Three-Dimensional Geometry

Identifying Basic 3D Shapes

Let’s learn about 3D shapes! A 3D shape is a shape that is solid. It is not flat. You can hold it in your hand.

3D shapes have length, width, and height. That means they take up space.

We see 3D shapes all around us every day. A ball, a box, and a can are all 3D shapes.

In this lesson, we will learn to name these basic 3D shapes:

• sphere
• cube
• cylinder
• rectangular prism
• cone

How are 3D shapes different from flat shapes?

A flat shape is a shape like a circle, square, or rectangle. Flat shapes are smooth and thin on paper.

A 3D shape is solid. You can stack some of them, roll some of them, and pick them up.

Think about it like this:

• A circle is flat, but a sphere is solid.
• A square is flat, but a cube is solid.
• A rectangle is flat, but a rectangular prism is solid.

1. Sphere

A sphere is round all over.

It has no flat sides. It can roll.

Real-world examples of a sphere:

• a ball
• a globe

If it looks like a ball, it may be a sphere.

2. Cube

A cube is like a box where all the sides are the same size.

It has flat faces. It does not roll like a sphere. It can stack.

Real-world examples of a cube:

• a dice
• a sugar cube
• a toy block

If it looks like a little square box, it may be a cube.

3. Cylinder

A cylinder has 2 flat circles and 1 curved side.

It can roll on its curved side. It can also stand on its flat ends.

Real-world examples of a cylinder:

• a can
• a paper towel roll
• a candle

If it looks like a can, it may be a cylinder.

4. Rectangular Prism

A rectangular prism is a box shape.

It has flat faces. Its faces are rectangles. It can stack.

Real-world examples of a rectangular prism:

• a cereal box
• a tissue box
• a book

If it looks like a long box, it may be a rectangular prism.

5. Cone

A cone has 1 flat circle on the bottom and comes to a point on top.

It can roll in a funny way because of its curved side. It can also stand on its flat bottom.

Real-world examples of a cone:

• an ice cream cone
• a party hat
• a traffic cone

If it has a point at the top, it may be a cone.

Let’s compare the shapes

• Sphere: round all over, rolls
• Cube: box shape, all sides the same size
• Cylinder: like a can, 2 flat ends and 1 curved side
• Rectangular prism: like a box, longer than a cube
• Cone: 1 flat bottom and 1 point

Helpful clues

• If it is shaped like a ball, think sphere.
• If it is shaped like a small box with equal sides, think cube.
• If it is shaped like a can, think cylinder.
• If it is shaped like a longer box, think rectangular prism.
• If it has a point, think cone.

Worked Example 1

Question: A soccer ball is round and can roll. What 3D shape is it?

Think: A shape that is round all over is a sphere.

Answer: A soccer ball is a sphere.

Worked Example 2

Question: A soup can has 2 flat circle ends and 1 curved side. What 3D shape is it?

Think: A shape with 2 flat circles and 1 curved side is a cylinder.

Answer: A soup can is a cylinder.

Worked Example 3

Question: A toy block looks like a small box. All its sides are the same size. What 3D shape is it?

Think: A box with all sides the same size is a cube.

Answer: The toy block is a cube.

Worked Example 4

Question: You see an ice cream cone. It has a flat circle on the bottom and a point on top. What 3D shape is it?

Think: A shape with 1 flat bottom and 1 point is a cone.

Answer: The ice cream cone is a cone.

Try thinking about these objects

1. A gift box shaped like a long box is a rectangular prism.
2. A basketball is a sphere.
3. A birthday hat is a cone.
4. A dice is a cube.
5. A can of beans is a cylinder.

What should you ask yourself?

• Is it round all over?
• Does it look like a box?
• Does it look like a can?
• Does it have a point?
• Can it roll, stack, or both?

These questions can help you choose the right 3D shape.

Summary

3D shapes are solid shapes that take up space. The basic 3D shapes you should know are sphere, cube, cylinder, rectangular prism, and cone.

You can find these shapes in real life. Balls are spheres, dice are cubes, cans are cylinders, boxes are rectangular prisms, and party hats are cones.

