Surface Area of Cubes and Cuboids: Identifying Prisms and Pyramids

The primary focus of this lesson is on identifying prisms and pyramids, with a specific emphasis on calculating the surface area of cubes and cuboids (rectangular prisms).
Lesson Objectives: - Develop the ability to find the surface area of a cube/cuboid. - Success Criteria: - Identification of the net of 3D solid shapes (Depth of Knowledge/DOK-1). - Solving word problems involving surface area in real-life contexts (DOK-3). - Comparing surface areas and providing justification for answers (DOK-3).

Surface Area: The total area that the surface of an object occupies.
Curved / Lateral Surface: The area of all the sides of a 3D object excluding its top and bottom bases.
Base: The bottom surface of a solid object.
Cube: A three-dimensional solid object bounded by six square faces, facets, or sides, with three meeting at each vertex.
Cuboid / Rectangular Prism: A convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube.
Square Unit: The unit of measurement for area (e.g., $cm^2$ , $m^2$ ).

Common 3D Shapes: - Cube: A shape where all sides and faces are equal squares. - Cuboid: A rectangular-shaped box. - Cylinder: A shape with two circular bases and a curved surface. - Triangular Prism: A prism with two triangular bases and three rectangular sides. - Cone: A shape with a circular base that tapers to a point (apex). - Pyramid: A shape with a polygonal base and triangular faces that meet at a common point. - Tetrahedron: A triangular pyramid with four triangular faces.
Activity: Investigation of Nets: - A net is a two-dimensional pattern that can be folded to make a three-dimensional shape. - Q1: For a cube with a side length of $4\,cm$ , the net consists of six squares, each measuring $4\,cm \times 4\,cm$ . On a grid where each square represents $1\,cm$ , this would occupy a specific cross-shaped or T-shaped arrangement of $16$ grid squares per face. - Q2: For a cuboid with dimensions $9\,cm$ (length), $2\,cm$ (width), and $1\,cm$ (height), the net must represent two faces of $9 \times 2$ , two faces of $9 \times 1$ , and two faces of $2 \times 1$ .

Surface Area of a Cuboid: - A cuboid has three pairs of identical faces: - Top and Bottom: $2 \times (l \times w)$ - Front and Back: $2 \times (h \times w)$ - Left and Right: $2 \times (l \times h)$ - The complete formula is: $Surface \, Area = 2lw + 2wh + 2lh$ - Alternatively written as: $Surface \, Area = 2(lw + wh + lh)$
Surface Area of a Cube: - A cube has six identical square faces. If the length of one side is $x$ : - The area of one face is $x \times x = x^2$ . - Since there are six faces, the formula is: $Surface \, Area = 6x^2$

Cuboid Demo 1: - Dimensions: $l = 5\,cm$ , $w = 2\,cm$ , $h = 1\,cm$ - Calculation: $SA = 2((5 \times 2) + (2 \times 1) + (5 \times 1)) = 2(10 + 2 + 5) = 2(17) = 34\,cm^2$
Cuboid "Your Turn" 1: - Dimensions: $l = 4\,cm$ , $w = 4\,cm$ , $h = 2\,cm$ - Calculation: $SA = 2((4 \times 4) + (4 \times 2) + (4 \times 2)) = 2(16 + 8 + 8) = 2(32) = 64\,cm^2$
Cube Demo 2: - Dimensions: $Side = 5\,cm$ - Calculation: $SA = 6 \times 5^2 = 6 \times 25 = 150\,cm^2$
Cube "Your Turn" 2: - Dimensions: $Side = 7\,cm$ - Calculation: $SA = 6 \times 7^2 = 6 \times 49 = 294\,cm^2$

UAE Link: Premium Dates Packaging: - Dates are a symbol of hospitality and tradition in the UAE, often served with Arabic coffee (Gahwa). - Problem: A cuboidal gift box for dates measures $20\,cm$ length, $12\,cm$ width, and $8\,cm$ height. - Task: Calculate the surface area for a themed paper cover. - Calculation: $SA = 2((20 \times 12) + (12 \times 8) + (20 \times 8)) = 2(240 + 96 + 160) = 2(496) = 992\,cm^2$
Comparison Task (Think-Pair-Share): - Box A (Cube): Side = $10\,cm$ . $SA = 6 \times 10^2 = 600\,cm^2$ - Box B (Cuboid): $12\,cm \times 8\,cm \times 6\,cm$ . - Calculation for Box B: $SA = 2((12 \times 8) + (8 \times 6) + (12 \times 6)) = 2(96 + 48 + 72) = 2(216) = 432\,cm^2$ - Comparison: The cube-shaped box requires more paper. It requires $600 - 432 = 168\,cm^2$ more wrapping paper.
Villa Water Tank (Abu Dhabi): - Problem: A cuboid water tank needs painting to protect it from the sun. - Dimensions: $2.5\,m$ long, $1.5\,m$ wide, and $1\,m$ high. - Calculation: $SA = 2((2.5 \times 1.5) + (1.5 \times 1) + (2.5 \times 1)) = 2(3.75 + 1.5 + 2.5) = 2(7.75) = 15.5\,m^2$

Wrapping Contest (DOK 2/3): - Ria has a cube box with side $6\,cm$ . $SA = 6 \times 6^2 = 216\,cm^2$ - Mai has a cereal box ( $20 \times 10 \times 5$ ). $SA = 2(200 + 50 + 100) = 2(350) = 700\,cm^2$ - Conclusion: Mai needs more wrapping paper.
Storage Container (DOK 2): - Dimensions: $30\,cm \times 25\,cm \times 18\,cm$ - Calculation: $SA = 2((30 \times 25) + (25 \times 18) + (30 \times 18)) = 2(750 + 450 + 540) = 2(1740) = 3480\,cm^2$
DOK 4 Extended Task (Biology/Math Link): - Context: A treasure chest in an aquarium is covered with algae. - Given: Algae population = $30,000,000$ cells. - Objective: Calculate population density based on the total surface area of the chest.

Problem: Find the surface area of a square-based prism with a base side of $3\,cm$ and a height of $8\,cm$ .
Calculation: - Two square bases: $2 \times (3 \times 3) = 18\,cm^2$ - Four rectangular sides: $4 \times (3 \times 8) = 96\,cm^2$ - Total Surface Area: $18 + 96 = 114\,cm^2$
Correct Answer: Option 1: $114\,cm^2$