Compound shapes with circles - TickTockMaths - Free Printable

To solve the problem, we need to calculate the area and perimeter of each compound shape. Let's go through each shape step by step.

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Shape 1: Blue Shape (Semicircle on top of a Trapezoid)

#### Given Dimensions:
- Semicircle diameter = 4 cm
- Trapezoid dimensions:
- Top base = 2.2 cm
- Bottom base = 4 cm
- Height = 2 cm

#### Step 1: Calculate the Area
1. Area of the Semicircle:
\[
\text{Radius of semicircle} = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
\[
\text{Area of full circle} = \pi r^2 = \pi (2)^2 = 4\pi \, \text{cm}^2
\]
\[
\text{Area of semicircle} = \frac{1}{2} \times 4\pi = 2\pi \, \text{cm}^2
\]

2. Area of the Trapezoid:
\[
\text{Area of trapezoid} = \frac{1}{2} \times (\text{Top base} + \text{Bottom base}) \times \text{Height}
\]
\[
\text{Area of trapezoid} = \frac{1}{2} \times (2.2 + 4) \times 2 = \frac{1}{2} \times 6.2 \times 2 = 6.2 \, \text{cm}^2
\]

3. Total Area:
\[
\text{Total area} = \text{Area of semicircle} + \text{Area of trapezoid}
\]
\[
\text{Total area} = 2\pi + 6.2 \approx 2 \times 3.1416 + 6.2 \approx 6.2832 + 6.2 \approx 12.5 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
1. Perimeter of the Semicircle:
\[
\text{Circumference of full circle} = 2\pi r = 2\pi \times 2 = 4\pi \, \text{cm}
\]
\[
\text{Arc length of semicircle} = \frac{1}{2} \times 4\pi = 2\pi \, \text{cm}
\]
\[
\text{Straight diameter of semicircle} = 4 \, \text{cm}
\]
\[
\text{Perimeter of semicircle part} = 2\pi + 4 \, \text{cm}
\]

2. Perimeter of the Trapezoid (excluding the shared side with the semicircle):
- The bottom base of the trapezoid is shared with the diameter of the semicircle, so it is not counted again.
- We need the lengths of the non-shared sides. Using the Pythagorean theorem for the slanted sides:
\[
\text{Slant height} = \sqrt{\left(\frac{\text{Difference in bases}}{2}\right)^2 + \text{Height}^2}
\]
\[
\text{Difference in bases} = 4 - 2.2 = 1.8 \, \text{cm}
\]
\[
\text{Slant height} = \sqrt{\left(\frac{1.8}{2}\right)^2 + 2^2} = \sqrt{(0.9)^2 + 2^2} = \sqrt{0.81 + 4} = \sqrt{4.81} \approx 2.19 \, \text{cm}
\]
- There are two slant heights, so:
\[
\text{Total slant height contribution} = 2 \times 2.19 \approx 4.38 \, \text{cm}
\]
- The top base of the trapezoid is 2.2 cm.

3. Total Perimeter:
\[
\text{Total perimeter} = \text{Arc length of semicircle} + \text{Straight diameter} + \text{Top base of trapezoid} + \text{Two slant heights}
\]
\[
\text{Total perimeter} = 2\pi + 4 + 2.2 + 4.38 \approx 2 \times 3.1416 + 4 + 2.2 + 4.38 \approx 6.2832 + 4 + 2.2 + 4.38 \approx 16.86 \, \text{cm}
\]
\[
\text{Total perimeter} \approx 16.9 \, \text{cm}
\]

#### Final Answer for Shape 1:
\[
\boxed{12.5 \, \text{cm}^2, 16.9 \, \text{cm}}
\]

---

Shape 2: Yellow Shape (Rectangle with a Semicircle Attached)

#### Given Dimensions:
- Rectangle dimensions:
- Length = 4 cm
- Width = 2 cm
- Semicircle diameter = 4 cm

#### Step 1: Calculate the Area
1. Area of the Rectangle:
\[
\text{Area of rectangle} = \text{Length} \times \text{Width} = 4 \times 2 = 8 \, \text{cm}^2
\]

2. Area of the Semicircle:
\[
\text{Radius of semicircle} = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
\[
\text{Area of full circle} = \pi r^2 = \pi (2)^2 = 4\pi \, \text{cm}^2
\]
\[
\text{Area of semicircle} = \frac{1}{2} \times 4\pi = 2\pi \, \text{cm}^2
\]

