Basic geometry

The key concepts of Geometry

1. BASIC 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 l 2 or l  ×  l , A= l 2 . 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 × r 2  or Area = (pi × d 2 )/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 cm 3

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 L 3  = 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 a 2

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 (cm 3 ) Its surface area would be 6 × 2.× 2 = 24 square centimeters (cm 2 ).

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  ×  r 2 , and the surface area is 2   × pi ×   r 2 + 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  × r 3 . The surface area S of the sphere is given by the formula S  = 4 ×  pi  × r 2 .

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 × 4 3  = 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 × 4 2  = 4 × 3.14 × 4 × 4 = 201 square centimeters .