When you look at an object, think about its shape. Ask yourself if it is round, box-shaped, can-shaped, or pointed. That will help you identify the 3D shape.

Defining Attributes of 3D Shapes

Today we will learn about 3D shapes. A 3D shape is a solid shape. That means it is not flat. It has length, width, and height, so it takes up space.

You can hold many 3D shapes in your hands. A box, a ball, and a can are all examples of 3D shapes.

We will learn how to describe 3D shapes by looking at their faces, edges, and vertices.

Flat shapes and solid shapes

A flat shape is a 2D shape. It lies flat like a drawing on paper. A square, circle, and triangle are flat shapes.

A solid shape is a 3D shape. It can be picked up and turned. It is not just on paper. It is like a real object.

• 2D shape: flat
• 3D shape: solid

Words to know

Face: a flat surface on a solid shape.

For example, a box has flat sides. Each flat side is a face.

Edge: the place where two faces meet.

On a box, you can feel the line where two flat sides touch. That line is an edge.

Vertex: a corner where edges meet.

The plural of vertex is vertices. That means more than one corner.

Important idea: Some 3D shapes have flat faces, edges, and vertices. Some 3D shapes have curved parts, so they may have fewer or none of these parts.

Let’s look at some common 3D shapes

Cube

A cube looks like a small box. A toy block is often a cube.

• It has 6 faces.
• It has 12 edges.
• It has 8 vertices.

All of its faces are flat. All of its corners are vertices.

Rectangular prism

A rectangular prism also looks like a box. A cereal box is a rectangular prism.

• It has 6 faces.
• It has 12 edges.
• It has 8 vertices.

Its faces are rectangles. It has flat faces, straight edges, and corners.

Sphere

A sphere looks like a ball.

• It has no flat faces.
• It has no edges.
• It has no vertices.

A sphere is all curved around. That is why it has no flat faces and no corners.

Cylinder

A cylinder looks like a can.

• It has 2 flat faces.
• It has 0 vertices.

A cylinder also has a curved side. It does not have corners, so it has no vertices.

Cone

A cone looks like an ice cream cone.

• It has 1 flat face.
• It has 1 vertex.

It has one flat bottom and one point at the top.

How to describe a 3D shape

1. Look for flat faces.
2. Look for edges where flat faces meet.
3. Look for vertices, or corners.
4. Notice if the shape has curved parts.

These clues help us tell shapes apart.

Worked Example 1

You are holding a ball. What 3D shape is it?

Step 1: Does it have any flat faces? No.

Step 2: Does it have any edges? No.

Step 3: Does it have any vertices? No.

Answer: It is a sphere.

Worked Example 2

You see a shape like a can. It has 2 flat faces and no corners. What shape is it?

Step 1: Count the flat faces. There are 2.

Step 2: Look for vertices. There are 0.

Step 3: Notice the curved side.

Answer: It is a cylinder.

Worked Example 3

A shape has 6 faces, 12 edges, and 8 vertices. It looks like a box. What could it be?

Step 1: 6 faces means it has many flat sides.

Step 2: 12 edges and 8 vertices match a box shape.

Answer: It could be a cube or a rectangular prism.

Worked Example 4

A shape has 1 flat face and 1 vertex. What shape is it?

Step 1: Find the flat face. There is 1.

Step 2: Find the corner or point. There is 1 vertex.

Answer: It is a cone.

Let’s compare shapes

• Cube: many flat faces, many edges, many vertices
• Rectangular prism: many flat faces, many edges, many vertices
• Sphere: no flat faces, no edges, no vertices
• Cylinder: 2 flat faces, no vertices
• Cone: 1 flat face, 1 vertex

Helpful tip

If a shape has corners, it has vertices.

If a shape has flat sides, those flat sides are faces.

If two flat faces touch, they make an edge.

Try thinking about real objects

• A dice is like a cube.
• A cereal box is like a rectangular prism.
• A basketball is like a sphere.
• A soup can is like a cylinder.
• An ice cream cone is like a cone.

