Ultimate Guide on Areas of Sector, Segment, and Surface Area of Combination of Solid

In geometry, many real-world objects are not made up of a single basic shape but rather consist of a combination of different solid geometrical forms. These composite structures often involve a blend of two or more simple geometric shapes, such as cylinders, cones, spheres, and hemispheres combined in various configurations. Understanding how to calculate the total surface area of these combinations is vital for accurately determining material usage and space coverage in various practical scenarios.

This concept is highly applicable in diverse fields, including architecture, where designs frequently incorporate domes on cylindrical bases, or engineering, where machine parts often integrate multiple geometric forms. Mastering the calculations of surface areas for combined solids helps students break down complex geometry problems into manageable components and apply fundamental surface area formulas in a systematic way. This skill not only strengthens mathematical reasoning but also sharpens logical thinking and spatial visualization abilities.

Key Concepts for Combinations of Solids

Total Surface Area (TSA): The sum of the areas of all external surfaces of a combined solid. It includes the area of both curved and flat surfaces.
Curved Surface Area (CSA): This refers to the total area of the curved surfaces of combined solids, excluding the areas of flat bases or tops.
Basic Solids Involved:
- Cylinder:
  - TSA: $\boldsymbol{2\pi r (h + r)}$
  - CSA: $\boldsymbol{2\pi rh}$
- Cone:
  - TSA: $\boldsymbol{\pi r (l + r)}$
  - CSA: $\boldsymbol{\pi r l}$
- Sphere:
  - TSA: $\boldsymbol{4 \pi r^2}$
- Hemisphere:
  - TSA: $\boldsymbol{3 \pi r^2}$
  - CSA: $\boldsymbol{2 \pi r^2}$
Composite Solids: Real-world objects often consist of different solid shapes combined together. Examples include ice cream cones, cylindrical water tanks with hemispherical tops, domed stadiums, and decorative lampshades. These combinations require specialized approaches to accurately calculate their surface areas by summing the relevant curved and flat surfaces while considering overlapping areas.

Formulas for Surface Area of a Combination of Solids

Calculating the total surface area of combined solids involves understanding each component’s individual contribution and adjusting for shared surfaces:

TSA of Combination: $\boldsymbol{TSA = \text{Sum of Surface Areas of Individual Solids} - \text{Common Surface Area}}$
CSA of a Cylinder with Hemisphere on Top:
- $\boldsymbol{CSA = 2\pi rh + 2\pi r^2}$
TSA of Cone on Hemisphere:
- $\boldsymbol{TSA = \pi r^2 + \pi r l + 2 \pi r^2}$
TSA of Cylinder with a Cone on Top:
- $\boldsymbol{TSA = 2\pi rh + \pi r l + \pi r^2}$
TSA of Sphere Mounted on Cylinder:
- $\boldsymbol{TSA = 2\pi rh + 4\pi r^2 - \pi r^2}$ (subtracting the overlapping base area of the sphere and cylinder)

Examples with Expanded Step-by-Step Solutions

Example 1: A toy is designed with a cone attached to a hemisphere. The cone has a height of 9 cm, and the radius for both the cone and hemisphere is 3.5 cm. Find the total surface area.

Solution:

CSA of Hemisphere: $\boldsymbol{2 \pi r^2}$
CSA of Cone: $\boldsymbol{\pi r l}$ (where l = slant height)
Slant Height Calculation: $\boldsymbol{l = \sqrt{h^2 + r^2} = \sqrt{9^2 + 3.5^2} = 9.67 \text{ cm}}$
Total Surface Area: $\boldsymbol{2 \pi (3.5)^2 + \pi (3.5)(9.67)}$
Final Answer: TSA ≈ $\boldsymbol{215.4 \text{ cm}^2}$

Example 2: Find the total surface area of a cylinder topped with a hemisphere. The cylinder’s height is 10 cm, and the radius of both the cylinder and hemisphere is 5 cm.

Solution:

CSA of Cylinder: $\boldsymbol{2 \pi r h}$
CSA of Hemisphere: $\boldsymbol{2 \pi r^2}$
Total Surface Area: $\boldsymbol{2 \pi (5)(10) + 2 \pi (5)^2}$
Final Answer: TSA ≈ $\boldsymbol{471.24 \text{ cm}^2}$

Example 3: A decorative lamp is constructed with a cone mounted on a cylinder. The cone has a slant height of 8 cm, and both the cone and cylinder share a radius of 4 cm. The cylinder has a height of 12 cm.

Solution:

CSA of Cylinder: $\boldsymbol{2 \pi r h}$
CSA of Cone: $\boldsymbol{\pi r l}$
Total Surface Area: $\boldsymbol{2 \pi (4)(12) + \pi (4)(8)}$
Final Answer: TSA ≈ $\boldsymbol{401.92 \text{ cm}^2}$

Example 4: A water tank consists of a sphere mounted on a cylindrical base. The cylinder has a height of 15 cm and a radius of 6 cm. Find the total surface area.

Solution:

CSA of Cylinder: $\boldsymbol{2 \pi r h}$
CSA of Sphere: $\boldsymbol{4 \pi r^2}$
Subtract Overlapping Area: $\boldsymbol{\pi r^2}$
Total Surface Area: $\boldsymbol{2 \pi (6)(15) + 4 \pi (6)^2 - \pi (6)^2}$
Final Answer: TSA ≈ $\boldsymbol{1508.52 \text{ cm}^2}$

Applications of Surface Area of a Combination of Solids

Architecture: Designing complex building structures that combine multiple geometric forms, such as cylindrical towers with spherical domes.
Manufacturing: Estimating materials needed for constructing industrial tanks, silos, and containers.
Medical Field: Creating prosthetic limbs and orthopedic devices that require custom shapes and accurate surface area measurements.
Art and Design: Constructing sculptures and installations that integrate various geometric forms.
Urban Planning: Designing fountains, monuments, and playgrounds that involve multiple geometric shapes.