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Lesson 2 of 4

Surface Areas and Volumes · Lesson 2 of 4

Surface Area of a Combination of Solids

“Calculating the surface area of a combination of solids is just 3D geometry's way of asking: "How much wrapping paper do you need for a present that makes no sense?"”

Learning Objectives

• Calculate the surface area of composite solids made by combining basic 3D shapes. • Identify which internal base areas disappear when two solids are joined together. • Apply the slant height formula l = √(r² + h²) accurately for conical sections. • Solve step-by-step NCERT board exam problems involving toys, decorative blocks, capsules, and hollowed-out solids.

When we combine two basic solids—such as placing a cone on top of a hemisphere or joining two hemispheres to a cylinder—we create a composite solid. To find the surface area of this new object, we must identify only the outer surfaces that are visible and exposed to the outside world.

Think About Painting or Coating

A helpful trick when finding combined surface areas is to ask yourself: 'If I were to paint this entire object with a brush, which surfaces would get covered with paint?' You would only paint the exposed outer faces. Any base that is glued inside the solid is hidden and receives no paint.

Core Principles for Combined Surface Area

General Rule for Composite Surface AreaLaTeX
Crucial Exam Tip

Never add the Total Surface Areas (TSA) of individual components! The base areas that touch each other disappear when the shapes are attached.

Type 1: Cone Surmounted on a Hemisphere (The Toy / Lattu)

A popular shape is a spinning top or toy consisting of a cone mounted on a hemisphere of equal radius.

Surface Area of Cone + HemisphereLaTeX
NCERT Example 1: Rasheed's Playing Top

Problem
A playing top (lattu) is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the total area to be colored. (Take π = 22/7)

  1. 1.Step 1: Find the radius r. Diameter = 3.5 cm, so r = 3.5 / 2 = 1.75 cm = 7/4 cm.
  2. 2.Step 2: Find the height of the conical part (h). Height of hemispherical part = radius = 1.75 cm. So, h = Total height - radius = 5 - 1.75 = 3.25 cm = 13/4 cm.
  3. 3.Step 3: Calculate slant height (l) of cone using l = √(r² + h²). l = √[(1.75)² + (3.25)²] = √(3.0625 + 10.5625) = √13.625 ≈ 3.7 cm.
  4. 4.Step 4: Calculate CSA of hemisphere = 2πr² = 2 × (22/7) × (1.75)² = 19.25 cm².
  5. 5.Step 5: Calculate CSA of cone = πrl = (22/7) × 1.75 × 3.7 = 20.35 cm².
  6. 6.Step 6: Total Area to be colored = CSA of hemisphere + CSA of cone = 19.25 + 20.35 = 39.6 cm² (approx).
Playing Top (Lattu) — Cone + Hemisphere r = 1.75 cm h = 3.25 cm l ≈ 3.7 cm Total Height = 5 cm d = 3.5 cm
Playing Top (Lattu) - Cone + Hemisphere

Type 2: Hemisphere Fixed on a Cube (Decorative Block)

When a hemisphere sits on top of a flat face of a cube, it covers a circular portion of that face while adding its own dome-shaped curved surface.

Surface Area of Cube + HemisphereLaTeX
NCERT Example 2: Decorative Block

Problem
A decorative block is made of two solids: a cube and a hemisphere. The base of the block is a cube with edge 5 cm, and the hemisphere fixed on top has a diameter of 4.2 cm. Find the total surface area of the block. (Take π = 22/7)

  1. 1.Step 1: Calculate Total Surface Area of the cube = 6 × (edge)² = 6 × (5)² = 150 cm².
  2. 2.Step 2: Radius of hemisphere (r) = 4.2 / 2 = 2.1 cm.
  3. 3.Step 3: Base area of hemisphere that covers the cube face = πr² = (22/7) × (2.1)² = 13.86 cm².
  4. 4.Step 4: Curved Surface Area (CSA) of hemisphere = 2πr² = 2 × 13.86 = 27.72 cm².
  5. 5.Step 5: TSA of decorative block = TSA of cube - Base area of hemisphere + CSA of hemisphere.
  6. 6.Step 6: TSA = 150 - 13.86 + 27.72 = 150 + 13.86 = 163.86 cm².
Decorative Block — Cube Surmounted by Hemisphere (Total Surface Area = TSA of Cube - Base Area of Hemisphere + CSA of Hemisphere) Edge (a) = 5 cm 5 cm Diameter (d) = 4.2 cm Hemisphere CSA = 2πr² = 27.72 cm² Covered Base Area = πr² = 13.86 cm² (Subtracted from Cube Surface) Surface Area Calculation: TSA = 6a² - πr² + 2πr² TSA = 150 - 13.86 + 27.72 Total Surface Area = 163.86 cm²
Decorative Block — Cube Surmounted by Hemisphere

Type 3: Cylinder with Hemispherical Ends (Capsule / Oil Tanker)