Looking at real objects can help you remember each shape and its parts.

Summary

3D shapes are solid shapes that take up space. We can describe them by their faces, edges, and vertices.

A face is a flat surface. An edge is where two faces meet. A vertex is a corner.

When you look at a 3D shape, ask: Does it have flat faces? Does it have edges? Does it have vertices? These questions will help you name and describe the shape.

Comparing 2D and 3D Shapes

Shapes are all around us. Some shapes are flat, and some shapes are solid.

In this lesson, we will learn how to tell the difference between 2D shapes and 3D shapes. We will also see that many 3D shapes have flat parts called faces, and those faces are often 2D shapes.

What does 2D mean?

2D means two-dimensional. A 2D shape is flat. It has length and width, but it does not have thickness.

You can draw a 2D shape on paper. It does not stick out.

• A circle is a 2D shape.
• A square is a 2D shape.
• A rectangle is a 2D shape.
• A triangle is a 2D shape.

What does 3D mean?

3D means three-dimensional. A 3D shape is solid. It is not flat.

A 3D shape has length, width, and height. That means it takes up space. You can often hold it in your hand.

• A cube is a 3D shape.
• A sphere is a 3D shape.
• A cylinder is a 3D shape.
• A cone is a 3D shape.

How can we tell them apart?

Here is an easy way to think about it:

• If a shape is flat, it is 2D.
• If a shape is solid and can be held, stacked, or rolled, it is 3D.

A square drawn on paper is flat, so it is 2D.

A box is solid, so it is 3D.

2D shapes and 3D shapes can be connected

Many 3D shapes have flat sides called faces. A face is a flat surface on a solid shape.

These faces are often 2D shapes.

For example:

• A cube has faces that are squares.
• A rectangular prism has faces that are rectangles.
• A cylinder has flat circular faces.
• A cone has one flat circular face.

This means a 3D shape can have 2D shapes as part of it.

Think about real objects

It helps to look at things you know.

• A sheet of paper is flat like a rectangle. That is like a 2D shape.
• A ball is solid like a sphere. That is a 3D shape.
• A can is shaped like a cylinder. That is a 3D shape.
• An ice cream cone is shaped like a cone. That is a 3D shape.
• A dice is shaped like a cube. That is a 3D shape.

Words we can use to compare shapes

• Flat: not thick, like paper
• Solid: has thickness and takes up space
• Face: a flat side on a 3D shape
• Roll: some 3D shapes can move by turning, like a ball
• Stack: some 3D shapes can sit on top of each other, like boxes

Worked Example 1

Question: Is a triangle 2D or 3D?

Think: Is it flat, or is it solid?

A triangle is flat. You can draw it on paper.

Answer: A triangle is 2D.

Worked Example 2

Question: Is a cube 2D or 3D?

Think: Can you hold it? Does it take up space?

A cube is solid. It has length, width, and height.

Answer: A cube is 3D.

Worked Example 3

Question: What 2D shape do you see on the faces of a cube?

Think: A face is a flat side. What flat shape is on each side of a cube?

Each face of a cube is a square.

Answer: A cube has square faces.

Worked Example 4

Question: Which one is 3D: a circle or a sphere?

Think: Which one is flat, and which one is like a ball?

A circle is flat, so it is 2D. A sphere is solid, so it is 3D.

Answer: The sphere is 3D.

Try comparing these pairs

• Square and cube: a square is 2D, a cube is 3D.
• Circle and cylinder: a circle is 2D, a cylinder is 3D.
• Triangle and cone: a triangle is 2D, a cone is 3D.
• Rectangle and rectangular prism: a rectangle is 2D, a rectangular prism is 3D.

Helpful tips

1. Ask, Is it flat? If yes, it is probably 2D.
2. Ask, Can I hold it? If yes, it is probably 3D.
3. Look for faces on a 3D shape. The faces may be 2D shapes.
4. Think about real objects you know.

Let’s remember

2D shapes are flat. Examples are circle, square, rectangle, and triangle.

3D shapes are solid. Examples are cube, sphere, cylinder, and cone.

