Perimeter of a Composite Figure - Free Printable

To find the perimeter of the composite figures, we need to calculate the total length of the outer boundary for each shape.

Figure 1: The L-Shaped Polygon

1. Identify the given sides:
* Left vertical side = $8\text{ cm}$
* Top horizontal side = $6\text{ cm}$
* Bottom horizontal side = $10\text{ cm}$
* Right vertical side (bottom part) = $2\text{ cm}$

2. Determine the missing sides:
* The inner vertical side: The total height on the left is $8\text{ cm}$. The bottom part of the right side is $2\text{ cm}$. The missing vertical segment connects the top part to the bottom part. So, the length of the inner vertical side is $8\text{ cm} - 2\text{ cm} = 6\text{ cm}$.
* The inner horizontal side: The total width at the bottom is $10\text{ cm}$. The top width is $6\text{ cm}$. The missing horizontal segment connects the top part to the bottom part. So, the length of the inner horizontal side is $10\text{ cm} - 6\text{ cm} = 4\text{ cm}$.

3. Calculate the perimeter:
* Perimeter = Sum of all outer sides
* Perimeter = $8\text{ cm}$ (left) + $6\text{ cm}$ (top) + $6\text{ cm}$ (inner vertical) + $4\text{ cm}$ (inner horizontal) + $2\text{ cm}$ (bottom right vertical) + $10\text{ cm}$ (bottom)
* Wait, let's re-evaluate the sides to ensure we are tracing the perimeter correctly.
* Starting from top-left and going clockwise:
* Top side: $6\text{ cm}$
* Inner vertical side (going down): $8\text{ cm} - 2\text{ cm} = 6\text{ cm}$
* Inner horizontal side (going right): $10\text{ cm} - 6\text{ cm} = 4\text{ cm}$
* Right vertical side (going down): $2\text{ cm}$
* Bottom side (going left): $10\text{ cm}$
* Left vertical side (going up): $8\text{ cm}$
* Sum: $6 + 6 + 4 + 2 + 10 + 8 = 36\text{ cm}$.

*Alternative method:* For rectilinear shapes like this, the perimeter is often equal to the perimeter of the bounding rectangle.
* Bounding rectangle width = $10\text{ cm}$
* Bounding rectangle height = $8\text{ cm}$
* Perimeter = $2 \times (10 + 8) = 2 \times 18 = 36\text{ cm}$.
* Both methods give $36\text{ cm}$.

Figure 2: The Ice Cream Cone Shape

1. Identify the given sides:
* The shape consists of a semi-circle on top and a triangle on the bottom.
* The two slanted sides of the triangle are marked with ticks, indicating they are equal in length. One is labeled $6\text{ cm}$, so the other is also $6\text{ cm}$.
* The dashed line represents the diameter of the semi-circle and the base of the triangle.

2. Determine the missing side (the curved part):
* We need the length of the arc of the semi-circle.
* The base of the triangle is the diameter of the semi-circle. However, the length of the base is not explicitly given.
* Let's look at the triangle again. It is an isosceles triangle with two sides of $6\text{ cm}$. Is it equilateral? The markings (single tick on each side) just mean those two are equal. Without an angle or the base length, we cannot calculate the arc length exactly unless we assume something.
* Let's re-examine the image. Ah, looking closely at the triangle, there are tick marks on the two slanted sides. There is *no* tick mark on the dashed line.
* Wait, is it possible the triangle is equilateral? If the triangle were equilateral, the base would be $6\text{ cm}$. If the base is $6\text{ cm}$, then the diameter of the semi-circle is $6\text{ cm}$.
* Let's assume the triangle is equilateral because that is a common setup in such problems when only one side length is given for an isosceles triangle and no other dimensions are provided. If the base is $6\text{ cm}$:
* Diameter ($d$) = $6\text{ cm}$.
* Radius ($r$) = $3\text{ cm}$.
* Circumference of a full circle = $\pi \times d = 3.14 \times 6 = 18.84\text{ cm}$.
* Length of the semi-circle arc = $18.84 / 2 = 9.42\text{ cm}$.
* Perimeter = Arc length + Side 1 + Side 2
* Perimeter = $9.42 + 6 + 6 = 21.42\text{ cm}$.

