Distance, Volume, and Surface Area Study

Distance, volume, and surface area calculations are fundamental for analyzing three-dimensional geometric objects. This section delves into measuring distances in 3D space, calculating volumes and surface areas, and determining angles in 3D shapes.

Guidance, Clarifications, and Syllabus Links:

• Distance in 3D Space:
  • Distance Formula:

    The distance between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is calculated as:

    \( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2} \)

  • Midpoint Formula:

    The midpoint of the line segment joining two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by:

    \( \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \)

• Volume of 3D Solids:
  • Right pyramid: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \)

• • • Right cone: \( V = \frac{1}{3} \pi r^2 h \)
    • Sphere: \( V = \frac{4}{3} \pi r^3 \)
    • Hemisphere: \( V = \frac{2}{3} \pi r^3 \)
    • Combinations of solids: Add or subtract volumes as required based on the configuration.
• Surface Area of 3D Solids:
  • Right cone: \( A = \pi r (r + l) \), where \( l \) is the slant height.
  • Sphere: \( A = 4 \pi r^2 \)
  • Hemisphere: \( A = 2 \pi r^2 + \pi r^2 \)
  • Composite solids: Add or subtract surface areas based on the shape configuration.

Examples:

• Example 1: Find the distance between the points \( (1, 2, 3) \) and \( (4, 6, 8) \).

  \( d = \sqrt{(4 – 1)^2 + (6 – 2)^2 + (8 – 3)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \)

• Example 2: Calculate the volume of a composite solid formed by placing a hemisphere with radius 3 cm on top of a cylinder with the same radius and height 7 cm.
  • Volume of hemisphere: \( V_h = \frac{2}{3} \pi 3^3 = 18\pi \)
  • Volume of cylinder: \( V_c = \pi r^2 h = \pi 3^2 7 = 63\pi \)
  • Total volume: \( V = V_h + V_c = 18\pi + 63\pi = 81\pi \)
• Example 3: Find the surface area of a sphere with radius 5 cm.

  \( A = 4 \pi r^2 = 4 \pi 5^2 = 100\pi \)