📏 Surface Area & Volume

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:

PropertyCubeCuboidFaces6 square faces6 rectangular facesEdges12 equal edges12 edges (4 length, 4 breadth, 4 height)Vertices88Edge lengthAll edges = \(a\)Length (\(l\)), Breadth (\(b\)), Height (\(h\))Surface Area\(6a²\)\(2(lb + bh + hl)\)Volume\(a³\)\(l × b × h\)
Formulas Summary:

MeasurementCube (side = a)Cuboid (l, b, h)Lateral Surface Area…

What is a Cylinder? A cylinder is a three-dimensional solid with two parallel circular bases connected by a curved surface. Examples: a soda can, a pipe, a candle.

Parts of a Cylinder:

• Radius (\(r\)): distance from center to edge of circular base
• Height (\(h\)): perpendicular distance between the two bases

Formulas for a Cylinder:

MeasurementFormulaCurved Surface Area (CSA)\(2π r h\)Total Surface Area (TSA)\(2π r h + 2π r² = 2π r (r + h)\)Volume\(π r² h\)
Understanding the Formulas:

• Curved surface area = circumference of base × height = \((2π r) × h\)
• Volume = area of base × height = \((π r²) × h\)

Comparison of Solids:

SolidTSA FormulaVolume FormulaCube\(6a²\)\(a³\)Cuboid\(2(lb + bh + hl)\)\(l × b × h\)Cylinder\(2π r(r + h)\)\(π r² h\)

What is a Prism? A prism is a three-dimensional solid with two parallel, identical polygonal bases connected by rectangular lateral faces. The shape is named after its base.

Types of Prisms:

Base ShapePrism NameExampleTriangleTriangular prismToblerone chocolate boxSquareSquare prism (cube if all edges equal)DiceRectangleRectangular prism (cuboid)ShoeboxPentagonPentagonal prismPencil (some types)HexagonHexagonal prismNut (some types)
Formulas for Prisms:

MeasurementGeneral FormulaLateral Surface Area (LSA)Perimeter of base × HeightTotal Surface Area (TSA)LSA + 2 × Area of baseVolumeArea of base × Height
Comparison: Cylinder and Prism

• A cylinder is like a prism with circular bases
• A cube is a prism with square bases
• A cuboid is a prism with rectangular bases

Problem Solving Tips:

• Identify the shape (cube, cuboid, cylinder, triangular prism, etc.)
• Write…