POINTS, LINES and SOLIDS |

**Introduction**

The 3-dimensional space we call reality is made up of **points**, **lines** and **solids**. **Points define lines**. **Lines** become the **sides of 2-dimensional figures** called polygons -- such as rectangles and triangles -- which in turn, become the **faces** and bases **of 3-dimensional solids** like buildings, storage tanks, monuments and pyramids of course. We name these solids by the shapes in them.

The base of the pyramid is a square, each face is a triangle.

Most of us use the word "line" incorrectly in everyday talk. Geometry, however, demands

that we use precise terms to define things, so let's learn to call lines,

points and solids by their correct names.

One term we'll discuss that we generally use correctly is a **ray**.

Once we've heard the expression "a ray of sunshine" we know instinctively what it means.

Each ray of sunshine originates at the sun (a point in space)

and continues on forever in one direction.

**Points and Lines**

A ** point** is a precise

Here, we see 3 points labeled **A**, **B** and **C**.

**
**

When we join **A** to **B**, **B** to **C** and **C** to **A**, we have

3 **line segments** that define a **triangle** **ABC**.

In the diagram below:

**CD** or **DC** is a** line** -- a

We name or

A

**.**

A** ray** is a part of a line that

We name or

**Note:** **QP** is a ** ray** which **has direction** so we must **name it in the correct order**.

We **can't call it** ray **PQ**. In more advanced math courses, we'll call this ray a vector.

.

**3-Dimensional Solids**

**There are two types** of 3-dimensional solids: those **based on polygons**, and those **based on circles**. The first category includes **prisms** of all shapes. The **circular solids** include **cylinders**, **cones**, and **spheres**.

**Polygonal solids **are composed of** **polygon-shaped ** faces**,

A** prism **has two identical or congruent bases shaped like a polygon.

A Prism is

A ** cube is like a pile of squares. **It has 6 square faces – top, bottom, and four sides.

All edges of a cube are equal in length. All sides meet at 90°.

We usually label the length " *l* " -- the width " *w* " and the height " *h* ".

Prisms can be based on any polygon, such as a hexagon, or octagon. In a way, we should call the American military headquarters in Washington DC a pentagonal prism, and not a pentagon – for its certainly exists in more than two dimensions.

In a** pyramid **all

When the **base is a square**, we call it a **square pyramid**.

It has **4 triangular faces** and a **square base**.

The length of an **edge joining the vertex to a corner of the base** is called the **slant heigh****t**.

The **perpendicular distance** between the **vertex and the base **is called the **height** or **altitude**.

A ** triangular pyramid **has

The base of a pyramid can be a pentagon, hexagon, an octagon – or any polygon.

**Circular Solids**

These solids are based on circles. Cans, the columns on our buildings, storage

tanks for liquids are all cylindrical in shape.

A ** cylinder**, is a

A ** cone, **is based on a circular flat surface.

The "sides" (curved surface) come to a

The **line joining** the **vertex** to the **circumference** of the circular **base** is called **slant height** (like in a pyramid), and the **vertical distance** between the **vertex and base** is called the **height**.

**SPHERES**

A ** sphere or orb** is the correct term for a ball or globe.

It has no flat surfaces.

Northern Hemisphere -- the upper half of the sphere on which we all live.

The **curved surface** of a sphere is made up of **4 circles**. The lacing

on a softball almost outlines the **4 circular areas** that make up the **surface**.

From this image, we can see how the sphere is made of 4 identical cones with bases on the

4 circular surfaces, with vertices meeting at the center of the sphere.

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Now get a pencil, an eraser and a note book, copy the questions,

do the practice exercise(s), then check your work with the solutions.

If you get stuck, review the examples in the lesson, then try again.

**Practice Exercise**

1) Match the words in the left column with the descriptions on the right.

a ray | a) a pile of circles. |

square pyramid | b) the line where 2 faces meet in a rectangular prism. |

rectangular prism | c) a point or corner where more than 2 edges meet. |

frustrum | d) a prism with 6 congruent rectangular faces. |

cylinder | e) a pointy solid with 4 triangular faces. |

cube | f) a 6 faced solid with rectangular faces. |

cone | g) a part of a line from one endpoint extending forever in one direction . |

face | h) a part of a line precisely defined by 2 endpoints. |

edge | i) a particular spot in space. |

vertex | j) a polygon-shaped side of a 3-dimensional solid. |

point | k) a cone with its head knocked off. |

line segment | l) a circular solid that comes to a point. |

2) Identify all the 3-dimensional solids in these images.

... ..

3) Think of a name for the base of the Washington Monument?

It's not a rectangular prism because the faces are not rectangles.

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**Solutions**

1) Match the words in the left column with the descriptions on the right.

a ray | g) a part of a line from one endpoint extending forever in one direction . |

square pyramid | e) a pointy solid with 4 triangular faces. |

rectangular prism | f) a 6 faced solid with rectangular faces. |

frustrum | k) a cone with its head knocked off. |

cylinder | a) a pile of circles. |

cube | d) a prism with 6 congruent rectangular faces. |

cone | l) a circular solid that get to the point. |

face | j) a polygon-shaped side of a 3-dimensional solid. |

edge | b) the line where 2 faces meet in a rectangular prism. |

vertex | c) a point or corner where more than 2 edges meet. |

point | i) a particular spot in space. |

line segment | h) a part of a line precisely defined by 2 endpoints. |

2) the **dome is a hemisphere**, the **columns are cylinders**, there's a **frustrum** under the dome, the **flag pole is a cylinder**, the **tornado is a cone**, the top of the Washington Monument is a **square** **pyramid**.

3) The faces of the base are trapezoids (trapeziums) -- so it is a **trapezoidal prism.**