Basic 3D Shapes and Their Formulas

The Foundation of 3D Geometry

Before we can build complex structures, we need to understand the bricks. In the next chapter, you will learn how to calculate the surface areas and volumes of combined solids—like a cone sitting on top of a cylinder, or a hemisphere scooped out of a wooden cube. But to do that successfully, you must first have absolute mastery over the basic, standalone 3D shapes.

Think of this chapter as your toolkit. We are going to look at the six fundamental 3D shapes: the cuboid, the cube, the cylinder, the cone, the sphere, and the hemisphere. For each one, we will not just memorize the formulas; we will understand why the formulas look the way they do. When you understand the logic behind a formula, you rarely forget it in the exam hall.

Let us start by clearing up the biggest source of confusion in mensuration: the difference between the types of areas and volumes.

Volume vs. Surface Area: What Are We Measuring?

When we talk about 3D shapes, we are generally interested in two things: how much space the object takes up, and how much material is needed to cover its outside.

The Two Types of Surface Area

This is where examiners love to set traps. Surface area is usually split into two categories:

1. 1Lateral Surface Area (LSA) or Curved Surface Area (CSA):

This refers to the area of the "sides" of the object, ignoring the top and the bottom.

• We use the term Lateral Surface Area (LSA) for shapes with flat sides, like cubes and cuboids. (Think of painting the four walls of a room, but ignoring the floor and the ceiling).
• We use the term Curved Surface Area (CSA) for shapes with round sides, like cylinders and cones. (Think of the paper label wrapped around a plastic water bottle).

1. 1Total Surface Area (TSA):

This is exactly what it sounds like—the area of all the surfaces combined.

• TSA = LSA (or CSA) + Area of the Top Base + Area of the Bottom Base.
• If you are painting a closed wooden box on all sides, you need the TSA.

Now, let us break down the six fundamental shapes one by one.

1. The Cuboid (The Box)

A cuboid is a 3D shape with six rectangular faces. A brick, a shoebox, and the room you are sitting in are all examples of cuboids. It has three dimensions: Length (l), Breadth (b), and Height (h).

Understanding the Formulas

• Volume: To find the space inside a cuboid, you simply multiply its three dimensions.

Volume = l × b × h

• Lateral Surface Area (LSA): Imagine a room. The LSA is the area of the four walls.

The front and back walls each have an area of l × h. The left and right walls each have an area of b × h. Adding them up: lh + lh + bh + bh = 2lh + 2bh. If we factor out the 2h, we get: LSA = 2h(l + b)

• Total Surface Area (TSA): To get the TSA, we take the four walls (LSA) and add the floor and the ceiling. The floor is l × b, and the ceiling is also l × b.

So, we add 2lb to our previous total. TSA = 2(lb + bh + hl)

• Length of the Longest Diagonal: If you want to fit the longest possible stick inside a rectangular room, it must go from one bottom corner to the opposite top corner.

Diagonal = √(l² + b² + h²)

Worked Example: Cuboid

Question: A room is 5 m long, 4 m wide, and 3 m high. Find the cost of whitewashing the four walls and the ceiling at the rate of ₹50 per m².

Step-by-Step Solution:

1. 1Identify the dimensions: l = 5, b = 4, h = 3.
2. 2Identify what needs to be painted: Four walls (LSA) + Ceiling. (We do not paint the floor).
3. 3Area of four walls (LSA) = 2h(l + b) = 2 × 3 × (5 + 4) = 6 × 9 = 54 m².
4. 4Area of the ceiling = l × b = 5 × 4 = 20 m².
5. 5Total area to be whitewashed = 54 + 20 = 74 m².
6. 6Total cost = Area × Rate = 74 × 50 = ₹3700.

2. The Cube (The Dice)

A cube is simply a special type of cuboid where all the sides are perfectly equal. Length = Breadth = Height. Let us call this single side length a. A Rubik's cube or a standard playing die are perfect examples.

Understanding the Formulas

Because l = b = h = a, the cuboid formulas become beautifully simple.

• Volume: a × a × a

Volume = a³

• Lateral Surface Area (LSA): A cube has 4 side walls. Each wall is a square with area a².

