29. Area, perimeter, and volume of 2D/3D shapes (triangles, circles, rectangles, cylinders, etc.) Surface area calculations problems

 29. Area, perimeter, and volume of 2D/3D shapes (triangles, circles, rectangles, cylinders, etc.) Surface area calculations problems - step by step solution procedure and detailed explanation for each problem. Give me 10 examples for each variety of problem. 

A. Perimeter Calculations 

Below, I’ll provide a detailed explanation of how to calculate the area, perimeter, and volume of various 2D and 3D shapes, along with surface area calculations for 3D shapes. For each type of problem (area, perimeter, volume, surface area), I’ll include 10 example problems with step-by-step solutions and explanations. The shapes covered will include triangles, circles, rectangles, cylinders, cubes, spheres, and rectangular prisms. Since the request is extensive, I’ll structure it clearly and concisely, ensuring each problem is solved systematically.

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1. Area Calculations

General Explanation

The area of a shape measures the amount of space enclosed within its boundaries, typically in square units (e.g., cm², m²). Different shapes have specific formulas:

• Triangle:

  A = \frac{1}{2} \cdot \text{base} \cdot \text{height}
• Circle:

  A = \pi \cdot r^2

  , where ( r ) is the radius
• Rectangle:

  A = \text{length} \cdot \text{width}
• Square:

  A = \text{side}^2

Example Problems

Problem 1: Triangle

Find the area of a triangle with base 6 cm and height 8 cm.

• Step 1: Use the formula

  A = \frac{1}{2} \cdot \text{base} \cdot \text{height}

  .
• Step 2: Substitute:

  A = \frac{1}{2} \cdot 6 \cdot 8 = \frac{1}{2} \cdot 48 = 24

  .
• Answer: Area = 24 cm².
• Explanation: The base and height are perpendicular, and the formula halves the product to account for the triangular shape.

Problem 2: Triangle

Triangle with base 10 m and height 5 m.

• A = \frac{1}{2} \cdot 10 \cdot 5 = \frac{1}{2} \cdot 50 = 25

  .
• Area = 25 m².

Problem 3: Circle

Find the area of a circle with radius 7 cm (use

\pi \approx 3.14

).

• A = \pi \cdot r^2 = 3.14 \cdot 7^2 = 3.14 \cdot 49 = 153.86

  .
• Area = 153.86 cm².
• Explanation: The radius is squared, then multiplied by

  \pi

  , giving the total enclosed area.

Problem 4: Circle

Circle with radius 3 m.

• A = 3.14 \cdot 3^2 = 3.14 \cdot 9 = 28.26

  .
• Area = 28.26 m².

Problem 5: Rectangle

Rectangle with length 12 cm and width 5 cm.

• A = \text{length} \cdot \text{width} = 12 \cdot 5 = 60

  .
• Area = 60 cm².
• Explanation: The area is the product of the two dimensions, covering the entire rectangular space.

Problem 6: Rectangle

Rectangle with length 8 m and width 3 m.

• A = 8 \cdot 3 = 24

  .
• Area = 24 m².

Problem 7: Square

Square with side 4 cm.

• A = \text{side}^2 = 4^2 = 16

  .
• Area = 16 cm².

Problem 8: Square

Square with side 9 m.

• A = 9^2 = 81

  .
• Area = 81 m².

Problem 9: Triangle

Triangle with base 15 cm and height 6 cm.

• A = \frac{1}{2} \cdot 15 \cdot 6 = \frac{1}{2} \cdot 90 = 45

  .
• Area = 45 cm².

Problem 10: Circle

Circle with radius 10 cm.

• A = 3.14 \cdot 10^2 = 3.14 \cdot 100 = 314

  .
• Area = 314 cm².

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2. Perimeter Calculations

General Explanation

The perimeter is the total distance around the boundary of a 2D shape, measured in linear units (e.g., cm, m). Formulas:

• Triangle:

  P = \text{side}_1 + \text{side}_2 + \text{side}_3
• Circle: Circumference

  C = 2 \cdot \pi \cdot r
• Rectangle:

  P = 2 \cdot (\text{length} + \text{width})
• Square:

  P = 4 \cdot \text{side}

Example Problems

Problem 1: Triangle

Find the perimeter of a triangle with sides 5 cm, 6 cm, and 7 cm.

• Step 1: Use

  P = \text{side}_1 + \text{side}_2 + \text{side}_3

  .
• Step 2:

  P = 5 + 6 + 7 = 18

  .
• Answer: Perimeter = 18 cm.
• Explanation: Sum all sides to get the total boundary length.

Problem 2: Triangle

Triangle with sides 8 m, 8 m, 10 m.

• P = 8 + 8 + 10 = 26

  .
• Perimeter = 26 m.

Problem 3: Circle

Find the circumference of a circle with radius 5 cm.

