Surface Area & Volume • Topic 1 of 3

Cubes, Cuboids, and Their Nets

What is a Cube? A cube is a three-dimensional solid shape with six square faces, all of equal size. All edges of a cube are equal in length.

What is a Cuboid? A cuboid is a three-dimensional solid shape with six rectangular faces. Opposite faces are identical. A cuboid has length ($l$), breadth ($b$), and height ($h$).

Nets of Solids: A net is a two-dimensional pattern that can be folded to form a three-dimensional solid. Nets help us understand how the faces of a solid are arranged.

Properties of Cubes and Cuboids:

Property	Cube	Cuboid
Faces	6 square faces	6 rectangular faces
Edges	12 equal edges	12 edges (4 length, 4 breadth, 4 height)
Vertices	8	8
Edge length	All edges = $a$	Length ($l$), Breadth ($b$), Height ($h$)
Surface Area	$6a^2$	$2(lb + bh + hl)$
Volume	$a^3$	$l \times b \times h$

Formulas Summary:

Measurement	Cube (side = a)	Cuboid (l, b, h)
Lateral Surface Area (LSA)	$4a^2$	$2h(l + b)$
Total Surface Area (TSA)	$6a^2$	$2(lb + bh + hl)$
Volume	$a^3$	$l \times b \times h$
Diagonal	$\sqrt{3}a$	$\sqrt{l^2 + b^2 + h^2}$

Solved Examples

Worked Example

Find the total surface area and volume of a cube with side 5 cm.

Solution- TSA = $6a^2 = 6 \times 5^2 = 6 \times 25 = 150$ cm² - Volume = $a^3 = 5^3 = 125$ cm³ - **Answer:** TSA = 150 cm², Volume = 125 cm³ *Example 2: A cuboid has dimensions: length = 8 cm, breadth = 6 cm, height = 4 cm. Find its total surface area and volume. Solution: - TSA = $2(lb + bh + hl) = 2(8×6 + 6×4 + 4×8)$ - $= 2(48 + 24 + 32) = 2 \times 104 = 208$ cm² - Volume = $l \times b \times h = 8 \times 6 \times 4 = 192$ cm³ - **Answer:** TSA = 208 cm², Volume = 192 cm³ *Example 3: A cuboid has a volume of 240 cm³. Its length is 10 cm and breadth is 6 cm. Find its height and total surface area. Solution: - Volume = $l \times b \times h = 10 \times 6 \times h = 60h = 240$ - $h = \frac{240}{60} = 4$ cm - TSA = $2(lb + bh + hl) = 2(10×6 + 6×4 + 4×10)$ - $= 2(60 + 24 + 40) = 2 \times 124 = 248$ cm² - **Answer:** Height = 4 cm, TSA = 248 cm²

Key Points

Cube: all edges equal; 6 square faces; Volume = $a^3$; TSA = $6a^2$
Cuboid: length, breadth, height; Volume = $l \times b \times h$; TSA = $2(lb + bh + hl)$
Lateral surface area excludes top and bottom: Cube = $4a^2$; Cuboid = $2h(l+b)$
A net is a 2D pattern that folds into a 3D shape
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Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The volume of a cube of edge $a$ is:

Explanation: $a^3$.

Q2.The surface area of a cube is:

Explanation: $6a^2$.

Q3.The volume of a cuboid is:

Explanation: $lbh$.

Q4.A net of a cube has how many square faces?

Explanation: Six.

Practice more MCQs → 📝 Assignment · 35 marks

Measurement	Cube (side = a)	Cuboid (l, b, h)
Lateral Surface Area (LSA)	\(4a^2\)	\(2h(l + b)\)
Total Surface Area (TSA)	\(6a^2\)	\(2(lb + bh + hl)\)
Volume	\(a^3\)	\(l \times b \times h\)
Diagonal	\(\sqrt{3}a\)	\(\sqrt{l^2 + b^2 + h^2}\)