CH – 12 SURFACE AREAS AND VOLUMES
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Master solid shapes step-by-step and build complete confidence to solve combination problems easily!
Chapter Overview
In this chapter, you will learn how to measure 3D shapes and real-world objects made by combining simple solids:
- Understanding different solid shapes: Revisiting basic 3D figures like cuboids, cubes, cylinders, cones, spheres, and hemispheres.
- Surface area of combined solids: Finding the outer area when two or more simple solids are joined together.
- Volume of combined solids: Calculating the total internal space of objects made of multiple shapes.
- Identifying visible and hidden surfaces: Knowing which parts remain on the outside and which parts get covered during joining.
- Breaking a complicated solid into simple shapes: Splitting complex real-life items (like toys, test tubes, and sheds) into basic individual shapes to solve them step-by-step.
🎯 Board Exam Focus
Practise the following core skills carefully to perform well in your exam:
- Identifying the simple shapes present inside a combined object.
- Breaking a combined solid into individual basic shapes.
- Choosing the correct formula for surface area or volume based on what is asked.
- Understanding the exact difference between Curved Surface Area (CSA) and Total Surface Area (TSA).
- Knowing which surfaces are visible on the outside and must be counted.
- Identifying hidden or joined surfaces so you do not accidentally add them.
- Adding volumes of combined solids correctly.
- Subtracting volumes when material is scooped out, hollowed, or carved.
- Writing correct units (cm² or m² for surface area; cm³ or m³ for volume).
Most Important Concepts
Quick Revision: Basic Solids
Before calculating combined figures, let us revise the basic 3D shapes:
- Cube: A box-like solid where all six flat faces are identical squares.
- Cuboid: A rectangular box with length, breadth, and height.
- Cylinder: A smooth solid with a curved side and two circular flat bases at the top and bottom.
- Cone: A solid with a flat circular base that tapers smoothly up to a single top point (vertex).
- Sphere: A completely round 3D ball where every point on the surface is at equal distance from the centre.
- Hemisphere: Exactly half of a solid sphere, having one curved surface and one flat circular face.
Plaintext
Visual Representation 1: Basic Solids
CUBE CUBOID CYLINDER CONE SPHERE HEMISPHERE
+---+ +-------+ .---. /\ .---. .---.
/ /| / /| / \ / \ / \ / \
+---+ | +-------+ | | r | / h \ | r | (_______)
| | + | | + | |---| | /---|--\ | O | \ r /
| |/ | |/ | h | / r \ \ \ / '---'
+---+ +-------+ '---' '------'--' '---'
(Edge a) (l, b, h) (r, h) (r, h, l) (Radius r) (Radius r)
🧠 Surface Area vs Volume
To avoid confusion, remember this basic distinction:
- Surface Area: The total measurement of the boundary or skin on the outside of an object. It is measured in square units (cm², m²).
- Volume: The amount of space occupied inside a 3D object, or its storage capacity. It is measured in cubic units (cm³, m³).
Everyday Example: Think of a juice box.
- The amount of cardboard material used to make the box wrapper is its Surface Area.
- The actual quantity of fruit juice filled inside the box is its Volume.
Plaintext
Visual Representation 2: Surface Area vs Volume
SURFACE AREA VOLUME
(Outer Boundary / Cardboard) (Space Inside / Juice)
+------------+ +------------+
/| /| /|:::::::::::|
/ | / | / |:::::::::::|
+--+---------+ | +--+---------+:|
| | OUTSIDE| | | | Juice |:|
| | SKIN | | | | Inside |:|
| +---------+--+ | +---------+-+
| / | / | /::::::::::/
|/ |/ |/::::::::::/
+------------+ +------------+
[ Square Units: cm² ] [ Cubic Units: cm³ ]
📌 Important Formulas
Cube
- Curved / Lateral Surface Area (LSA): 4a²
- Total Surface Area (TSA): 6a²
- Volume: a³(where a is the edge length of the cube)
Cuboid
- Curved / Lateral Surface Area (LSA): 2h(l + b)
- Total Surface Area (TSA): 2(lb + bh + hl)
- Volume: l × b × h(where l = length, b = breadth, h = height)
Cylinder
- Curved Surface Area (CSA): 2πrh
- Total Surface Area (TSA): 2πrh + 2πr² = 2πr(r + h)
- Volume: πr²h(where r = base radius, h = vertical height)
Cone
- Slant Height (l): √(r² + h²)
- Curved Surface Area (CSA): πrl
- Total Surface Area (TSA): πrl + πr² = πr(l + r)
- Volume: (1/3)πr²h(Slant height (l) is the slanted distance from the top vertex along the side to the circular base boundary)
Sphere
- Surface Area (CSA = TSA): 4πr²
- Volume: (4/3)πr³(A sphere has only one continuous surface, so its CSA and TSA are identical)
Hemisphere
- Curved Surface Area (CSA): 2πr²
- Total Surface Area (TSA): 2πr² + πr² = 3πr²
- Volume: (2/3)πr³(where r is the radius of the hemisphere)
🧠 Curved Surface Area (CSA) and Total Surface Area (TSA)
- Curved Surface Area (CSA): Measures only the curved side of the solid. It ignores top and bottom flat bases.
