The key concepts of Geometry

2. What is Geometry?
-is the study of points,
lines, angles,
surfaces, and solids.

3. BASIC TERMS
Point(s):
A point is a location in space. We may think of a
point as a "dot" on a piece of paper. We identify this
point with a number or an upper case letter. A point
has no length or width, it just specifies an exact
location.

4. BASIC TERMS
Lines
A line is a collection of points that extend
forever. We write the name of a line passing through
two different points A and F as "line AF" or as , the
two-headed arrow over AF signifying a line passing
through points A and F.

5. BASIC TERMS
Line Segment(s):
A line segment is part of a line. The following is a
segment. A segment has two endpoints. A line
segment does not extend forever, but has two
distinct endpoints. We write the name of a line
segment with endpoints A and F as "line segment
AF" or as “FA” . Note how there are no arrow heads
on the line over AF such as when we denote a line or
a ray.

6. BASIC TERMS
Ray(s):
A ray is a collection of points that begin at one
point (an endpoint) and extend forever on one
direction.The point where the ray begins is known as
its endpoint. We write the name of a ray with
endpoint A and passing through a point F as "ray
AF". Note how the arrow head denotes the direction
the ray extends in: there is no arrow head over the
endpoint.

7. BASIC TERMS
Endpoint(s):
An endpoint is a point used to define a line
segment or ray. A line segment has two endpoints; a
ray has one.

8. BASIC TERMS
Plane:
A plane is a flat surface like a piece of paper. It
extends in all directions. We can use arrows to show
that it extends in all directions forever. The following
is a plane.

9. BASIC TERMS
Parallel Line(s):
Two lines in the same plane which never
intersect are called parallel lines. We say that two
line segments are parallel if the lines that they lie on
are parallel. If line 1 is parallel to line 2, we write this
as line 1 || line 2
When two line segments DC and AB lie on parallel
lines, we write this as segment DC || segment AB.

10. BASIC TERMS
Intersecting Lines
When lines meet in space or on a plane, we say
that they are intersecting lines The term intersect
is used when lines, rays, line segments or figures
meet, that is, they share a common point. The point
they share is called the point of intersection.

11. ANGLE AND ANGLE TERMS
What is an Angle?
Two rays that share the same endpoint
form an angle. The point where the rays
intersect is called the vertex of the angle.
The two rays are called the sides of the
angle.

12. ANGLE AND ANGLE TERMS
Here are some examples of angles.

13. ANGLE AND ANGLE TERMS
We can specify an angle by using a
point on each ray and the vertex. The angle
below may be specified as angle ABC or as
angle CBA; you may also see this written as
ABC or as CBA. Note how the vertex point is
always given in the middle.

14. ANGLE AND ANGLE TERMS
Degrees: Measuring Angles
We measure the size of an angle using
degrees

15. ANGLE AND ANGLE TERMS
Acute Angles
An acute angle is an angle measuring
between 0 and 90 degrees.

16. ANGLE AND ANGLE TERMS
Obtuse Angles
An obtuse angle is an angle measuring
between 90 and 180 degrees.

17. ANGLE AND ANGLE TERMS
Right Angles
A right angle is an angle measuring 90
degrees. Two lines or line segments that
meet at a right angle are said to be
perpendicular. Note that any two right
angles are supplementary angles (a right
angle is its own angle supplement).

18. ANGLE AND ANGLE TERMS
Right Angles

19. ANGLE AND ANGLE TERMS
Complementary Angles
Two angles are called complementary
angles if the sum of their degree
measurements equals 90 degrees. One of the
complementary angles is said to be the
complement of the other.

20. ANGLE AND ANGLE TERMS
Complementary Angles

21. ANGLE AND ANGLE TERMS
Supplementary Angles
Two angles are called supplementary
angles if the sum of their degree
measurements equals 180 degrees. One of
the supplementary angles is said to be the
supplement of the other.

