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How do pharmacists determine the exact gelatin coating of a pharmaceutical capsule that has a cylindrical body capped by two hemispherical domes? Adva...
How do pharmacists determine the exact gelatin coating of a pharmaceutical capsule that has a cylindrical body capped by two hemispherical domes? Advanced mensuration calculates combined 3D shapes.
Why This Chapter Matters
In Class 10 Mathematics, "Surface Areas and Volumes" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Before You Begin (Prerequisites)
- Surface area and volume of cylinders, cones, spheres from Class 9.
- Algebraic simplification.
- Conversion of units.
What You Will Learn (Core Objectives)
- Calculate the surface area of combinations of solid shapes (cylinder with hemispherical ends, cone on hemisphere).
- Calculate the volume of combinations of solid shapes.
- Analyze conversion of solids from one shape into another (melting and recasting without volume loss).
- Solve real-world industrial container and tent problems.
- Differentiate between interior capacity and exterior volume.
Chapter Roadmap & Progression
1
1. Surface Area of Combined Solids
2
2. Volume of Combined Solids
3
3. Melting and Recasting (Conservat...
Complete Concept Guide (100% Curriculum Coverage)
1. Surface Area of Combined Solids
When combining solids (like a cone mounted on a hemisphere to make a toy), the base of the cone and the flat top of the hemisphere are hidden inside! The total surface area is: $$\mathbf{\text{TSA of Toy} = \text{CSA of Cone } (\pi rl) + \text{CSA of Hemisphere } (2\pi r^2)}$$
2. Volume of Combined Solids
Unlike surface area, Volume is strictly additive! The total volume of any composite solid is simply the sum of the volumes of its individual components: $$\mathbf{\text{Total Volume} = V_{\text{Cone}} + V_{\text{Hemisphere}} = \frac{1}{3}\pi r^2h + \frac{2}{3}\pi r^3}$$
3. Melting and Recasting (Conservation of Volume)
When a solid is melted and recast into another shape (e.g. melting metallic spheres into a cylinder), the total volume remains identical: $\text{Volume of Original Solid} = \text{Volume of New Solid}$.
Visual Learning & Conceptual Map
Surface Areas and Volumes Master Matrix
Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture
1. Surface Area of Combined Solids • 2. Volume of Combined Solids
Chapter Summary & 10 Key Takeaways
Takeaway 1
Additive Volume: Volume of composite solid equals sum of component volumes.
Takeaway 2
Surface Rule: Only exposed exterior faces are counted in composite surface area.
Takeaway 3
Melting Law: Volume is conserved during melting and recasting.
Takeaway 4
Pharmaceutical Capsule: Cylinder with two hemispherical ends ($TSA = 2\pi rh + 4\pi r^2$).
Takeaway 5
Capacity: Internal volume measuring liquid volume.
Check Your Understanding (Diagnostic Practice Questions)
Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.
1
A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find its total surface area.
Reveal Answer & Explanation
Answer: Height of cone $h = 15.5 - 3.5 = 12\text{ cm}$. Slant height $l = \sqrt{3.5^2 + 12^2} = \sqrt{12.25 + 144} = 12.5\text{ cm}$. $\text{TSA} = \pi rl + 2\pi r^2 = \frac{22}{7}(3.5)[12.5 + 2(3.5)] = 11[12.5 + 7] = 11(19.5) = 214.5\text{ cm}^2$.
TSA = 214.5 cm^2.
2
A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, surmounted by another cylinder of height 60 cm and radius 8 cm. Find its mass if $1\text{ cm}^3$ of iron has 8 g mass.
Reveal Answer & Explanation
Answer: $V_1 = \pi(12^2)(220) = 31,680\pi$. $V_2 = \pi(8^2)(60) = 3,840\pi$. $\text{Total Volume} = 35,520\pi \approx 111,532.8\text{ cm}^3$. $\text{Mass} = 111,532.8 \times 8\text{ g} = 892,262\text{ g} \approx 892.26\text{ kg}$.
Mass = 892.26 kg.
3
Metallic spheres of radii 6 cm, 8 cm, and 10 cm are melted to form a single solid sphere. Find the radius of the resulting sphere.
Reveal Answer & Explanation
Answer: Conservation of volume: $\frac{4}{3}\pi R^3 = \frac{4}{3}\pi(6^3 + 8^3 + 10^3) \implies R^3 = 216 + 512 + 1000 = 1728 \implies R = \sqrt[3]{1728} = 12\text{ cm}$.
Radius = 12 cm.
4
A medicine capsule is shaped like a cylinder with two hemispheres stuck to each of its ends. If total length is 14 mm and diameter is 5 mm, find its surface area.
Reveal Answer & Explanation
Answer: Radius $r = 2.5\text{ mm}$, cylinder height $h = 14 - 5 = 9\text{ mm}$. $\text{Surface Area} = 2\pi rh + 4\pi r^2 = 2\pi r(h + 2r) = 2\left(\frac{22}{7}\right)(2.5)(9 + 5) = \frac{110}{7} \times 14 = 220\text{ mm}^2$.
Surface area = 220 mm^2.
5
Why is the base of the cone not included when calculating the surface area of a cone mounted on a hemisphere?
Reveal Answer & Explanation
Answer: Because the base of the cone is joined and glued to the flat surface of the hemisphere, burying it inside the solid where it is not exposed to the outside.
Base is hidden inside the solid.
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