- Total Surface Area (TSA): Measures the entire outer surface, including the curved side plus all flat top or bottom circular ends.
When to include flat bases?
- Include a circular base if it is completely exposed to the outside air (e.g., a closed cylindrical tin drum).
- Do NOT include a circular base if it is open (e.g., a hollow pipe or tent) or if it is joined/attached to another solid.
Plaintext
Visual Representation 3: CSA vs TSA of a Cylinder
ONLY CURVED SIDE CURVED SIDE + TOP + BOTTOM
(CSA) (TSA)
.---. .---. <-- Top Base (πr²)
/ \ / \
| | <-- Curved | | <-- Curved
| | Surface | | Surface
| | (2πrh) | | (2πrh)
\ / \ /
'---' '---' <-- Bottom Base (πr²)
Formula = 2πrh Formula = 2πrh + πr² + πr²
= 2πr(h + r)
🔥 The Most Important Rule for Combined Solids
ONLY COUNT THE SURFACES THAT ARE VISIBLE FROM THE OUTSIDE.
When two solids are glued, welded, or attached together, the faces that touch each other get covered. They are inside the combined solid and cannot be seen or painted.
- Golden Rule: Never simply add the Total Surface Area (TSA) of individual solids!
- Always identify which surfaces are exposed to the outside, and add only those visible areas.
Simple Example: If a hemisphere is stuck onto a cylinder to make a test tube, the touching circular bases disappear inside. Therefore, total surface area = CSA of cylinder + CSA of hemisphere.
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Visual Representation 4: Visible vs Hidden Surfaces
SEPARATE SOLIDS JOINED COMBINED SOLID
(Bases are visible) (Joined Base is HIDDEN!)
.---. .---.
/ \ / \ <-- VISIBLE (CSA of Cone)
/ \ / \
'=========' <-- Circular Base '=========' <-- HIDDEN / JOINED BASE!
.---------. <-- Circular Base | | (Do NOT count in area!)
| | | | <-- VISIBLE (CSA of Cylinder)
| | | |
'---------' '---------'
📌 Surface Area of a Combination of Solids
To solve surface area questions of combined figures accurately, follow these five steps:
- Step 1: Look at the complete shape: Observe the overall shape given in the question.
- Step 2: Break the object into simple solids: Split the structure into basic shapes like Cylinder + Hemisphere, Cone + Hemisphere, or Cylinder + Two Hemispheres.
- Step 3: Identify visible surfaces: Ask yourself: “If I dip this toy into paint, which surfaces will get wet?”
- Step 4: Use the correct surface area formulas: Write down the individual CSA or base area formulas for exposed parts only.
- Step 5: Add only the required areas: Sum the visible components to calculate total surface area.
📌 Volume of a Combination of Solids
Calculating volume is simpler than surface area because internal joining faces do not change space capacity.
- Joined Solids: If two or more solids are joined together to form an object, the space inside simply adds up.Total Volume = Volume of Solid 1 + Volume of Solid 2
- Scooped Out / Carved Solids: If a hole, depression, or cavity is cut out from a solid block, space is removed.Remaining Volume = Volume of Original Solid − Volume of Removed Solid
🧠 How to Break a Difficult Shape into Simple Shapes
Follow this practical strategy whenever you face a complex visual problem:
Plaintext
Visual Representation 5: Step-by-Step Problem Solving
[ 1. LOOK AT COMPLETE OBJECT ]
│
▼
[ 2. BREAK INTO SIMPLE SOLIDS ] ─── (e.g., Cylinder + Cone)
│
▼
[ 3. DECIDE: AREA OR VOLUME? ]
│
┌────────┴────────┐
▼ ▼
[ SURFACE AREA ] [ VOLUME ]
Count exposed Add space (or subtract
surfaces only if carved out)
│ │
└────────┬────────┘
▼
[ 4. APPLY FORMULAS & CALCULATE ]
📌 Important Combined Shapes
1. Cylinder + Two Hemispheres (e.g., Capsule or Storage Tank)
- Description: A cylindrical middle tube with two hemispherical caps fixed on both ends.