22. ANGLE AND ANGLE TERMS
Supplementary Angles

23. FIGURES and POLYGONS
Examples:

24. PERIMETER
Finding the perimeter of a shape means that you are
looking for the distance around the outside of that
shape.
Example #1
Find the distance around for the following irregular
polygon
Distance around = 5 inches + 4 inches + 2 inches + 3
inches + 6 inches = 20 inches

25. AREA of Shapes
By definition, the area of shapes is the amount of
space inside those shapes.You can find the area of
any two-dimensional shape or any shape that has a
width and a length
The area can only be found for flat surfaces, so it
does not make sense to say, " What is the area of a
box ? “
To get the amount of space inside a figure, we
use a square to represent 1 unit and we say that the
area is measured in square units

26. AREA of Shapes
Area of a Square
If l is the side-length of a square, the area of
the square is l2 or l × l, A= l2.
Example:
What is the area of a square having side-
length 4cm?
The area is the square of the side-length,
which is 4 × 4 = 16cm.

27. AREA of Shapes
Area of a Rectangle
The area of a rectangle is the product of its
width and length. A= L x W
Example:
What is the area of a rectangle having a
length of 6mm and a width of 2mm?
The area is the product of these two side-
lengths, which is 6 × 2 = 12mm.

28. AREA of Shapes
Area of a Triangle
Consider a triangle with base length b and
height h.
The area of the triangle is 1/2 × b × h.
A=1/2(bh)

29. AREA of Shapes
Area & Perimeter of a Circle
Perimeter = 2 × pi × r or Perimeter = pi × d
Area = pi × r2 or Area = (pi × d2)/4

30. Understanding Volume
What is volume or capacity? In math, capacity is the
amount a container will hold when full
Capacity is generally measured in milliliters, liters,
or kiloliters. Take a look at the following container,
which is a rectangular prism.
If the length, width, and height
of this three-dimensional
container, measures 5 cm, 10
cm, and 20 cm respectively, the
volume is 5 cm × 10 cm × 20 cm
= 1000 cm3

31. Surface Area
Surface Area
The surface area of a space figure is the
total area of all the faces of the figure.

32. Surface Area
Surface Area
What is the surface area of a box whose length is 8,
width is 3, and height is 4? This box has 6 faces: two
rectangular faces are 8 by 4, two rectangular faces
are 4 by 3, and two rectangular faces are 8 by 3.
Adding the areas of all these faces, we get the
surface area of the box:
8 × 4 + 8 × 4 + 4 × 3 + 4 × 3 + 8 × 3 + 8 × 3 =
32 + 32 + 12 + 12 +24 + 24=
136.
S.A. = 2 l w + 2lh + 2wh

33. SPACE FIGURES and BASIC SOLIDS
Cube
A cube is a three-dimensional figure having
six matching square sides. If L is the length
of one of its sides, the volume of the cube is
L3 = L × L × L. A cube has six square-
shaped sides. The surface area of a cube is
six times the area of one of these sides.
S.A.= 6 x a2

34. SPACE FIGURES and BASIC SOLIDS
What is the volume and
surface are of a cube having a side-length of
2.1 cm?
Its volume would be 2× 2 × 2 = 9 cubic
centimeters (cm3)
Its surface area would be 6 × 2.× 2 = 24
square centimeters(cm2).

35. SPACE FIGURES and BASIC SOLIDS
Cylinder
A cylinder is a space figure having two
congruent circular bases that are parallel. If
L is the length of a cylinder, and r is the
radius of one of the bases of a cylinder, then
the volume of the cylinder is L × pi × r2,
and the surface area is 2 × pi × r2 + 2 × pi
× r × h.

36. SPACE FIGURES and BASIC SOLIDS
Cylinder
The figure pictured
below is a cylinder.
The grayed lines are
edges hidden from
view.

37. SPACE FIGURES and BASIC SOLIDS
Sphere
A sphere is a space figure having all of its
points the same distance from its center. The
distance from the center to the surface of the
sphere is called its radius. Any cross-section of
a sphere is a circle.
If r is the radius of a sphere, the volume V of
the sphere is given by the formula
V = 4/3 × pi ×r3.
The surface area S of the sphere is given by the
formula S = 4 × pi ×r2.

38. SPACE FIGURES and BASIC SOLIDS
Sphere

39. SPACE FIGURES and BASIC SOLIDS
Sphere
To the nearest tenth, what is the volume and
surface area of a sphere having a radius of
4cm?
Using an estimate of 3.14 for pi,
the volume would be
4/3 × 3.14 × 43 = 4/3 × 3.14 × 4 × 4 × 4 = 268
cubic centimeters.
Using an estimate of 3.14 for pi, the surface
area would be
4 × 3.14 × 42 = 4 × 3.14 × 4 × 4 = 201 square
centimeters.