• C = 2 \cdot 3.14 \cdot 5 = 31.4

  .
• Circumference = 31.4 cm.
• Explanation: The circumference is the perimeter of a circle, calculated using the radius.

Problem 4: Circle

Circle with radius 12 m.

• C = 2 \cdot 3.14 \cdot 12 = 75.36

  .
• Circumference = 75.36 m.

Problem 5: Rectangle

Rectangle with length 10 cm and width 4 cm.

• P = 2 \cdot (10 + 4) = 2 \cdot 14 = 28

  .
• Perimeter = 28 cm.
• Explanation: Add length and width, then double to account for opposite sides.

Problem 6: Rectangle

Rectangle with length 15 m and width 6 m.

• P = 2 \cdot (15 + 6) = 2 \cdot 21 = 42

  .
• Perimeter = 42 m.

Problem 7: Square

Square with side 7 cm.

• P = 4 \cdot 7 = 28

  .
• Perimeter = 28 cm.

Problem 8: Square

Square with side 11 m.

• P = 4 \cdot 11 = 44

  .
• Perimeter = 44 m.

Problem 9: Triangle

Triangle with sides 3 cm, 4 cm, 5 cm.

• P = 3 + 4 + 5 = 12

  .
• Perimeter = 12 cm.

Problem 10: Circle

Circle with radius 8 cm.

• C = 2 \cdot 3.14 \cdot 8 = 50.24

  .
• Circumference = 50.24 cm.

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3. Volume Calculations

General Explanation

The volume of a 3D shape measures the space it occupies, in cubic units (e.g., cm³, m³). Formulas:

• Cylinder:

  V = \pi \cdot r^2 \cdot h
• Cube:

  V = \text{side}^3
• Rectangular Prism:

  V = \text{length} \cdot \text{width} \cdot \text{height}
• Sphere:

  V = \frac{4}{3} \cdot \pi \cdot r^3

Example Problems

Problem 1: Cylinder

Find the volume of a cylinder with radius 4 cm and height 10 cm.

• Step 1: Use

  V = \pi \cdot r^2 \cdot h

  .
• Step 2:

  V = 3.14 \cdot 4^2 \cdot 10 = 3.14 \cdot 16 \cdot 10 = 502.4

  .
• Answer: Volume = 502.4 cm³.
• Explanation: The base area (

  \pi \cdot r^2

  ) is multiplied by height to get the total volume.

Problem 2: Cylinder

Cylinder with radius 6 m and height 5 m.

• V = 3.14 \cdot 6^2 \cdot 5 = 3.14 \cdot 36 \cdot 5 = 565.2

  .
• Volume = 565.2 m³.

Problem 3: Cube

Cube with side 5 cm.

• V = 5^3 = 125

  .
• Volume = 125 cm³.
• Explanation: All sides are equal, so the volume is the side length cubed.

Problem 4: Cube

Cube with side 8 m.

• V = 8^3 = 512

  .
• Volume = 512 m³.

Problem 5: Rectangular Prism

Rectangular prism with length 10 cm, width 6 cm, height 4 cm.

• V = 10 \cdot 6 \cdot 4 = 240

  .
• Volume = 240 cm³.
• Explanation: Multiply all three dimensions to find the total space.

Problem 6: Rectangular Prism

Rectangular prism with length 12 m, width 3 m, height 7 m.

• V = 12 \cdot 3 \cdot 7 = 252

  .
• Volume = 252 m³.

Problem 7: Sphere

Sphere with radius 3 cm.

• V = \frac{4}{3} \cdot 3.14 \cdot 3^3 = \frac{4}{3} \cdot 3.14 \cdot 27 = 113.04

  .
• Volume = 113.04 cm³.
• Explanation: The formula accounts for the spherical shape, scaling the radius cubed.

Problem 8: Sphere

Sphere with radius 5 m.

• V = \frac{4}{3} \cdot 3.14 \cdot 5^3 = \frac{4}{3} \cdot 3.14 \cdot 125 = 523.33

  .
• Volume = 523.33 m³.

Problem 9: Cylinder

Cylinder with radius 2 cm and height 15 cm.

• V = 3.14 \cdot 2^2 \cdot 15 = 3.14 \cdot 4 \cdot 15 = 188.4

  .
• Volume = 188.4 cm³.

Problem 10: Rectangular Prism

Rectangular prism with length 9 cm, width 5 cm, height 6 cm.

• V = 9 \cdot 5 \cdot 6 = 270

  .
• Volume = 270 cm³.

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4. Surface Area Calculations

General Explanation

The surface area of a 3D shape is the total area of all its surfaces, in square units. Formulas:

• Cylinder:

  SA = 2 \cdot \pi \cdot r^2 + 2 \cdot \pi \cdot r \cdot h
• Cube:

  SA = 6 \cdot \text{side}^2
• Rectangular Prism:

  SA = 2 \cdot (\text{length} \cdot \text{width} + \text{length} \cdot \text{height} + \text{width} \cdot \text{height})
• Sphere:

  SA = 4 \cdot \pi \cdot r^2

Example Problems

Problem 1: Cylinder

Find the surface area of a cylinder with radius 3 cm and height 7 cm.

