Grade 9 Surface Area and Volume

Cuboid: A 3D shape with 6 rectangular faces. It has length (l), breadth (b), and height (h).

Surface Area: The total area of all the faces of the 3D shape.

Faces of a Cuboid: A cuboid has 3 pairs of identical rectangular faces:

• Top and Bottom: Area = l * b
• Front and Back: Area = l * h
• Left and Right Sides: Area = b * h

Total Surface Area (TSA) Formula: The sum of the areas of all 6 faces.
TSA = 2 * (Area of Top/Bottom) + 2 * (Area of Front/Back) + 2 * (Area of Sides)
TSA = 2 * (l * b) + 2 * (l * h) + 2 * (b * h)
TSA = 2(lb + lh + bh)

Lateral Surface Area (LSA) / Area of Four Walls: The sum of the areas of the four side faces (excluding top and bottom).
LSA = 2 * (l * h) + 2 * (b * h) = 2h(l + b)

Volume: The amount of space a 3D object occupies. It is measured in cubic units (e.g., cm³, m³).

Volume of a Cuboid Formula: The product of its length, breadth, and height.
Volume (V) = length * breadth * height
V = l * b * h

Example: Find the volume of a cuboid with length 5 cm, breadth 3 cm, and height 2 cm.
l = 5, b = 3, h = 2.

Step 1: Use the formula V = l * b * h.
V = 5 * 3 * 2

Step 2: Perform the calculation.
V = 15 * 2
V = 30

Result: The volume of the cuboid is 30 cubic cm (30 cm³).

Cube: A special type of cuboid where all sides (edges) are equal in length. Let the side length be 'a'.

Faces of a Cube: A cube has 6 identical square faces. The area of each square face is side * side = a * a = a².

Total Surface Area (TSA) Formula: The sum of the areas of all 6 identical faces.
TSA = 6 * (Area of one face) = 6 * a²

Lateral Surface Area (LSA) / Area of Four Walls: The sum of the areas of the four side faces (excluding top and bottom).
LSA = 4 * (Area of one face) = 4 * a²

Example: Find the total surface area of a cube with side length 4 cm.
a = 4.

Step 1: Use the formula TSA = 6 * a².
TSA = 6 * 4²

Step 2: Calculate the area.
TSA = 6 * 16
TSA = 96

Result: The total surface area of the cube is 96 square cm (96 cm²).

Cube: A cuboid with equal sides (edge length 'a').

Volume of a Cube Formula: The product of its length, breadth, and height, which are all equal to 'a'.
Volume (V) = side * side * side
V = a * a * a = a³

Example: Find the volume of a cube with side length 5 cm.
a = 5.

Step 1: Use the formula V = a³.
V = 5³

Step 2: Calculate the volume.
V = 5 * 5 * 5
V = 125

Result: The volume of the cube is 125 cubic cm (125 cm³).

Cylinder: A 3D shape with two parallel circular bases and a curved surface connecting them. It has a radius (r) and height (h).

Curved Surface Area (CSA): The area of the curved part of the cylinder. Imagine unrolling the curved surface into a rectangle; its length would be the circumference of the base (2 * pi * r) and its width would be the height (h).
CSA = Circumference of base * height = (2 * pi * r) * h
CSA = 2 * pi * r * h

Area of Bases: Each circular base has an area of pi * r². Since there are two bases (top and bottom), their combined area is 2 * pi * r².

Total Surface Area (TSA) Formula: The sum of the curved surface area and the area of the two bases.
TSA = CSA + Area of two bases
TSA = 2 * pi * r * h + 2 * pi * r²
TSA = 2 * pi * r (h + r)

Volume of a Cylinder Formula: The product of the area of its base and its height.
Volume (V) = Area of base * height

Area of Base: The base is a circle with radius 'r'. The area of the base is pi * r².

Formula: Substitute the area of the base into the volume formula.
V = (pi * r²) * h
V = pi * r² * h

Example: Find the volume of a cylinder with radius 3 cm and height 10 cm (Use pi ≈ 3.14).
r = 3, h = 10, pi ≈ 3.14.

Step 1: Use the formula V = pi * r² * h.
V = 3.14 * 3² * 10

Step 2: Perform the calculation.
V = 3.14 * 9 * 10
V = 3.14 * 90
V = 282.6

Result: The volume of the cylinder is approximately 282.6 cubic cm (282.6 cm³).