Surface Area of a Cuboid and a Cube

Surface Area of a Cuboid and a Cube

Understanding Three-Dimensional Shapes

When we move from flat, two-dimensional figures like squares and rectangles to the world of three dimensions, we encounter solid shapes or solids. These objects occupy space and have three measurements: length, breadth, and height. In our daily lives, we see these shapes everywhere — a matchbox, a room, a dice, a gift box — all are examples of three-dimensional objects.

In this lesson, we focus on two fundamental solids: the cuboid and the cube. Understanding how to calculate their surface areas is essential, because surface area tells us how much material is needed to cover the solid completely. Whether you're wrapping a gift, painting a room's walls, or designing packaging, surface area calculations are at the heart of the task.

What is a Cuboid?

A cuboid is a three-dimensional solid bounded by six rectangular faces. Think of a brick, a book, or a shoebox. Each face is a rectangle, and opposite faces are identical. A cuboid has:

• 8 vertices (corners)
• 12 edges (line segments where two faces meet)
• 6 faces (flat rectangular surfaces)

The three dimensions of a cuboid are typically called length (l), breadth (b), and height (h).

{{VISUAL: diagram: a labeled cuboid with vertices marked A, B, C, D, E, F, G, H, showing length l, breadth b, and height h clearly marked along the edges}}

{{KEY: type=definition | title=Cuboid | text=A cuboid is a three-dimensional solid with six rectangular faces, where opposite faces are equal and parallel. It has 8 vertices, 12 edges, and 6 faces.}}

Surface Area of a Cuboid

Total Surface Area

To find the total surface area (TSA) of a cuboid, we need to calculate the area of all six faces and add them together.

Let's observe the faces of a cuboid with dimensions l, b, and h:

• Top and bottom faces: Each has area = l × b. There are 2 such faces, so total area = 2lb
• Front and back faces: Each has area = b × h. There are 2 such faces, so total area = 2bh
• Left and right faces: Each has area = l × h. There are 2 such faces, so total area = 2lh

Adding all these together:

Total Surface Area = 2lb + 2bh + 2lh = 2(lb + bh + lh)

{{FORMULA: expr=TSA = 2(lb + bh + lh) | symbols=TSA:Total Surface Area (square units), l:length (units), b:breadth (units), h:height (units)}}

{{KEY: type=concept | title=Total Surface Area of a Cuboid | text=The total surface area of a cuboid is the sum of the areas of all six rectangular faces. It is given by TSA = 2(lb + bh + lh), where l, b, and h are the length, breadth, and height respectively.}}

Lateral Surface Area

Sometimes we need to find only the area of the four vertical faces, excluding the top and bottom. This is called the lateral surface area (LSA) or curved surface area.

For a cuboid:

Lateral Surface Area = 2(l + b)h = 2h(l + b)

This formula is particularly useful when calculating the area of walls in a room (excluding floor and ceiling).

{{KEY: type=definition | title=Lateral Surface Area | text=The lateral surface area of a cuboid is the total area of its four vertical faces, excluding the top and bottom. LSA = 2h(l + b), where l is length, b is breadth, and h is height.}}

What is a Cube?

A cube is a special type of cuboid where all edges are equal. In other words, l = b = h = a (where a is the edge length). A dice, a Rubik's cube, and sugar cubes are everyday examples.

{{VISUAL: diagram: a labeled cube with all edges marked as 'a', showing one vertex labeled O and edges OA, OB, OC extending along three perpendicular axes}}

Since a cube is a regular solid with all faces as identical squares, its surface area formulas become simpler.

Total Surface Area of a Cube

A cube has 6 square faces, each with area = a × a = a².

Total Surface Area of a Cube = 6a²

{{KEY: type=concept | title=Surface Area of a Cube | text=A cube has six identical square faces. If the edge length is a, then the total surface area is TSA = 6a², which is simply six times the area of one face.}}

Lateral Surface Area of a Cube

The lateral surface area includes only the four vertical faces:

Lateral Surface Area of a Cube = 4a²

Worked Examples

Example 1: Finding Total Surface Area of a Cuboid

Problem: A rectangular box has dimensions: length = 12 cm, breadth = 8 cm, height = 5 cm. Find its total surface area.

Solution:

Given: l = 12 cm, b = 8 cm, h = 5 cm

Using the formula:

TSA = 2(lb + bh + lh)

Substitute the values:

TSA = 2[(12 × 8) + (8 × 5) + (12 × 5)]
TSA = 2[96 + 40 + 60]
TSA = 2 × 196
TSA = 392 cm²

The box requires 392 square centimeters of material to be completely covered.