Many 3D shapes have 2D faces. For example, a cube has square faces, and a cylinder has circular faces.

When you compare 2D and 3D shapes, look for the big difference: flat or solid.

Composing 3D Shapes means putting solid shapes together to make a new solid shape or a bigger structure.

Solid shapes are shapes you can hold in your hand. They are not flat. They have length, width, and height.

When we compose 3D shapes, we stack, build, or connect them. We can use shapes like cubes, rectangular prisms, cylinders, cones, and spheres.

Let’s remember some 3D shapes:

• Cube – looks like a box where all sides are the same size
• Rectangular prism – looks like a cereal box or brick
• Cylinder – looks like a can
• Cone – looks like an ice cream cone
• Sphere – looks like a ball

Some 3D shapes are easy to stack. Some are not.

• Shapes with flat faces can often stack well.
• Shapes with curved surfaces may roll or slide.

For example, a cube has flat faces, so it stacks nicely. A sphere is round, so it rolls and does not stack well.

When we build with 3D shapes, we also think about whether the structure is stable. Stable means it stays up and does not easily fall over.

Ways to make a structure stable:

• Put bigger shapes on the bottom.
• Use flat faces to connect shapes.
• Do not put rolling shapes like spheres on the bottom.
• Stack carefully so the shapes balance.

Main idea: Flat faces help shapes connect and stack. Curved surfaces can make building harder.

Let’s look at how shapes can be put together.

If we put 2 cubes together, we get a bigger solid shape made of cubes.

If we put a cone on top of a cylinder, it can look like a rocket or a tower.

If we put a sphere on the bottom of a stack, the shapes may fall because the sphere can roll.

Worked Example 1

You have 2 cubes. You put one cube on top of the other cube. What do you make?

Answer: You make a taller solid shape made from 2 cubes.

Why does this work? Cubes have flat top and bottom faces, so they stack easily.

Worked Example 2

You have 1 cylinder and 1 cone. You put the cone on top of the cylinder. Will this make a good tower?

Answer: Yes, it can make a good tower.

Why? The top of the cylinder is flat, and the bottom of the cone is flat. Flat faces help the shapes fit together.

Worked Example 3

You have 1 sphere and 1 cube. You put the cube on top of the sphere. Is the structure stable?

Answer: No, it is not very stable.

Why? The sphere is round and can roll. The cube may slide or fall off.

Worked Example 4

You want to build a strong shape using 3 solids: 2 rectangular prisms and 1 cylinder. Which shape should go on the bottom?

Answer: A rectangular prism should go on the bottom.

Why? A rectangular prism has a flat face and makes a stronger base. The cylinder can go on top if it is balanced on a flat face.

Let’s think while building:

1. What shapes do I have?
2. Which shapes have flat faces?
3. Which shape should go on the bottom?
4. Will my structure stay up, or will it roll or fall?

Here are some good building choices:

• Cube on cube
• Rectangular prism on rectangular prism
• Cone on cylinder
• Cube next to cube to make a bigger shape

Here are some tricky building choices:

• Sphere on sphere
• Cube on sphere
• Cylinder on its curved side

You can also compose shapes by putting them side by side, not just on top.

For example, if 2 cubes are joined side by side, they make a longer solid shape. If several rectangular prisms are put together, they can make a wall, bridge, or step shape.

Sometimes we use many shapes to build something from real life.

• A tower might use cubes and cylinders.
• A rocket might use a cylinder and a cone.
• A snowman shape might use spheres, but it would not be easy to stack without help.

Remember: When we compose 3D shapes, we are making new solid shapes or structures from smaller solid shapes.

We look at the parts of each shape to help us build:

• Flat faces help with stacking.
• Curved surfaces may roll.
• A strong bottom helps keep the structure stable.

Summary

Composing 3D shapes means putting solid shapes together to build something new. Shapes with flat faces, like cubes and rectangular prisms, are easier to stack. Round shapes, like spheres, can roll, so they are harder to use in stable structures. To build well, choose a strong bottom and connect shapes carefully.