G17e - Perimeter and area of composite shapes made up of polygons ... - Free Printable

Problem Analysis

The task involves finding the perimeter and area of two composite shapes, each consisting of a polygon and a sector of a circle. Let's solve this step by step for both shapes.

---

Shape 1:

#### Description
- The shape consists of a rectangle with a semicircle on top.
- Dimensions:
- Rectangle: Length = 16 cm, Width = 6 cm.
- Semicircle: Diameter = 16 cm (same as the length of the rectangle).

#### (a) Perimeter
The perimeter of the shape includes:
1. The two widths of the rectangle.
2. The curved part of the semicircle (half the circumference of a full circle).
3. The straight part of the semicircle (the diameter, which is already part of the rectangle and does not need to be counted again).

##### Step 1: Calculate the curved part of the semicircle
The radius of the semicircle is:
$$
r = \frac{\text{Diameter}}{2} = \frac{16}{2} = 8 \, \text{cm}.
$$
The circumference of a full circle is:
$$
C = 2\pi r = 2\pi(8) = 16\pi \, \text{cm}.
$$
The curved part of the semicircle is half of this:
$$
\text{Curved part} = \frac{16\pi}{2} = 8\pi \, \text{cm}.
$$

##### Step 2: Add the remaining sides of the rectangle
The two widths of the rectangle are:
$$
2 \times 6 = 12 \, \text{cm}.
$$

##### Step 3: Total perimeter
The total perimeter is the sum of the curved part of the semicircle and the two widths of the rectangle:
$$
\text{Perimeter} = 8\pi + 12.
$$
Using \( \pi \approx 3.14 \):
$$
8\pi \approx 8 \times 3.14 = 25.12.
$$
Thus:
$$
\text{Perimeter} \approx 25.12 + 12 = 37.12 \, \text{cm}.
$$

#### (b) Area
The area of the shape includes:
1. The area of the rectangle.
2. The area of the semicircle.

##### Step 1: Calculate the area of the rectangle
The area of the rectangle is:
$$
\text{Area of rectangle} = \text{Length} \times \text{Width} = 16 \times 6 = 96 \, \text{cm}^2.
$$

##### Step 2: Calculate the area of the semicircle
The area of a full circle is:
$$
A = \pi r^2 = \pi (8)^2 = 64\pi \, \text{cm}^2.
$$
The area of the semicircle is half of this:
$$
\text{Area of semicircle} = \frac{64\pi}{2} = 32\pi \, \text{cm}^2.
$$
Using \( \pi \approx 3.14 \):
$$
32\pi \approx 32 \times 3.14 = 100.48 \, \text{cm}^2.
$$

##### Step 3: Total area
The total area is the sum of the area of the rectangle and the area of the semicircle:
$$
\text{Area} = 96 + 32\pi.
$$
Using \( \pi \approx 3.14 \):
$$
\text{Area} \approx 96 + 100.48 = 196.48 \, \text{cm}^2.
$$

#### Final Answers for Shape 1
$$
\boxed{37.12 \, \text{cm}, 196.48 \, \text{cm}^2}
$$

---

Shape 2:

#### Description
- The shape consists of a trapezoid with a quarter-circle on one side.
- Dimensions:
- Trapezoid: Top base = 4 cm, Bottom base = 9 cm, Height = 4 cm, Slant sides = 3.5 cm each.
- Quarter-circle: Radius = 4 cm (same as the height of the trapezoid).

#### (a) Perimeter
The perimeter of the shape includes:
1. The three sides of the trapezoid (top base, bottom base, and one slant side).
2. The curved part of the quarter-circle (one-fourth of the circumference of a full circle).
3. The straight part of the quarter-circle (the radius, which is already part of the trapezoid and does not need to be counted again).

##### Step 1: Calculate the curved part of the quarter-circle
The radius of the quarter-circle is:
$$
r = 4 \, \text{cm}.
$$
The circumference of a full circle is:
$$
C = 2\pi r = 2\pi(4) = 8\pi \, \text{cm}.
$$
The curved part of the quarter-circle is one-fourth of this:
$$
\text{Curved part} = \frac{8\pi}{4} = 2\pi \, \text{cm}.
$$

##### Step 2: Add the remaining sides of the trapezoid
The three sides of the trapezoid are:
- Top base: 4 cm.
- Bottom base: 9 cm.
- One slant side: 3.5 cm.

##### Step 3: Total perimeter
The total perimeter is the sum of the curved part of the quarter-circle and the three sides of the trapezoid:
$$
\text{Perimeter} = 2\pi + 4 + 9 + 3.5 = 2\pi + 16.5.
$$
Using \( \pi \approx 3.14 \):
$$
2\pi \approx 2 \times 3.14 = 6.28.
$$
Thus:
$$
\text{Perimeter} \approx 6.28 + 16.5 = 22.78 \, \text{cm}.
$$

#### (b) Area
The area of the shape includes:
1. The area of the trapezoid.
2. The area of the quarter-circle.

##### Step 1: Calculate the area of the trapezoid
The area of a trapezoid is given by:
$$
\text{Area of trapezoid} = \frac{1}{2} \times (\text{Top base} + \text{Bottom base}) \times \text{Height}.
$$
Substituting the values:
$$
\text{Area of trapezoid} = \frac{1}{2} \times (4 + 9) \times 4 = \frac{1}{2} \times 13 \times 4 = 26 \, \text{cm}^2.
$$

##### Step 2: Calculate the area of the quarter-circle
The area of a full circle is:
$$
A = \pi r^2 = \pi (4)^2 = 16\pi \, \text{cm}^2.
$$
The area of the quarter-circle is one-fourth of this:
$$
\text{Area of quarter-circle} = \frac{16\pi}{4} = 4\pi \, \text{cm}^2.
$$
Using \( \pi \approx 3.14 \):
$$
4\pi \approx 4 \times 3.14 = 12.56 \, \text{cm}^2.
$$

##### Step 3: Total area
The total area is the sum of the area of the trapezoid and the area of the quarter-circle:
$$
\text{Area} = 26 + 4\pi.
$$
Using \( \pi \approx 3.14 \):
$$
\text{Area} \approx 26 + 12.56 = 38.56 \, \text{cm}^2.
$$

#### Final Answers for Shape 2
$$
\boxed{22.78 \, \text{cm}, 38.56 \, \text{cm}^2}
$$

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Final Combined Answer

For Shape 1:
$$
\boxed{37.12 \, \text{cm}, 196.48 \, \text{cm}^2}
$$
For Shape 2:
$$
\boxed{22.78 \, \text{cm}, 38.56 \, \text{cm}^2}
$$