- Visible Surfaces: Curved surface of cylinder + Curved surface of left hemisphere + Curved surface of right hemisphere.
- Hidden Surfaces: Both circular joints between the cylinder and hemispheres.
- Total Surface Area Formula: 2πrh + 2πr² + 2πr² = 2πrh + 4πr²
- Total Volume Formula: πr²h + (2/3)πr³ + (2/3)πr³ = πr²h + (4/3)πr³
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Visual Representation 6: Cylinder + Two Hemispheres
Hemisphere 1 Cylinder Hemisphere 2
(CSA = 2πr²) (CSA = 2πrh) (CSA = 2πr²)
.---. .------------------. .---.
/ | | | | \
| | | | | |
\ | | | | /
'---' '------------------' '---'
\ / \ /
Hidden Joint Hidden Joint
COMBINED OBJECT:
.---.------------------------------.---.
/ | | \
| | | |
\ | | /
'---'------------------------------'---'
<-- r --><----------- h -----------><-- r -->
2. Cone + Hemisphere (e.g., Playing Top / Toy)
- Description: A cone placed flat on top of a hemispherical base with equal radius.
- Visible Surfaces: Curved surface of the cone + Curved surface of the hemisphere.
- Hidden Surfaces: The circular base where the cone meets the hemisphere.
- Total Surface Area Formula: πrl + 2πr²
- Total Volume Formula: (1/3)πr²h + (2/3)πr³
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Visual Representation 7: Cone + Hemisphere
/\
/ \ <-- Cone (CSA = πrl)
/ h \
/ l \
'=========' <-- Hidden Joined Circular Base!
\ r /
\ / <-- Hemisphere (CSA = 2πr²)
'---'
3. Cuboidal Block with Hemispherical Top / Depression
- Top Fixed on Cube: Total surface area = (TSA of cube) − (Base area of hemisphere) + (CSA of hemisphere) = 6a² − πr² + 2πr² = 6a² + πr².
- Depression Scooped Out of Cube: When a hemisphere is carved out from a cube face, the top circular area is lost, but a new curved inner hemispherical bowl is exposed.
- Surface Area = 6a² − πr² + 2πr² = 6a² + πr²
- Remaining Volume = (Volume of cube) − (Volume of hemisphere) = a³ − (2/3)πr³
🧠 How to Know Whether to Add or Subtract?
For Volume Questions:
- Add Volumes: When two or more solids are attached or joined to form a bigger object.
- Subtract Volume: When material is removed, scooped out, drilled, or hollowed out.
For Surface Area Questions:
- Do NOT simply add TSAs: Never add the full TSAs of attached shapes.
- Add CSAs of Exposed Parts: Add up all individual curved or flat surfaces that are visible from the outside.
- Carved-out Shapes: When a cavity is scooped out, surface area increases because a new internal face is now exposed to the outside air!
🧠 Concepts Students Find Difficult
- Confusing CSA and TSA: Applying TSA formulas directly to individual parts without checking if their circular bases are covered.
- Forgetting hidden surfaces: Including base areas that lie hidden inside the joint.
- Using diameter instead of radius: Forgetting to divide given diameter by 2 before calculations.
- Confusing slant height (l) and vertical height (h): Using height h instead of slant height l in cone CSA formula (πrl).
- Adding areas when material is scooped out: Thinking that scooping out a hemisphere decreases total surface area, whereas it actually exposes more surface area.
- Adding volumes incorrectly: Adding or subtracting incorrect parameters while keeping units mismatched.
- Forgetting unit power: Writing cm or cm³ for area instead of cm².
⚠️ Common Mistakes Students Make
- Adding TSAs of both shapes directly:
- Mistake: Writing TSA of combined toy = TSA of cone + TSA of hemisphere.
- Fix: Standard rule: Combined TSA = CSA of cone + CSA of hemisphere.
- Using height instead of slant height for cone area:
- Mistake: Using πrh for cone surface area.
- Fix: Cone CSA requires slant height l. Always find l = √(r² + h²) first.
- Forgetting to halve the diameter:
- Mistake: Using r = 7 cm when given diameter = 7 cm.
- Fix: Always write r = d / 2 immediately upon reading the question.
- Subtracting surface area when scooping out cavity:
- Mistake: Subtracting hemisphere area from cube area when a hemisphere is carved out.
- Fix: Scooping creates extra inner surface area! Add CSA of hemisphere and subtract only the open top circle.
- Calculating total height of cone part incorrectly:
- Mistake: Taking total toy height as the height of the cone.
- Fix: Cone height = (Total Height) − (Radius of Hemisphere).
- Writing wrong units:
- Mistake: Expressing volume in cm² or surface area in cm³.
