( 9 + 4 = 13 \) m (Wait, no, actually, looking at the markings, the vertical side is 7 m, and the horizontal side: let's re - examine. Wait, the shape is a rectangle - like composite. Let's find the perimeter first. For perimeter, we can use the property of rectangles (since the indentations will have their opposite sides equal).

The vertical sides: the total vertical length. Let's see, the right - hand side is 7 m. The left - hand side: the lower part has a segment of 4 m, and the upper part should be equal to the right - hand side? Wait, no, let's calculate the perimeter by adding all outer sides.
Wait, another approach: the composite shape can be thought of as a rectangle with length \( 9 + 4=13 \) m and width \( 7 \) m? No, wait, maybe it's better to calculate the perimeter by considering the sum of all outer sides.
Wait, actually, the perimeter of a composite shape with right angles can be calculated as if it were a rectangle, because the horizontal and vertical segments will add up.
Let's find the length and width:

• Horizontal length: Let's assume the bottom horizontal segment (the one with 4 m and the top horizontal segment with 9 m) sum up. Wait, the vertical side is 7 m. Let's calculate the perimeter:

The formula for the perimeter of a rectangle is \( P = 2\times(l + w) \). Let's find \( l \) and \( w \).
Looking at the horizontal direction: the total length \( l=9 + 4 = 13 \) m? Wait, no, maybe the vertical side is 7 m, and the horizontal side: let's see the vertical segments. The right - hand vertical side is 7 m. The left - hand vertical side: the lower part is 4 m, and the upper part should be equal to 7 m? No, that can't be. Wait, I think I made a mistake. Let's look at the markings: the equal marks. The horizontal segments with three marks are equal, and the vertical segments with two marks are equal.
Wait, let's re - draw mentally:

• The top horizontal segment (with three marks) and the bottom horizontal segment (with three marks) are equal.
• The left - hand vertical segment (with two marks) and the right - hand vertical segment (with two marks) are equal. Wait, the right - hand vertical segment is 7 m. So the left - hand vertical segment is also 7 m.
• The horizontal segments: the top horizontal (with three marks) and the bottom horizontal (with three marks) have length, let's see, the top horizontal has a segment of 9 m, and the bottom horizontal has a segment of 4 m? No, maybe the total horizontal length is \( 9+4 = 13 \) m, and the vertical length is 7 m. Wait, no, let's calculate the perimeter.

Perimeter \( P=2\times((9 + 4)+7)=2\times(13 + 7)=2\times20 = 40 \) m? Wait, no, that's not right. Wait, maybe the length is \( 9+4 = 13 \) m and the width is 7 m. Wait, let's check the area.
Area of the composite shape: If we consider it as a rectangle with length \( l = 9 + 4=13 \) m and width \( w = 7 \) m, then the area \( A=l\times w=13\times7 = 91 \) \( m^{2} \). Wait, but let's verify.
Wait, maybe the correct way:

• Perimeter:

The horizontal sides: the top and bottom. The top horizontal side: let's say the length is \( 9 + 4 = 13 \) m, and the bottom horizontal side is also 13 m.
The vertical sides: the left and right. The right vertical side is 7 m, and the left vertical side is 7 m.
So perimeter \( P=2\times(13 + 7)=2\times20 = 40 \) m.

• Area:

The area of the composite shape (since it's a rectangle - like shape) is \( A=(9 + 4)\times7=13\times7 = 91 \) \( m^{2} \). Wait, but let's check again. Maybe the length is \( 9+4 = 13 \) and width is 7. So perimeter is 40 m and area is 91 \( m^{2} \).

Step 2: Recalculate (Correcting the Earlier Mistake)

Wait, actually, looking at the diagram again, the vertical side is 7 m, and the horizontal side: the bottom horizontal segment is 4 m, and the top horizontal segment is 9 m. But the vertical segments: the right - hand vertical is 7 m, and the left - hand vertical: the lower part is 4 m, and the upper part is equal to the right - hand vertical? No, that's wrong.
Wait, a better approach:
For perimeter:
The sum of all horizontal sides: the top horizontal (let's call it \( h_1 \)) and the bottom horizontal (\( h_2 \)). Since it's a composite with right angles, \( h_1=h_2 \).
The sum of all vertical sides: the left vertical (\( v_1 \)) and the right vertical (\( v_2 \)), \( v_1 = v_2 \).
From the diagram, the right vertical side \( v_2 = 7 \) m.
The horizontal segments: one part is 9 m, another is 4 m. So the total horizontal length \( l=9 + 4 = 13 \) m.
The vertical length \( w = 7 \) m.
Perimeter \( P = 2\times(l + w)=2\times(13 + 7)=2\times20 = 40 \) m.
Area \( A=l\times w=13\times7 = 91 \) \( m^{2} \).