Surface Area of CapsuleLaTeX
Worked Example: Medicine Capsule

Problem
A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area. (Take π = 22/7)

  1. 1.Step 1: Radius of capsule (r) = 5 / 2 = 2.5 mm.
  2. 2.Step 2: Height/Length of cylindrical part (h) = Total length - 2 × radius = 14 - (2.5 + 2.5) = 14 - 5 = 9 mm.
  3. 3.Step 3: CSA of cylindrical part = 2πrh = 2 × (22/7) × 2.5 × 9 = 990 / 7 mm².
  4. 4.Step 4: CSA of two hemispherical ends = 2 × (2πr²) = 4 × (22/7) × (2.5)² = 550 / 7 mm².
  5. 5.Step 5: Total Surface Area = 990/7 + 550/7 = 1540 / 7 = 220 mm².
Medicine Capsule — Cylinder with Hemispherical Ends (Total Surface Area = CSA of Cylinder + 2 × CSA of Hemispherical Ends) Total Length = 14 mm Cylinder Length (h) = 14 - (2.5 + 2.5) = 9 mm r = 2.5 mm Diameter (d) = 5 mm Cylindrical Part CSA = 2πrh Hemispherical Cap (2 ends) CSA = 2 × (2πr²) = 4πr² Surface Area Calculation: TSA = 2πrh + 2(2πr²) = 2πr(h + 2r) TSA = 2 × (22/7) × 2.5 × (9 + 5) Total Surface Area = 220 mm²
Medicine Capsule — Cylinder with Hemispherical Ends

Type 4: Scooped-Out / Cavity Solids

Important Paradox

When a cavity (like a cone or hemisphere) is scooped or hollowed out of a solid cylinder, the VOLUME decreases, but the SURFACE AREA increases! This is because carving out a cavity exposes new inner walls that can be touched.

Worked Example: Wooden Article with Scooped Ends

Problem
A wooden article is made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 10 cm and its base radius is 3.5 cm, find the total surface area of the article. (Take π = 22/7)

  1. 1.Step 1: Identify exposed surfaces: CSA of outer cylinder + CSA of top hollow hemisphere + CSA of bottom hollow hemisphere.
  2. 2.Step 2: Radius (r) = 3.5 cm, Height (h) = 10 cm.
  3. 3.Step 3: CSA of cylinder = 2πrh = 2 × (22/7) × 3.5 × 10 = 220 cm².
  4. 4.Step 4: CSA of two hollowed hemispheres = 2 × (2πr²) = 4 × (22/7) × (3.5)² = 154 cm².
  5. 5.Step 5: Total Surface Area = 220 + 154 = 374 cm².
Scooped-Out Solid — Cylinder with Hemispherical Cavities (Total Surface Area = Outer CSA of Cylinder + CSA of Top Cavity + CSA of Bottom Cavity) Height (h) = 10 cm r = 3.5 cm Top Hemispherical Cavity Exposed Inner CSA = 2πr² Cylinder Outer Wall Exposed Outer CSA = 2πrh Bottom Hemispherical Cavity Exposed Inner CSA = 2πr² Surface Area Calculation: TSA = 2πrh + 2πr² + 2πr² = 2πr(h + 2r) TSA = 220 + 154 Total Surface Area = 374 cm²
Scooped-Out Solid — Cylinder with Hemispherical Cavities

Quiz

Quick check

A cone is mounted on a hemisphere of the same radius. Which expression gives the exposed surface area of the combined solid?

Quick check

A cone has radius 3 cm and vertical height 4 cm. What is its slant height?

Quick check

A hemisphere of radius r is fixed on one face of a cube with edge a. Which expression gives the exposed surface area of the combination?

Quick check

A capsule consists of a cylinder of radius r and height h with a hemisphere attached at each end. What is its total exposed surface area?

Quick check

Why can the surface area of a solid increase when a cavity is scooped out of it?

Practice Problems

Practice Problems
  1. 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
  2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
  3. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
  4. A hemispherical depression is cut out from one face of a cubical wooden block of edge 'a' such that the diameter of the hemisphere is equal to the edge. Find the surface area of the remaining solid in terms of 'a'.
  5. A tent is in the shape of a cylinder surmounted by a conical top. Height and diameter of cylindrical part are 2.1 m and 4 m, and slant height of top is 2.8 m. Find the area of canvas required and its cost at ₹500/m².
  6. From a solid cylinder of height 2.4 cm and diameter 1.4 cm, a conical cavity of same height and diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm².

Key Takeaways

Key Takeaways

• Composite Surface Area = Sum of all visible exposed external/internal curved and flat surfaces. • Hidden contacting faces must be subtracted or omitted. • Scooping out a cavity INCREASES surface area while DECREASING volume. • Always compute slant height l = √(r² + h²) when dealing with conical sections.

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Surface Areas and Volumes

Next · Lesson 3

Volume of a Combination of Solids