Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify and classify various 3D objects (prisms, pyramids, etc.).
• Calculate the surface area of rectangular prisms and cubes.
• Calculate the volume of rectangular prisms and cubes.
• Describe the nets of rectangular prisms and cubes.
• Relate 2D shapes to the faces of 3D objects.

Lesson summary

Geometry helps us understand the world around us. From the design of our homes and schools to the shapes of everyday objects like cell phones and soccer balls, geometry is everywhere. This week, we will delve deeper into 2D shapes and 3D objects, focusing on understanding their properties and how to identify them. Knowing these shapes and their properties helps us solve practical problems, like calculating the amount of paint needed for a wall or understanding how different shapes fit together in construction. Imagine building a RDP house; you need to understand shapes to build strong walls and a stable roof!

Lesson notes

3D Objects: A World of Shapes 3D objects, also known as solid figures, have three dimensions: length, width, and height.

We'll explore several important types: Prisms: A prism has two identical parallel faces (bases) that are polygons (like triangles, squares, or pentagons) and other faces that are parallelograms (usually rectangles). The name of the prism comes from the shape of its base.

Rectangular Prism: The bases are rectangles. Think of a brick or a shoebox.

Cube: A special rectangular prism where all sides are squares. Think of a die (singular of dice).

Triangular Prism: The bases are triangles. Think of a Toblerone chocolate box.

Pyramids: A pyramid has a polygon as its base and triangular faces that meet at a single point called the apex. Again, the name comes from the shape of the base.

Square Pyramid: The base is a square. Think of the pyramids of Giza.

Triangular Pyramid: The base is a triangle (also called a tetrahedron).

Cylinder: Has two circular bases that are parallel and connected by a curved surface. Think of a tin can or a water pipe.

Cone: Has a circular base and a curved surface that tapers to a single point called the apex. Think of an ice cream cone.

Sphere: A perfectly round 3D object where every point on the surface is the same distance from the center. Think of a soccer ball or a marble. Faces, Edges, and Vertices: Face: A flat surface of a 3D object.

Edge: A line segment where two faces meet.

Vertex (plural: Vertices): A point where three or more edges meet.

Example: A cube has 6 faces, 12 edges, and 8 vertices. Nets of 3D Objects A net is a 2D pattern that can be folded to form a 3D object. Imagine unfolding a box; the resulting flat shape is its net. Different objects can have multiple different nets.

Rectangular prism net: a rectangle for the middle with smaller rectangles and squares on the top and bottom.

Cube net: six squares joined so that it will fold to form a cube. Surface Area The surface area of a 3D object is the total area of all its faces.

Rectangular Prism: The surface area (SA) is calculated as: SA = 2(lw + lh + wh), where l = length, w = width, and h = height. Why this formula? A rectangular prism has 6 faces. There are three pairs of faces that are identical in size. The formula accounts for the area of each of these three pairs.

Cube: Since all sides are equal (let's call the side length 's'), the surface area is: SA = 6s², where s = side length. Why this formula? A cube has 6 identical square faces, each with an area of s * s = s².

Example 1: Surface Area of a Rectangular Prism A brick used in construction is 22 cm long, 11 cm wide, and 7 cm high. What is its surface area?

Solution: Identify the dimensions: l = 22 cm, w = 11 cm, h = 7 cm.

Use the formula: SA = 2(lw + lh + wh)

Substitute the values: SA = 2((22 11) + (22 7) + (11 * 7))

Calculate: SA = 2(242 + 154 + 77) SA = 2(473) SA = 946 cm² Therefore, the surface area of the brick is 946 cm².

Example 2: Surface Area of a Cube A sugar cube has sides of 1 cm. What is its surface area?

Solution: Identify the side length: s = 1 cm Use the formula: SA = 6s² Substitute the value: SA = 6 * (1)² Calculate: SA = 6 * 1 SA = 6 cm² Therefore, the surface area of the sugar cube is 6 cm². Volume The volume of a 3D object is the amount of space it occupies.

Rectangular Prism: The volume (V) is calculated as: V = lwh, where l = length, w = width, and h = height. Why this formula? Volume is essentially the base area (l * w) multiplied by the height (h). It tells you how many unit cubes fit inside the prism.

Cube: Since all sides are equal (side length 's'), the volume is: V = s³, where s = side length. Why this formula? It's length width height, but since all sides are the same, its s s s = s³.

Example 3: Volume of a Rectangular Prism A container used to transport fruit is 50 cm long, 30 cm wide, and 20 cm high. What is its volume?

Solution: Identify the dimensions: l = 50 cm, w = 30 cm, h = 20 cm Use the formula: V = lwh Substitute the values: V = 50 30 20 Calculate: V = 30000 cm³ Therefore, the volume of the container is 30000 cm³.

Example 4: Volume of a Cube A box used to store spices has sides of 8 cm. What is its volume?

Solution: Identify the side length: s = 8 cm Use the formula: V = s³ Substitute the value: V = 8³ Calculate: V = 512 cm³ Therefore, the volume of the box is 512 cm³. Guided Practice (With Solutions)

Question 1: Identify the following 3D object and state the number of faces, edges and vertices. The object has six square faces.

Solution: Object: Cube Faces: 6 Edges: 12 Vertices: 8

Commentary: Recognizing square faces helps identify a cube. Remembering that cubes have 6 faces, 12 edges and 8 vertices will help with future problems.

Question 2: A rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 4 cm. Calculate its surface area.