Part (b)

Step 1: Analyze the Composite Shape

The composite shape consists of a trapezoid and a rectangle? Wait, no, let's see:

• The shape has a rectangular part (with length 5 in and height 6 in) and a trapezoidal part (with bases \( 7 + 5=12 \) in and 15 in, and height \( 10 - 6 = 4 \) in). Wait, no, let's re - examine.

Wait, the total height of the shape is 10 in. The rectangular part is on top with height 6 in and width 5 in. The lower part is a trapezoid with the top base \( 7+5 = 12 \) in, bottom base 15 in, and height \( 10 - 6=4 \) in.

Step 2: Calculate the Area

• Area of the rectangle: \( A_{rect}=length\times width=5\times6 = 30 \) \( in^{2} \)
• Area of the trapezoid: The formula for the area of a trapezoid is \( A_{trap}=\frac{(a + b)}{2}\times h \), where \( a = 12 \) in, \( b = 15 \) in, and \( h = 4 \) in.

\( A_{trap}=\frac{(12 + 15)}{2}\times4=\frac{27}{2}\times4 = 27\times2=54 \) \( in^{2} \)

• Total area \( A = A_{rect}+A_{trap}=30 + 54 = 84 \) \( in^{2} \)

Step 3: Calculate the Perimeter

To calculate the perimeter, we need to find the length of the slant side. Wait, the left - hand side is a slant side. Let's find the difference in the horizontal bases of the trapezoid: \( 15-(7 + 5)=15 - 12 = 3 \) in. The height of the trapezoid is \( 10 - 6 = 4 \) in. So the slant side (let's call it \( s \)) can be found using the Pythagorean theorem: \( s=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5 \) in.
Now, let's sum up all the sides:

• Top side of the rectangle: 5 in
• Right - hand side of the rectangle: 6 in
• Right - hand vertical side of the lower part: 10 - 6 = 4 in? No, wait, the right - hand side of the entire shape is 10 in (vertical). Wait, let's list all the sides:

1. Top horizontal side of the rectangle: 5 in
2. Right - hand vertical side: 10 in
3. Bottom horizontal side: 15 in
4. Slant side: 5 in (calculated above)
5. Top horizontal side of the trapezoid (the one with 7 in and 5 in): 7+5 = 12 in
6. Left - hand vertical side of the rectangle: 6 in

Wait, let's add them up:
\( P=5 + 10+15 + 5+12 + 6=53 \) in? Wait, no, let's re - check:

• From the top - right corner, go down 10 in (vertical side: 10 in)
• Then left 15 in (bottom horizontal side: 15 in)
• Then up the slant side: 5 in (we found using Pythagoras: horizontal difference 3 in, vertical difference 4 in, so slant side 5 in)
• Then right along the top of the trapezoid: 7 + 5=12 in
• Then down the left - hand vertical side of the rectangle: 6 in
• Then right along the top of the rectangle: 5 in

Wait, adding these: \( 10+15 + 5+12 + 6+5=53 \) in. Wait, but let's check the area again.
Alternative way to calculate area:
The composite shape can be seen as a rectangle (5 in by 10 in) plus a trapezoid? No, wait, no. Wait, the area of the composite shape:

• The rectangular part: 5 in (width) and 6 in (height): \( A_1 = 5\times6=30 \)
• The trapezoidal part: the two parallel sides are \( 15 \) in and \( 7 + 5 = 12 \) in, and the height is \( 10 - 6 = 4 \) in.

\( A_2=\frac{(12 + 15)}{2}\times4=\frac{27}{2}\times4 = 54 \)
Total area \( A=30 + 54 = 84 \) \( in^{2} \)

Final Answers

Part (a)

• Perimeter: \(\boldsymbol{40}\) m
• Area: \(\boldsymbol{91}\) \( m^{2} \)

Part (b)

• Perimeter: \(\boldsymbol{53}\) in
• Area: \(\boldsymbol{84}\) \( in^{2} \)