{{VISUAL: diagram: a labeled cuboid for Example 1 showing length 12 cm, breadth 8 cm, height 5 cm with all dimensions clearly marked}}

Example 2: Finding Edge of a Cube from Surface Area

Problem: The total surface area of a cube is 294 cm². Find the length of its edge.

Solution:

Given: TSA = 294 cm²

Using the formula for a cube:

6a² = 294

a² = 294 ÷ 6 = 49

a = √49 = 7 cm

The edge of the cube is 7 cm.

{{KEY: type=exam | title=Reverse Problems are Common | text=CBSE often asks questions where surface area is given and you must find dimensions. Always rearrange the formula correctly and remember that edge length cannot be negative.}}

Example 3: Lateral Surface Area Application

Problem: A room is 8 m long, 6 m wide, and 4 m high. Find the cost of whitewashing its four walls at the rate of ₹50 per square meter.

Solution:

This is a lateral surface area problem because we only paint the walls, not the floor or ceiling.

Given: l = 8 m, b = 6 m, h = 4 m

LSA = 2h(l + b)
LSA = 2 × 4 × (8 + 6)
LSA = 8 × 14
LSA = 112 m²

Cost = 112 × 50 = ₹5,600

{{VISUAL: diagram: a rectangular room viewed from inside showing four walls, with length 8 m, breadth 6 m, height 4 m, and shaded walls representing the area to be whitewashed}}

Comparison Table: Cuboid vs. Cube

{{ZOOM: title=Why "Lateral" and not "Curved"? | text=For cuboids and cubes, we use the term "lateral surface area" because the surfaces are flat (planes). The term "curved surface area" is reserved for solids with curved faces like cylinders, cones, and spheres — which we will study in the coming pages.}}

{{KEY: type=points | title=Quick Revision Points | text=- A cuboid has 6 rectangular faces; a cube has 6 square faces.

• Total surface area includes all faces; lateral surface area excludes top and bottom.
• For a cube with edge a: TSA = 6a² and LSA = 4a².
• Always write units as square units (cm², m², etc.) for area.}}

Practice Questions

1. Find the total surface area of a cuboid with dimensions 10 cm × 7 cm × 5 cm.
2. A cube has an edge of 9 cm. Find its total and lateral surface areas.
3. The lateral surface area of a cuboid is 250 m². If its length and breadth are 10 m and 5 m, find its height.
4. How much cardboard is required to make a closed box of dimensions 15 cm × 12 cm × 8 cm?

Mastering surface area formulas of cuboids and cubes is the foundation for understanding all three-dimensional mensuration in Class 9.

Surface Area and Volume of a Right Circular Cylinder

Page 2: Surface Area and Volume of a Right Circular Cylinder

Understanding the Right Circular Cylinder

A right circular cylinder is one of the most common three-dimensional shapes we encounter in daily life — from water bottles and cans to pipes and drums. The term "right" means that the axis of the cylinder is perpendicular to its circular bases, and "circular" tells us that the cross-section is a circle.

Let's visualize this shape clearly before diving into formulas. A cylinder has two parallel circular bases that are congruent (identical in size and shape) and are connected by a curved surface. The perpendicular distance between these two bases is called the height (denoted as h), and the radius of each circular base is denoted as r.

{{VISUAL: diagram: a right circular cylinder with labeled radius r on the top circular base, height h marked along the vertical axis, and the curved surface clearly shown}}

{{KEY: type=definition | title=Right Circular Cylinder | text=A solid geometric figure with two parallel, congruent circular bases connected by a curved surface, where the line joining the centers of the two bases is perpendicular to the base planes.}}

Surface Area of a Cylinder

The total surface area of a cylinder consists of three distinct parts:

• The area of the top circular base
• The area of the bottom circular base
• The area of the curved surface (lateral surface)

Curved Surface Area (CSA)

Imagine unwrapping the curved surface of a cylinder and laying it flat. What shape would you get? You'd get a rectangle! The length of this rectangle equals the circumference of the base circle (2πr), and its width equals the height (h) of the cylinder.

{{VISUAL: diagram: a cylinder being unwrapped to show the curved surface as a rectangle with length 2πr and width h, with labeled dimensions}}

Therefore, the Curved Surface Area (also called Lateral Surface Area) is:

{{FORMULA: expr=CSA = 2πrh | symbols=CSA:curved surface area (sq units), r:radius of base (units), h:height (units), π:mathematical constant (≈ 3.14 or 22/7)}}

Total Surface Area (TSA)

To find the Total Surface Area, we add the areas of both circular bases to the curved surface area:

• Area of top base = πr²
• Area of bottom base = πr²
• Curved surface area = 2πrh

Total Surface Area = 2πr² + 2πrh = 2πr(r + h)

{{KEY: type=concept | title=Total Surface Area of Cylinder | text=The total surface area of a right circular cylinder is the sum of the areas of its two circular bases and its curved surface, given by TSA = 2πr(r + h), where r is the radius and h is the height.}}

Summary Table

Volume of a Cylinder

The volume of a cylinder measures the amount of space it occupies — essentially, how much substance it can hold. Think of filling a cylindrical tank with water; the volume tells us the maximum capacity.