- Fix: Always write cm² / m² for surface area, and cm³ / m³ for volume.
- Forgetting to square or cube the radius:
- Mistake: Writing πr²h as π × r × h or πr³ as π × r.
- Fix: Carefully check exponents in formulas before multiplying values.
🔥 Priority Revision
⭐⭐⭐ HIGH-PRIORITY PRACTICE
- Surface area of Cone + Hemisphere combinations (Toys / Tops).
- Surface area of Cylinder + Two Hemispheres combinations (Capsules / Tanks).
- Hemisphere mounted on or scooped out of a Cubical block.
- Correctly calculating slant height l = √(r² + h²).
⭐⭐ IMPORTANT PRACTICE
- Volume of Cuboid + Half-Cylinder shed structures.
- Volume of cylindrical glass with raised hemispherical bottom.
- Volume of solid cone + hemisphere placed inside a water-filled cylinder.
- Pen stand with conical depressions (subtracting volumes).
⭐ ADDITIONAL REVISION
- Cylinder with conical ends (Engineering models).
- Wooden article scooped out at both cylindrical ends.
- Finding mass of multi-part poles using density conversion.
⚡ Surface Areas and Volumes Quick Revision Sheet
- CSA vs TSA: CSA = curved boundary only; TSA = complete outer surface.
- Golden Rule for Combined Area: Sum exposed surfaces only; ignore hidden touching faces.
- Golden Rule for Combined Volume: Total Volume = Sum of individual volumes (Subtract if material removed).
- Slant Height of Cone: l = √(r² + h²)
- Cube: TSA = 6a² | Volume = a³
- Cuboid: TSA = 2(lb + bh + hl) | Volume = lbh
- Cylinder: CSA = 2πrh | TSA = 2πr(r + h) | Volume = πr²h
- Cone: CSA = πrl | TSA = πr(l + r) | Volume = (1/3)πr²h
- Sphere: Area = 4πr² | Volume = (4/3)πr³
- Hemisphere: CSA = 2πr² | TSA = 3πr² | Volume = (2/3)πr³
- Units: Area → cm² or m² | Volume → cm³ or m³
📝 Test Yourself
- A solid is composed of a cylinder of radius 7 cm and height 10 cm, topped by a hemisphere of radius 7 cm. State which circular surface is hidden inside the combined solid.
- Find the total surface area of a solid cube of side 6 cm which has a hemisphere of radius 2.1 cm fixed on its top face. (Take π = 22/7)
- A wooden toy is in the shape of a cone mounted on a hemisphere of common radius 3.5 cm. If the slant height of the cone is 9.5 cm, calculate the total outer surface area of the toy. (Take π = 22/7)
- A medicine capsule has a cylindrical body with two hemispherical ends. The total length of the capsule is 12 mm and its radius is 2 mm. Calculate the outer surface area of the capsule. (Take π = 22/7)
- A hollow cylinder open at both ends has height 14 cm and radius 3 cm. Find its curved surface area. (Take π = 22/7)
- A solid iron ornament is made by joining a cone of radius 3 cm and height 4 cm onto a hemisphere of radius 3 cm. Calculate the total volume of the ornament in terms of π.
- A cylindrical container of radius 5 cm and height 12 cm is filled with liquid. A solid hemisphere of radius 3 cm is submerged into the liquid. Find the volume of liquid remaining in the cylinder. (Use π = 3.14)
- A wooden block in the form of a cuboid measuring 10 cm × 8 cm × 6 cm has a hemispherical cavity of radius 3 cm carved out from its top face. Write the expression to calculate the total exposed surface area of the remaining block.
- An industrial storage tank consists of a cylinder of length 8 m with hemispherical caps attached to both ends. If the overall length of the tank is 14 m, determine the radius of the hemispherical ends and calculate the total internal volume of the tank. (Take π = 22/7)
- A hemispherical bowl of inner radius 9 cm is completely filled with soup. If the soup is poured into small cylindrical cups of radius 3 cm and height 3 cm, calculate how many cups can be filled completely.
🎯 Before Your Exam
☐ I know the difference between surface area and volume. ☐ I know all basic 3D formulas for cube, cuboid, cylinder, cone, sphere, and hemisphere. ☐ I understand the difference between CSA and TSA. ☐ I can identify simple 3D shapes inside a complex solid. ☐ I can break a combined solid into smaller basic shapes. ☐ I know which surfaces are visible on the outside. ☐ I know which surfaces are hidden and must be excluded from total surface area. ☐ I can add or subtract volumes correctly for combined or carved solids. ☐ I remember to write correct units (cm² or cm³) in my final answer.
Stay calm, draw neat diagrams, apply step-by-step logic, and you will score full marks in this chapter!