Chapter Notes: 3-D Shapes
1. Welcome to the World of 3-D Shapes!
Hello, super mathematicians! Have you ever built a tower with blocks, kicked a football, or eaten an ice cream cone? If you have, you're already an expert at using 3-D shapes!
"3-D" stands for 3-Dimensional. It means these shapes are not flat like a drawing on paper. They are solid objects that you can pick up, hold, and look at from all sides. They have height, width, and depth.
In this chapter, we're going on an adventure to explore these amazing shapes that are all around us!
2. Meet the 3-D Shapes!
Let's get to know the main members of the 3-D shape family. We will also investigate if they can stack on top of each other or if they can roll.
The Sphere
This shape is perfectly round! It has no flat surfaces or corners.
Examples: A football, a marble, a planet like Earth.
Investigation:
• Can it roll? Yes, it can roll in any direction!
• Can it stack? No, you can't stack spheres on top of each other easily.
The Cone
A cone has one flat, circular base and a pointy top called an apex.
Examples: An ice cream cone, a party hat, a traffic cone.
Investigation:
• Can it roll? Yes, but it will roll around in a circle.
• Can it stack? You can't stack another shape on its pointy top.
The Cylinder
A cylinder has two flat, identical circular bases and one curved side.
Examples: A can of soup, a tube of crisps, a glue stick.
Investigation:
• Can it roll? Yes, it can roll along its curved side.
• Can it stack? Yes, cylinders stack very well on their flat bases!
The Pyramid
A pyramid has one flat base and triangular sides that all meet at a single pointy top (the apex).
Examples: The great pyramids in Egypt, some types of tents.
Investigation:
• Can it roll? No, it has flat faces.
• Can it stack? Not on its pointy top.
The Prism
A prism has two flat, identical bases and flat sides (usually rectangles).
Examples: A cereal box, a bar of chocolate like Toblerone, a block.
Investigation:
• Can it roll? No, it has flat faces.
• Can it stack? Yes, prisms are excellent for stacking!
Key Takeaway
Shapes with curved surfaces (like spheres and cones) can roll. Shapes with flat faces (like prisms and pyramids) can be stacked.
3. The Parts of a 3-D Shape
Solid shapes like prisms and pyramids have special parts. Learning their names helps us describe them accurately. Don't worry if this seems tricky at first, we'll use a simple box shape (a cuboid) to help us understand.
• Faces: These are the flat surfaces of a 3-D shape. A dice has 6 faces.
• Edges: An edge is where two faces meet. Think of it as a straight line or a fold on a box.
• Vertices: A vertex is a corner where edges meet. It's the "pointy bit". The word for more than one vertex is vertices.
Some shapes, like a cylinder, have both flat faces (the circular ends) and a curved surface (the part that goes around). A sphere has only one curved surface.
Quick Review Box
Face = Flat surface
Edge = Where two faces meet (a line)
Vertex = A corner (a point)
4. A Closer Look at Prisms and Pyramids
Did you know we can give prisms and pyramids special first names? The secret is to look at the shape of their base (the bottom part)!
Naming Prisms
A prism is named after the shape of its two identical bases.
• Triangular Prism: Its bases are triangles. (Think of a tent!)
• Quadrilateral Prism: Its bases are quadrilaterals (4-sided shapes). A cuboid (like a brick) and a cube (where all faces are equal squares) are special types of quadrilateral prisms.
Naming Pyramids
A pyramid is named after the shape of its single base.
• Triangular Pyramid: Its base is a triangle.
• Square-based Pyramid: Its base is a square. (This is also a type of quadrilateral pyramid!)
• Pentagonal Pyramid: Its base is a pentagon (a 5-sided shape).
Key Takeaway
To name a prism or a pyramid, just look at its base! The name of the base shape is the first name of the 3-D shape.
5. Unfolding Shapes: An Introduction to Nets
What if you could carefully cut along the edges of a cardboard box and lay it completely flat? That flat shape is called a net. A net is the 2-D pattern that can be folded to make a 3-D shape.
Let's look at some nets:
Net of a Cube: This is one of the most common nets. If you cut this out and fold it, you'd get a perfect cube!
Net of a Cuboid: It looks similar to a cube's net, but some of the rectangular faces will be longer.
Net of a Cylinder: This one is surprising! It's made of two circles and one rectangle. The rectangle wraps around to form the curved surface.
Did you know?
There are 11 different ways to draw a net for a cube! Can you try to find some of them on a piece of paper?
6. What's Inside? A Peek at Cross-Sections
Imagine you have a magic knife that can slice perfectly through any 3-D shape. The new flat surface you create is called a cross-section.
Think about slicing a loaf of bread. Every slice is a cross-section of the loaf!
• Prisms and Cylinders: If you slice a prism or a cylinder parallel to its base (the same way you slice bread), the cross-section will be the exact same shape and size as the base. Every slice is identical!
• Pyramids and Cones: If you slice a pyramid or a cone parallel to its base, the cross-section will be the same shape as the base, but it will be smaller. The closer to the pointy top you slice, the smaller the cross-section gets.
7. Become a Shape Detective! (Enrichment)
Let's find some secret patterns in prisms and pyramids by counting their faces, vertices, and edges. You've got this!
The Prism Pattern
Let's look at a triangular prism (its base has 3 sides):
• Vertices (corners): 6
• Faces (flat sides): 5
• Edges (lines): 9
Now let's look at a cuboid/quadrilateral prism (its base has 4 sides):
• Vertices: 8
• Faces: 6
• Edges: 12
Can you see the pattern? If the base has N sides:
Number of Vertices = N x 2
Number of Faces = N + 2
Number of Edges = N x 3
The Pyramid Pattern
Let's look at a square-based pyramid (its base has 4 sides):
• Vertices: 5
• Faces: 5
• Edges: 8
Can you see the pattern here? If the base has N sides:
Number of Vertices = N + 1
Number of Faces = N + 1
Number of Edges = N x 2
Key Takeaway
Maths is full of amazing patterns! By looking closely, we can find rules that work for whole families of shapes. Great job, shape detective!