*Let's reconsider if there's another interpretation.*
Maybe the "6cm" label applies to the radius? No, it's clearly next to the slanted side.
Maybe the dashed line is not the diameter? No, it looks like the diameter.
Is it possible the triangle is a right-angled isosceles triangle? If so, base = $\sqrt{6^2 + 6^2} = \sqrt{72} \approx 8.48$.
Is it possible the triangle is just isosceles and we are missing info?
Let's look at the tick marks again. The slanted sides have single tick marks. The dashed line has a plus sign (+) in the middle, which usually indicates the midpoint. This confirms the dashed line is bisected, but doesn't give a length.
However, in many geometry problems of this level, if a triangle looks equilateral and has two sides marked equal, and no other info is given, it's often intended to be equilateral. Let's proceed with the assumption that the base is $6\text{ cm}$.

*Calculation with base = 6 cm:*
* Arc length = $\frac{1}{2} \times \pi \times 6 = 3\pi \approx 9.42\text{ cm}$.
* Straight sides = $6 + 6 = 12\text{ cm}$.
* Total Perimeter = $12 + 9.42 = 21.42\text{ cm}$.

*Let's try to see if I missed a number.*
Looking at the image again... no other numbers.
Is it possible the "6cm" refers to the radius? The text is placed next to the side. It's most likely the side length.
What if the triangle is a 30-60-90 or something? No, that's too complex without angles.
The most logical assumption for a student problem is that the triangle is equilateral, making the diameter $6\text{ cm}$.

Let's double check the first figure calculation.
$8 + 6 + (8-2) + (10-6) + 2 + 10$
$8 + 6 + 6 + 4 + 2 + 10 = 36$. Correct.

Let's double check the second figure.
If the triangle is equilateral, the perimeter is $6 + 6 + \frac{1}{2} \pi (6) = 12 + 3\pi$.
Using $\pi \approx 3.14$: $12 + 9.42 = 21.42$.
Using $\pi \approx 3.14159$: $12 + 9.4247 = 21.4247$.
Let's round to one or two decimal places. $21.4\text{ cm}$ or $21.42\text{ cm}$.

Wait, is it possible the "6cm" label is for the radius?
The label is next to the slanted side. It is extremely unlikely to be the radius unless the side *is* the radius, which would mean the triangle is formed by two radii and the chord. But the shape is a semi-circle on top of a triangle. The slanted sides are tangent? No, they meet at a point.
If the slanted sides are radii, then the triangle is formed by two radii and the chord? No, the semi-circle is attached to the top. The dashed line is the diameter. The slanted sides meet at the bottom vertex.
If the slanted sides are $6\text{ cm}$, and the triangle is equilateral, then the diameter is $6\text{ cm}$.
If the slanted sides are $6\text{ cm}$, and the triangle is a right isosceles triangle (vertex angle 90), then diameter = $\sqrt{72} \approx 8.49$. Arc = $\frac{1}{2} \pi (8.49) \approx 13.3$. Perimeter = $12 + 13.3 = 25.3$.
Given the visual appearance, the triangle looks equilateral (all sides roughly equal). The angles look like 60 degrees.
So, assumption: Triangle is equilateral. Base = $6\text{ cm}$.

Summary of steps for final output:
1. L-Shape:
* Add all outer lengths.
* Left: $8$
* Top: $6$
* Inner Vertical: $8 - 2 = 6$
* Inner Horizontal: $10 - 6 = 4$
* Right Vertical: $2$
* Bottom: $10$
* Total: $8 + 6 + 6 + 4 + 2 + 10 = 36\text{ cm}$.

2. Ice Cream Shape:
* Assume the triangle is equilateral because two sides are $6\text{ cm}$ and no other info is given.
* This means the diameter of the semi-circle is $6\text{ cm}$.
* Calculate the arc length of the semi-circle: $\frac{1}{2} \times \pi \times d = \frac{1}{2} \times 3.14 \times 6 = 9.42\text{ cm}$.
* Add the two straight sides of the triangle: $6 + 6 = 12\text{ cm}$.
* Total Perimeter: $12 + 9.42 = 21.42\text{ cm}$.

I will provide the answers for both figures.

Final Answer:
Figure 1 (L-shape): $36\text{ cm}$
Figure 2 (Ice cream shape): $\approx 21.42\text{ cm}$ (assuming the triangle is equilateral)

Wait, I should check if there is any other standard interpretation.
Could the "6cm" on the right figure be the radius?
If the radius is $6\text{ cm}$, then the diameter is $12\text{ cm}$.
Then the slanted sides... if the triangle is equilateral, the sides would be $12\text{ cm}$. But the label says $6\text{ cm}$.
So the label "6cm" must be the side length.
If the side length is $6\text{ cm}$ and it's equilateral, diameter is $6\text{ cm}$.
This seems the most consistent.