• Step 1: Use

  SA = 2 \cdot \pi \cdot r^2 + 2 \cdot \pi \cdot r \cdot h

  .
• Step 2:

  SA = 2 \cdot 3.14 \cdot 3^2 + 2 \cdot 3.14 \cdot 3 \cdot 7 = 2 \cdot 3.14 \cdot 9 + 2 \cdot 3.14 \cdot 21 = 56.52 + 131.88 = 188.4

  .
• Answer: Surface Area = 188.4 cm².
• Explanation: The formula adds the areas of two circular bases and the curved surface.

Problem 2: Cylinder

Cylinder with radius 5 m and height 10 m.

• SA = 2 \cdot 3.14 \cdot 5^2 + 2 \cdot 3.14 \cdot 5 \cdot 10 = 157 + 314 = 471

  .
• Surface Area = 471 m².

Problem 3: Cube

Cube with side 6 cm.

• SA = 6 \cdot 6^2 = 6 \cdot 36 = 216

  .
• Surface Area = 216 cm².
• Explanation: A cube has six equal square faces, so multiply the area of one face by 6.

Problem 4: Cube

Cube with side 4 m.

• SA = 6 \cdot 4^2 = 6 \cdot 16 = 96

  .
• Surface Area = 96 m².

Problem 5: Rectangular Prism

Rectangular prism with length 8 cm, width 5 cm, height 3 cm.

• SA = 2 \cdot (8 \cdot 5 + 8 \cdot 3 + 5 \cdot 3) = 2 \cdot (40 + 24 + 15) = 2 \cdot 79 = 158

  .
• Surface Area = 158 cm².
• Explanation: Calculate the area of each pair of opposite faces and sum them.

Problem 6: Rectangular Prism

Rectangular prism with length 10 m, width 4 m, height 6 m.

• SA = 2 \cdot (10 \cdot 4 + 10 \cdot 6 + 4 \cdot 6) = 2 \cdot (40 + 60 + 24) = 2 \cdot 124 = 248

  .
• Surface Area = 248 m².

Problem 7: Sphere

Sphere with radius 4 cm.

• SA = 4 \cdot 3.14 \cdot 4^2 = 4 \cdot 3.14 \cdot 16 = 200.96

  .
• Surface Area = 200.96 cm².
• Explanation: The formula scales the radius squared to cover the entire spherical surface.

Problem 8: Sphere

Sphere with radius 6 m.

• SA = 4 \cdot 3.14 \cdot 6^2 = 4 \cdot 3.14 \cdot 36 = 452.16

  .
• Surface Area = 452.16 m².

Problem 9: Cylinder

Cylinder with radius 2 cm and height 12 cm.

• SA = 2 \cdot 3.14 \cdot 2^2 + 2 \cdot 3.14 \cdot 2 \cdot 12 = 25.12 + 150.72 = 175.84

  .
• Surface Area = 175.84 cm².

Problem 10: Rectangular Prism

Rectangular prism with length 7 cm, width 6 cm, height 5 cm.

• SA = 2 \cdot (7 \cdot 6 + 7 \cdot 5 + 6 \cdot 5) = 2 \cdot (42 + 35 + 30) = 2 \cdot 107 = 214

  .
• Surface Area = 214 cm².

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Summary of Formulas Used

• Area:
  • Triangle:

    \frac{1}{2} \cdot \text{base} \cdot \text{height}
  • Circle:

    \pi \cdot r^2
  • Rectangle:

    \text{length} \cdot \text{width}
  • Square:

    \text{side}^2
• Perimeter:
  • Triangle: Sum of all sides
  • Circle:

    2 \cdot \pi \cdot r
  • Rectangle:

    2 \cdot (\text{length} + \text{width})
  • Square:

    4 \cdot \text{side}
• Volume:
  • Cylinder:

    \pi \cdot r^2 \cdot h
  • Cube:

    \text{side}^3
  • Rectangular Prism:

    \text{length} \cdot \text{width} \cdot \text{height}
  • Sphere:

    \frac{4}{3} \cdot \pi \cdot r^3
• Surface Area:
  • Cylinder:

    2 \cdot \pi \cdot r^2 + 2 \cdot \pi \cdot r \cdot h
  • Cube:

    6 \cdot \text{side}^2
  • Rectangular Prism:

    2 \cdot (\text{length} \cdot \text{width} + \text{length} \cdot \text{height} + \text{width} \cdot \text{height})
  • Sphere:

    4 \cdot \pi \cdot r^2

Each problem was solved step-by-step, with explanations to clarify the application of the formulas. If you need additional shapes (e.g., cones, pyramids) or more problems, let me know!