Lesson Notes By Weeks and Term v5 - Grade 7
Geometry of 2D shapes and 3D objects (Grade 7) – Week 2 focus
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Geometry of 2D shapes and 3D objects (Grade 7) – Week 2 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 7
Term: 3rd Term
Week: 2
Theme: General lesson support
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This week, we delve deeper into the fascinating world of geometry, focusing on understanding and working with 3D objects. Geometry is crucial because it helps us understand the world around us. From the shape of a soccer ball (a sphere) to the design of buildings (often rectangular prisms), geometry is everywhere. Understanding geometric shapes and their properties is essential for various careers, including architecture, engineering, design, and even carpentry. In South Africa, knowledge of geometry is vital for developing infrastructure and understanding spatial relationships in our environment.
Let's explore the world of 3D objects! 2.1 3D Objects and Their Properties: Cube: A cube is a 3D object with six identical square faces. All edges of a cube are equal in length. Think of a Rubik's Cube or a perfectly square sugar cube.
Faces: 6 Edges: 12 Vertices (corners): 8 Rectangular Prism (Cuboid): A rectangular prism has six rectangular faces. Opposite faces are identical. Think of a shoebox or a brick. Note that a cube is a special type of rectangular prism where all the faces are squares.
Faces: 6 Edges: 12 Vertices: 8 Triangular Prism: A triangular prism has two triangular faces and three rectangular faces. Imagine a Toblerone chocolate bar shape.
Faces: 5 Edges: 9 Vertices: 6 Sphere: A sphere is a perfectly round 3D object where every point on the surface is equidistant from the center. Think of a soccer ball. It has NO faces, edges, or vertices in the same way that polyhedra (shapes with flat faces) do.
Cylinder: A cylinder has two circular faces and one curved surface. Think of a tin can or a roll of toilet paper. It has two circular faces and a curved surface.
Cone: A cone has one circular face and one curved surface that tapers to a point (the apex). Think of an ice cream cone or a traffic cone. It has one circular face and a curved surface.
Pyramid: A pyramid has a polygon base and triangular faces that meet at a point (the apex). The shape of the base determines the type of pyramid (e.g., a square pyramid has a square base). Think of the pyramids of Egypt or a party hat. Faces depend on the base shape. A square pyramid has 5 faces (1 square, 4 triangles), 8 edges, and 5 vertices. 2.2 Nets of 3D Objects: A net is a 2D shape that can be folded to form a 3D object. Drawing nets helps us visualize how the faces of a 3D object fit together.
Cube Net: A cube net consists of six squares connected in such a way that they can be folded to form a cube. There are multiple possible nets for a cube. A common one is a "T" shape with the square base, four squares forming the sides and the sixth square for the top.
Rectangular Prism Net: A rectangular prism net consists of six rectangles connected in such a way that they can be folded to form a rectangular prism. Again, there are multiple possible nets. 2.3 Surface Area: The surface area of a 3D object is the total area of all its faces.
Surface Area of a Cube: Since a cube has six identical square faces, the surface area is 6 times the area of one face. If the side length of the cube is 's', then the surface area is: Surface Area = 6 s * s = 6s² Surface Area of a Rectangular Prism: A rectangular prism has three pairs of identical rectangular faces. Let the length, width, and height be 'l', 'w', and 'h' respectively.
The surface area is: Surface Area = 2 (lw + lh + w*h) 2.4 Volume: The volume of a 3D object is the amount of space it occupies.
Volume of a Cube: The volume of a cube is found by multiplying the length, width, and height. Since all sides are equal ('s'), the volume is: Volume = s s * s = s³ Volume of a Rectangular Prism: The volume of a rectangular prism is found by multiplying the length, width, and height. Volume = l w * h Example 1: Surface Area and Volume of a Cube Problem: A sugar cube has a side length of 1 cm. Calculate its surface area and volume.
Solution: Surface Area = 6s² = 6 (1 cm)² = 6 * 1 cm² = 6 cm² Volume = s³ = (1 cm)³ = 1 cm³ Example 2: Surface Area and Volume of a Rectangular Prism Problem: A brick has a length of 20 cm, a width of 10 cm, and a height of 5 cm. Calculate its surface area and volume.
Solution: Surface Area = 2 (lw + lh + wh) = 2 (20cm10cm + 20cm5cm + 10cm5cm) = 2 (200 cm² + 100 cm² + 50 cm²) = 2 * 350 cm² = 700 cm² Volume = l w h = 20 cm 10 cm * 5 cm = 1000 cm³ Example 3: Problem Solving with Surface Area Problem: A rectangular prism shaped container needs to be painted. It is 3m long, 2m wide, and 1.5m high. If 1 litre of paint covers 4m², how many litres of paint are needed to paint the entire container once?
Solution: Surface Area = 2 (lw + lh + wh) = 2 (3m2m + 3m1.5m + 2m1.5m) = 2 (6 m² + 4.5 m² + 3 m²) = 2 * 13.5 m² = 27 m² Paint needed = Total Surface Area / Coverage per litre = 27 m² / 4 m²/litre = 6.75 litres. Since you can't buy 0.75 of a litre, you'll need to buy 7 litres. Guided Practice (With Solutions)
Question 1: A cube has a side length of 3 cm. Draw a possible net of this cube. What is its surface area?
Solution: Draw a "T" shaped net of the cube, with each square having sides of 3cm. Surface Area = 6s² = 6 (3 cm)² = 6 * 9 cm² = 54 cm² Question 2: A rectangular prism has a length of 5 cm, a width of 4 cm, and a height of 2 cm. Calculate its volume.
Solution: Volume = l w h = 5 cm 4 cm * 2 cm = 40 cm³ Question 3: Imagine you are making a gift box. The box is a cube with sides of 10cm. How much cardboard (in square centimeters) do you need to make the box? This is the surface area.