Let's write down the solution clearly.

Step 1: Perimeter of the L-shaped figure
- The perimeter is the total distance around the outside of the shape.
- We have the following side lengths:
- Left side: $8\text{ cm}$
- Top side: $6\text{ cm}$
- Bottom side: $10\text{ cm}$
- Bottom-right vertical side: $2\text{ cm}$
- We need to find the lengths of the two inner "corner" sides.
- The inner vertical side length is the total height ($8\text{ cm}$) minus the bottom vertical part ($2\text{ cm}$). So, $8 - 2 = 6\text{ cm}$.
- The inner horizontal side length is the total width ($10\text{ cm}$) minus the top width ($6\text{ cm}$). So, $10 - 6 = 4\text{ cm}$.
- Now, add all the sides together:
$8 + 6 + 6 + 4 + 2 + 10 = 36\text{ cm}$.

Step 2: Perimeter of the ice cream cone figure
- This shape is made of a semi-circle on top and a triangle on the bottom.
- The perimeter includes the curved top part and the two straight sides at the bottom. The dashed line is inside the shape, so we don't count it.
- The two straight sides are marked as equal and one is labeled $6\text{ cm}$. So, both are $6\text{ cm}$.
- To find the length of the curved part (the arc), we need the diameter of the semi-circle. The diameter is the base of the triangle.
- Since the triangle has two equal sides of $6\text{ cm}$ and no other information is given, we assume it is an equilateral triangle. This means the base (diameter) is also $6\text{ cm}$.
- The circumference of a full circle is $C = \pi \times d$.
- The length of the semi-circle arc is half of that: $\frac{1}{2} \times \pi \times 6$.
- Using $\pi \approx 3.14$:
Arc length = $0.5 \times 3.14 \times 6 = 9.42\text{ cm}$.
- Total perimeter = Arc length + Side 1 + Side 2
- Total perimeter = $9.42 + 6 + 6 = 21.42\text{ cm}$.

The prompt asks for "the problem" solved. Since there are two distinct figures, I will provide the solution for both.

Final check of the calculations.
$8+6+4+2+10+8$? No.
Sides:
1. Left: 8
2. Top: 6
3. Inner Vertical: 6 (8-2)
4. Inner Horizontal: 4 (10-6)
5. Right Vertical: 2
6. Bottom: 10
Sum: $8+6+6+4+2+10 = 36$. Correct.

Figure 2:
Sides: 6, 6.
Arc: $\frac{1}{2} \pi (6) = 3\pi \approx 9.42$.
Sum: $12 + 9.42 = 21.42$. Correct.

I will formulate the response now.

Explanation:
Figure 1: The L-Shaped Polygon
To find the perimeter, we add up the lengths of all the outer sides.
1. Identify known sides:
* Left side = $8\text{ cm}$
* Top side = $6\text{ cm}$
* Bottom side = $10\text{ cm}$
* Short right side = $2\text{ cm}$
2. Find missing sides:
* The vertical side in the "crook" of the L is the difference between the total height and the short right side: $8\text{ cm} - 2\text{ cm} = 6\text{ cm}$.
* The horizontal side in the "crook" is the difference between the total bottom width and the top width: $10\text{ cm} - 6\text{ cm} = 4\text{ cm}$.
3. Calculate total perimeter:
* $8 + 6 + 6 + 4 + 2 + 10 = 36\text{ cm}$.

Figure 2: The Ice Cream Cone Shape
This shape is a combination of a semi-circle and a triangle. The perimeter is the distance around the outside.
1. Identify straight sides:
* The two slanted sides are marked as equal. One is $6\text{ cm}$, so the other is $6\text{ cm}$.
2. Identify the curved side:
* The curve is a semi-circle. Its diameter is the base of the triangle.
* Since the triangle has two sides of $6\text{ cm}$ and looks equilateral, we assume the base (diameter) is also $6\text{ cm}$.
* The circumference of a full circle is $\pi \times \text{diameter}$. For a semi-circle, it is half of that.
* Arc length = $\frac{1}{2} \times 3.14 \times 6\text{ cm} \approx 9.42\text{ cm}$.
3. Calculate total perimeter:
* Add the two straight sides and the curved arc: $6 + 6 + 9.42 = 21.42\text{ cm}$.

Final Answer:
Figure 1: $36\text{ cm}$
Figure 2: $\approx 21.42\text{ cm}$