3. Total Area:
\[
\text{Total area} = \text{Area of rectangle} + \text{Area of semicircle}
\]
\[
\text{Total area} = 8 + 2\pi \approx 8 + 2 \times 3.1416 \approx 8 + 6.2832 \approx 14.3 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
1. Perimeter of the Semicircle:
\[
\text{Arc length of semicircle} = 2\pi \, \text{cm}
\]
\[
\text{Straight diameter of semicircle} = 4 \, \text{cm}
\]
\[
\text{Perimeter of semicircle part} = 2\pi + 4 \, \text{cm}
\]

2. Perimeter of the Rectangle (excluding the shared side with the semicircle):
- The bottom side of the rectangle is shared with the diameter of the semicircle, so it is not counted again.
- The remaining sides are:
\[
\text{Two widths} = 2 \times 2 = 4 \, \text{cm}
\]
\[
\text{One length} = 4 \, \text{cm}
\]

3. Total Perimeter:
\[
\text{Total perimeter} = \text{Arc length of semicircle} + \text{Straight diameter} + \text{Two widths} + \text{One length}
\]
\[
\text{Total perimeter} = 2\pi + 4 + 4 + 4 \approx 2 \times 3.1416 + 4 + 4 + 4 \approx 6.2832 + 12 \approx 18.3 \, \text{cm}
\]

#### Final Answer for Shape 2:
\[
\boxed{14.3 \, \text{cm}^2, 18.3 \, \text{cm}}
\]

---

Shape 3: Red Shape (Rectangle with Two Quarter-Circles)

#### Given Dimensions:
- Rectangle dimensions:
- Length = 10 cm
- Width = 2 cm
- Quarter-circle diameters:
- Top quarter-circle diameter = 2 cm
- Bottom quarter-circle diameter = 1 cm

#### Step 1: Calculate the Area
1. Area of the Rectangle:
\[
\text{Area of rectangle} = \text{Length} \times \text{Width} = 10 \times 2 = 20 \, \text{cm}^2
\]

2. Area of the Top Quarter-Circle:
\[
\text{Radius of top quarter-circle} = \frac{\text{Diameter}}{2} = \frac{2}{2} = 1 \, \text{cm}
\]
\[
\text{Area of full circle} = \pi r^2 = \pi (1)^2 = \pi \, \text{cm}^2
\]
\[
\text{Area of top quarter-circle} = \frac{1}{4} \times \pi = \frac{\pi}{4} \, \text{cm}^2
\]

3. Area of the Bottom Quarter-Circle:
\[
\text{Radius of bottom quarter-circle} = \frac{\text{Diameter}}{2} = \frac{1}{2} = 0.5 \, \text{cm}
\]
\[
\text{Area of full circle} = \pi r^2 = \pi (0.5)^2 = 0.25\pi \, \text{cm}^2
\]
\[
\text{Area of bottom quarter-circle} = \frac{1}{4} \times 0.25\pi = \frac{0.25\pi}{4} = 0.0625\pi \, \text{cm}^2
\]

4. Total Area:
\[
\text{Total area} = \text{Area of rectangle} + \text{Area of top quarter-circle} + \text{Area of bottom quarter-circle}
\]
\[
\text{Total area} = 20 + \frac{\pi}{4} + 0.0625\pi \approx 20 + \frac{3.1416}{4} + 0.0625 \times 3.1416
\]
\[
\text{Total area} \approx 20 + 0.7854 + 0.1963 \approx 20.98 \, \text{cm}^2
\]
\[
\text{Total area} \approx 21.0 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
1. Perimeter of the Top Quarter-Circle:
\[
\text{Circumference of full circle} = 2\pi r = 2\pi \times 1 = 2\pi \, \text{cm}
\]
\[
\text{Arc length of top quarter-circle} = \frac{1}{4} \times 2\pi = \frac{\pi}{2} \, \text{cm}
\]
\[
\text{Two radii of top quarter-circle} = 2 \times 1 = 2 \, \text{cm}
\]
\[
\text{Perimeter of top quarter-circle part} = \frac{\pi}{2} + 2 \, \text{cm}
\]

2. Perimeter of the Bottom Quarter-Circle:
\[
\text{Circumference of full circle} = 2\pi r = 2\pi \times 0.5 = \pi \, \text{cm}
\]
\[
\text{Arc length of bottom quarter-circle} = \frac{1}{4} \times \pi = \frac{\pi}{4} \, \text{cm}
\]
\[
\text{Two radii of bottom quarter-circle} = 2 \times 0.5 = 1 \, \text{cm}
\]
\[
\text{Perimeter of bottom quarter-circle part} = \frac{\pi}{4} + 1 \, \text{cm}
\]

