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 location or spot in space. We use a dot -- not a potato! -- to represent it, then we name or label it with an upper case (Capital) letter.

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 path of points extending forever in opposite directions.
We name or label a line by indicating any two points on it.
A line segment is a part of a line precisely defined and named by two endpoints.
ST is a line segment. X is the midpoint so SX = XT.

.

A ray is a part of a line that extends indefinitely in one direction from one endpoint.
We name or label it by the
endpoint and any other point on it. QP is a ray.
Intersecting lines cross each other at a single point. Lines EF and GH intersect at J.
Perpendicular lines meet at
right angles forming four 90° angles at their intersection.
OP is perpendicular to
QR at point S. The small black square is used to indicate 90°.

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, vertices (corners) -- points where more than 2 sides meet , and edges, the lines where two sides meet.

A prism has two identical or congruent bases shaped like a polygon.
A Prism is named for the shape of the base.

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 faces except the base, meet at a single top point or vertex.

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

A triangular pyramid has 3 triangular faces and a triangular base.
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 pile of circles. A can is a cylinder. The base and top are circles, the sides or curved surface is a rectangle that has been wrapped around the circles. We measure the radius and circumference of the circular base and the height of the rectanglular "sides".

A cone, is based on a circular flat surface.
The "sides" (curved surface) come to a point called the vertex.

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. Half a sphere is called a hemisphere -- like the
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.

.

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.

.

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.