The volume can be understood as the area of the base multiplied by the height:

{{FORMULA: expr=V = πr²h | symbols=V:volume (cubic units), r:radius of base (units), h:height (units), π:mathematical constant (≈ 3.14 or 22/7)}}

This makes intuitive sense: if you stack circular discs of area πr² to a height h, the total space occupied is πr² × h.

{{KEY: type=points | title=Key Properties of Cylinders | text=- Volume depends on both the square of the radius and the height.

• Doubling the radius increases volume by 4 times (since r appears as r²).
• Doubling the height doubles the volume (linear relationship).
• Both surface area and volume use π, so keep consistent approximation (22/7 or 3.14).}}

Worked Examples

Example 1: Finding Surface Area

Problem: A cylindrical pillar has a radius of 7 cm and a height of 20 cm. Find its curved surface area and total surface area. (Use π = 22/7)

Solution:

Given:

• r = 7 cm
• h = 20 cm

Step 1: Calculate Curved Surface Area (CSA)

CSA = 2πrh

CSA = 2 × (22/7) × 7 × 20

CSA = 2 × 22 × 20

CSA = 880 cm²

Step 2: Calculate Total Surface Area (TSA)

TSA = 2πr(r + h)

TSA = 2 × (22/7) × 7 × (7 + 20)

TSA = 2 × 22 × 27

TSA = 1188 cm²

The curved surface area is 880 cm² and the total surface area is 1188 cm².

{{VISUAL: diagram: a labeled cylinder showing radius 7 cm and height 20 cm with calculations annotated}}

Example 2: Finding Volume

Problem: A cylindrical water tank has a diameter of 1.4 m and a height of 2 m. Find the volume of water it can hold. (Use π = 22/7)

Solution:

Given:

• Diameter = 1.4 m, so r = 1.4/2 = 0.7 m
• h = 2 m

Calculate Volume:

V = πr²h

V = (22/7) × (0.7)² × 2

V = (22/7) × 0.49 × 2

V = (22/7) × 0.98

V = 22 × 0.14

V = 3.08 m³

The tank can hold 3.08 cubic meters of water.

{{ZOOM: title=Why diameter is often given instead of radius | text=In real-world problems, diameters are easier to measure physically (using calipers or tape), so problems often provide diameter. Always remember to divide by 2 to get the radius before applying formulas, since all cylinder formulas use r, not d.}}

Example 3: Application Problem

Problem: A hollow cylindrical pipe has an outer radius of 10 cm, inner radius of 8 cm, and length of 50 cm. Find the total surface area of the pipe.

Solution:

This is a hollow cylinder, so we need to consider:

• Outer curved surface area
• Inner curved surface area
• Two ring-shaped ends

Given:

• Outer radius R = 10 cm
• Inner radius r = 8 cm
• Height h = 50 cm

Step-by-step:

1. Outer CSA = 2πRh = 2π(10)(50) = 1000π cm²

2. Inner CSA = 2πrh = 2π(8)(50) = 800π cm²

3. Area of two ring-shaped ends = 2(πR² - πr²) = 2π(R² - r²)

   = 2π(100 - 64) = 2π(36) = 72π cm²

4. Total Surface Area = 1000π + 800π + 72π = 1872π

   = 1872 × (22/7) = 5880 cm²

{{VISUAL: diagram: cross-section of a hollow cylinder showing outer radius R = 10 cm, inner radius r = 8 cm, with the ring-shaped end visible}}

{{KEY: type=exam | title=Hollow Cylinder Questions | text=Hollow cylinder problems frequently appear in CBSE exams. Remember to add both inner and outer curved surfaces plus the two ring-shaped ends. Don't forget the factor of 2 for the two ends.}}

Practice Problems

Try solving these on your own to build confidence:

1. Find the total surface area of a cylinder with radius 5 cm and height 12 cm.

2. A cylindrical tank has a volume of 1540 cm³ and radius 7 cm. Find its height.

3. The curved surface area of a cylinder is 264 cm² and its height is 12 cm. Find the radius.

4. How many liters of water can a cylindrical container hold if its radius is 35 cm and height is 1 m? (1 liter = 1000 cm³)

Hint for Problem 4: Convert all measurements to the same unit (cm) before calculating!