3. Perimeter of the Rectangle (excluding the shared sides with the quarter-circles):
- The top and bottom sides of the rectangle are partially replaced by the quarter-circles, so only the two vertical sides remain:
\[
\text{Two vertical sides} = 2 \times 10 = 20 \, \text{cm}
\]

4. Total Perimeter:
\[
\text{Total perimeter} = \text{Arc length of top quarter-circle} + \text{Two radii of top quarter-circle} + \text{Arc length of bottom quarter-circle} + \text{Two radii of bottom quarter-circle} + \text{Two vertical sides of rectangle}
\]
\[
\text{Total perimeter} = \frac{\pi}{2} + 2 + \frac{\pi}{4} + 1 + 20
\]
\[
\text{Total perimeter} \approx \frac{3.1416}{2} + 2 + \frac{3.1416}{4} + 1 + 20
\]
\[
\text{Total perimeter} \approx 1.5708 + 2 + 0.7854 + 1 + 20 \approx 25.3562 \, \text{cm}
\]
\[
\text{Total perimeter} \approx 25.4 \, \text{cm}
\]

#### Final Answer for Shape 3:
\[
\boxed{21.0 \, \text{cm}^2, 25.4 \, \text{cm}}
\]

---

Shape 4: Green Shape (Square with a Quarter-Circle Cut Out)

#### Given Dimensions:
- Square side = 1 cm
- Quarter-circle radius = 1 cm

#### Step 1: Calculate the Area
1. Area of the Square:
\[
\text{Area of square} = \text{Side}^2 = 1^2 = 1 \, \text{cm}^2
\]

2. Area of the Quarter-Circle:
\[
\text{Area of full circle} = \pi r^2 = \pi (1)^2 = \pi \, \text{cm}^2
\]
\[
\text{Area of quarter-circle} = \frac{1}{4} \times \pi = \frac{\pi}{4} \, \text{cm}^2
\]

3. Total Area:
\[
\text{Total area} = \text{Area of square} - \text{Area of quarter-circle}
\]
\[
\text{Total area} = 1 - \frac{\pi}{4} \approx 1 - \frac{3.1416}{4} \approx 1 - 0.7854 \approx 0.2146 \, \text{cm}^2
\]
\[
\text{Total area} \approx 0.2 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
1. Perimeter of the Square (excluding the cut-out side):
- Three sides of the square remain:
\[
\text{Three sides} = 3 \times 1 = 3 \, \text{cm}
\]

2. Perimeter of the Quarter-Circle:
\[
\text{Circumference of full circle} = 2\pi r = 2\pi \times 1 = 2\pi \, \text{cm}
\]
\[
\text{Arc length of quarter-circle} = \frac{1}{4} \times 2\pi = \frac{\pi}{2} \, \text{cm}
\]
\[
\text{Two radii of quarter-circle} = 2 \times 1 = 2 \, \text{cm}
\]
\[
\text{Perimeter of quarter-circle part} = \frac{\pi}{2} + 2 \, \text{cm}
\]

3. Total Perimeter:
\[
\text{Total perimeter} = \text{Three sides of square} + \text{Arc length of quarter-circle} + \text{Two radii of quarter-circle}
\]
\[
\text{Total perimeter} = 3 + \frac{\pi}{2} + 2 \approx 3 + \frac{3.1416}{2} + 2 \approx 3 + 1.5708 + 2 \approx 6.5708 \, \text{cm}
\]
\[
\text{Total perimeter} \approx 6.6 \, \text{cm}
\]

#### Final Answer for Shape 4:
\[
\boxed{0.2 \, \text{cm}^2, 6.6 \, \text{cm}}
\]

---

Shape 5: Gray Shape (Triangle on Top of a Semicircle)

#### Given Dimensions:
- Triangle dimensions:
- Base = 4 cm
- Height = 2 cm
- Semicircle diameter = 4 cm

#### Step 1: Calculate the Area
1. Area of the Triangle:
\[
\text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 4 \times 2 = 4 \, \text{cm}^2
\]

2. Area of the Semicircle:
\[
\text{Radius of semicircle} = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \, \text{cm}
\]
\[
\text{Area of full circle} = \pi r^2 = \pi (2)^2 = 4\pi \, \text{cm}^2
\]
\[
\text{Area of semicircle} = \frac{1}{2} \times 4\pi = 2\pi \, \text{cm}^2
\]