LSA = 4a²

• Total Surface Area (TSA): A cube has 6 faces in total (4 walls + 1 top + 1 bottom). Each face is a square with area a².

TSA = 6a²

• Length of the Longest Diagonal: Substituting a into the cuboid diagonal formula gives √(a² + a² + a²) = √(3a²).

Diagonal = a√(3)

Worked Example: Cube

Question: The total surface area of a solid cube is 216 cm². Find its volume.

Step-by-Step Solution:

1. 1We know TSA of a cube = 6a².
2. 2Set up the equation: 6a² = 216.
3. 3Divide by 6: a² = 36.
4. 4Take the square root: a = 6 cm. (Now we have the side length).
5. 5Calculate the volume: Volume = a³ = 6 × 6 × 6 = 216 cm³.

(Note: It is a mathematical coincidence that the numerical values of TSA and Volume are the same for a cube of side 6. Do not assume this is always true!)

3. The Right Circular Cylinder (The Pipe)

A right circular cylinder is a shape with two identical circular bases (top and bottom) connected by a curved surface. Think of a soup can, a water pipe, or a stack of coins. It has two key dimensions: the radius of the base (r) and the height (h).

Understanding the Formulas

• Volume: The volume of any uniform solid is the (Area of its Base) × (Height). The base is a circle with area π r². Multiply that by height h.

Volume = π r² h

• Curved Surface Area (CSA): Imagine taking a tin can, cutting it straight down the side, and unrolling it. The curved surface flattens out into a rectangle. The height of this rectangle is h. The length of this rectangle is exactly the circumference of the circular base, which is 2π r. Area of a rectangle is length × height.

CSA = 2π r h

• Total Surface Area (TSA): To get the TSA, take the CSA and add the two circular bases (top and bottom). Each base has an area of π r².

TSA = 2π r h + π r² + π r² = 2π r h + 2π r². Factoring out 2π r: TSA = 2π r(h + r)

Worked Example: Cylinder

Question: A cylindrical water tank has a base radius of 7 m and a height of 10 m. How many liters of water can it hold? (Use π = 22/7)

Step-by-Step Solution:

1. 1Identify dimensions: r = 7 m, h = 10 m.
2. 2Calculate Volume in m³: Volume = π r² h = (22/7) × 7 × 7 × 10.
3. 3The 7 in the denominator cancels with one 7 in the numerator: 22 × 7 × 10 = 1540 m³.
4. 4Convert to liters. (Remember: 1 m³ = 1000 liters).
5. 5Capacity = 1540 × 1000 = 1,540,000 liters.

4. The Right Circular Cone (The Ice Cream Cone)

A cone has a circular base that tapers smoothly to a single point at the top (the apex). Think of a traditional party hat or a traffic cone.

A cone introduces a new, very important dimension. Along with the base radius (r) and the straight vertical height (h), a cone has a slant height (l). The slant height is the distance from the edge of the circular base straight up the side of the cone to the tip.

Because the radius, the vertical height, and the slant height form a right-angled triangle inside the cone, they are connected by Pythagoras' theorem: l² = r² + h² or l = √(r² + h²)

Understanding the Formulas

• Volume: If you have a cylinder and a cone with the exact same base radius and height, it takes exactly three cones full of water to fill the cylinder. Therefore, the volume of a cone is one-third the volume of a cylinder.

Volume = 1⁄3π r² h

• Curved Surface Area (CSA): The area of the slanted, curved part of the cone (like the biscuit part of an ice cream cone). Notice that it uses the slant height (l), not the vertical height (h).

CSA = π r l

• Total Surface Area (TSA): The CSA plus the flat circular base at the bottom.

TSA = π r l + π r². Factoring out π r: TSA = π r(l + r)

Worked Example: Cone

Question: Find the curved surface area of a right circular cone whose vertical height is 4 cm and base radius is 3 cm. (Use π = 3.14)

Step-by-Step Solution:

1. 1Identify given dimensions: h = 4 cm, r = 3 cm.
2. 2We need CSA, which requires slant height (l). We must calculate l first.
3. 3l = √(r² + h²) = √(3² + 4²) = √(9 + 16) = √(25) = 5 cm.
4. 4Now calculate CSA = π r l = 3.14 × 3 × 5.
5. 5CSA = 3.14 × 15 = 47.1 cm².