3. Total Area:
\[
\text{Total area} = \text{Area of triangle} + \text{Area of semicircle}
\]
\[
\text{Total area} = 4 + 2\pi \approx 4 + 2 \times 3.1416 \approx 4 + 6.2832 \approx 10.3 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
1. Perimeter of the Semicircle:
\[
\text{Arc length of semicircle} = 2\pi \, \text{cm}
\]
\[
\text{Straight diameter of semicircle} = 4 \, \text{cm}
\]
\[
\text{Perimeter of semicircle part} = 2\pi + 4 \, \text{cm}
\]

2. Perimeter of the Triangle (excluding the shared side with the semicircle):
- The base of the triangle is shared with the diameter of the semicircle, so it is not counted again.
- We need the lengths of the two slant sides. Using the Pythagorean theorem:
\[
\text{Slant height} = \sqrt{\left(\frac{\text{Base}}{2}\right)^2 + \text{Height}^2}
\]
\[
\text{Slant height} = \sqrt{\left(\frac{4}{2}\right)^2 + 2^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 \, \text{cm}
\]
- There are two slant heights, so:
\[
\text{Total slant height contribution} = 2 \times 2.83 \approx 5.66 \, \text{cm}
\]

3. Total Perimeter:
\[
\text{Total perimeter} = \text{Arc length of semicircle} + \text{Straight diameter} + \text{Two slant heights}
\]
\[
\text{Total perimeter} = 2\pi + 4 + 5.66 \approx 2 \times 3.1416 + 4 + 5.66 \approx 6.2832 + 4 + 5.66 \approx 15.94 \, \text{cm}
\]
\[
\text{Total perimeter} \approx 15.9 \, \text{cm}
\]

#### Final Answer for Shape 5:
\[
\boxed{10.3 \, \text{cm}^2, 15.9 \, \text{cm}}
\]

---

Shape 6: Purple Shape (Right Triangle Inside a Circle)

#### Given Dimensions:
- Right triangle dimensions:
- Legs = 3 cm and 4 cm
- Hypotenuse = 5 cm
- Circle radius = 3 cm (since the hypotenuse is the diameter)

#### Step 1: Calculate the Area
1. Area of the Circle:
\[
\text{Radius of circle} = \frac{\text{Diameter}}{2} = \frac{5}{2} = 2.5 \, \text{cm}
\]
\[
\text{Area of circle} = \pi r^2 = \pi (2.5)^2 = 6.25\pi \, \text{cm}^2
\]

2. Area of the Right Triangle:
\[
\text{Area of triangle} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 3 \times 4 = 6 \, \text{cm}^2
\]

3. Total Area:
\[
\text{Total area} = \text{Area of circle} - \text{Area of triangle}
\]
\[
\text{Total area} = 6.25\pi - 6 \approx 6.25 \times 3.1416 - 6 \approx 19.6345 - 6 \approx 13.6 \, \text{cm}^2
\]

#### Step 2: Calculate the Perimeter
1. Perimeter of the Circle (excluding the triangle sides):
- The arc length of the circle that is not part of the triangle is the circumference minus the hypotenuse:
\[
\text{Circumference of circle} = 2\pi r = 2\pi \times 2.5 = 5\pi \, \text{cm}
\]
\[
\text{Arc length} = 5\pi - 5 \, \text{cm}
\]

2. Perimeter of the Triangle (excluding the hypotenuse):
- The two legs of the triangle are:
\[
\text{Two legs} = 3 + 4 = 7 \, \text{cm}
\]

3. Total Perimeter:
\[
\text{Total perimeter} = \text{Arc length} + \text{Two legs}
\]
\[
\text{Total perimeter} = 5\pi - 5 + 7 \approx 5 \times 3.1416 - 5 + 7 \approx 15.708 - 5 + 7 \approx 17.7 \, \text{cm}
\]

#### Final Answer for Shape 6:
\[
\boxed{13.6 \, \text{cm}^2, 17.7 \, \text{cm}}
\]

---

Final Answers for All Shapes:

1. \(\boxed{12.5 \, \text{cm}^2, 16.9 \, \text{cm}}\)
2. \(\boxed{14.3 \, \text{cm}^2, 18.3 \, \text{cm}}\)
3. \(\boxed{21.0 \, \text{cm}^2, 25.4 \, \text{cm}}\)
4. \(\boxed{0.2 \, \text{cm}^2, 6.6 \, \text{cm}}\)
5. \(\boxed{10.3 \, \text{cm}^2, 15.9 \, \text{cm}}\)
6. \(\boxed{13.6 \, \text{cm}^2, 17.7 \, \text{cm}}\)