5. The Sphere (The Ball)

A sphere is a perfectly round 3D object, like a football or a globe. Every point on the surface of a sphere is the exact same distance from the center. This distance is the radius (r). A sphere has no flat edges, no top, and no bottom.

Understanding the Formulas

Because a sphere is one continuous curved surface, it does not have separate LSA and TSA. It just has one "Surface Area."

• Surface Area: The surface area of a sphere is exactly equal to the area of four circles with the same radius.

Surface Area = 4π r²

• Volume: The formula for the volume of a sphere is derived using calculus, but for exam purposes, it is a straightforward memorization.

Volume = 4⁄3π r³

Worked Example: Sphere

Question: Find the surface area of a sphere of radius 10.5 cm. (Use π = 22/7)

Step-by-Step Solution:

1. 1Identify dimension: r = 10.5 cm. (Tip: Write 10.5 as 21/2 to make cancellation easier).
2. 2Surface Area = 4π r² = 4 × (22/7) × (21/2) × (21/2).
3. 3Cancel the 4 in the numerator with the 2 × 2 in the denominator.
4. 4We are left with: (22/7) × 21 × 21.
5. 5Cancel the 7 with the 21 to get 3: 22 × 3 × 21 = 66 × 21 = 1386 cm².

6. The Hemisphere (The Bowl)

If you take a solid sphere (like a solid wooden ball) and slice it exactly in half, you get two hemispheres.

This slicing action does something very important: it exposes a brand new, flat circular face at the top (or bottom, depending on how you hold it). This is why a hemisphere, unlike a full sphere, does have a difference between its Curved Surface Area and its Total Surface Area.

Understanding the Formulas

• Volume: Exactly half the volume of a full sphere. Half of 4/3 is 2/3.

Volume = 2⁄3π r³

• Curved Surface Area (CSA): The area of the curved, bowl-like part. It is exactly half the surface area of a full sphere. Half of 4π r² is 2π r².

CSA = 2π r²

• Total Surface Area (TSA): This is the classic exam trap. The TSA of a solid hemisphere is the curved part (2π r²) PLUS the flat circular face that was exposed when we cut the sphere (π r²).

2π r² + π r² = 3π r². TSA = 3π r²

Worked Example: Hemisphere

Question: A solid hemisphere has a radius of 7 cm. Find its total surface area. (Use π = 22/7)

Step-by-Step Solution:

1. 1Identify dimension: r = 7 cm.
2. 2The keyword is "solid" and "total surface area". We must use the TSA formula.
3. 3TSA = 3π r² = 3 × (22/7) × 7 × 7.
4. 4Cancel one 7 in the numerator and denominator: 3 × 22 × 7.
5. 566 × 7 = 462 cm².

How to Approach Any Mensuration Problem

When you sit in the exam, do not just rush to plug numbers into a calculator. Follow this reliable sequence to avoid silly mistakes:

Quick Revision: The Ultimate Cheat Sheet

Here is your consolidated list of all the formulas covered in this chapter. Review this list the night before your exam.

• Cuboid
• Volume: l × b × h
• LSA (4 walls): 2h(l + b)
• TSA: 2(lb + bh + hl)
• Diagonal: √(l² + b² + h²)

• Cube
• Volume: a³
• LSA: 4a²
• TSA: 6a²
• Diagonal: a√(3)

• Right Circular Cylinder
• Volume: π r² h
• CSA: 2π r h
• TSA: 2π r(h + r)

• Right Circular Cone
• Slant Height (l): √(r² + h²)
• Volume: 1⁄3π r² h
• CSA: π r l
• TSA: π r(l + r)

• Sphere
• Volume: 4⁄3π r³
• Surface Area: 4π r²

• Hemisphere
• Volume: 2⁄3π r³
• CSA (hollow bowl): 2π r²
• TSA (solid half-ball): 3π r²