PLANE & SOLID GEOMETRY DEFINITION OF : Axiom – a statement of truth which is itted without proof. Theorem – a statement of truth which must be established by proof. Corollary – a statement of truth which follows with little or no proof from a theorem. Postulate – the construction or drawing of lines and figures the possibility of which is itted without proof. Hypothesis – that part of a theorem which is assumed to be true. Conclusion – that part of a theorem which is to be proved. Converse of a theorem – another theorem wherein the hypothesis and conclusion of the first are reversed, i.e. the hypothesis becomes the conclusion and the conclusion becomes the hypothesis.

PLANE GEOMETRY TRIANGLES: B a

c h

C

b

A

Area of triangle:

1 ab sin C 2 1 A  bc sin A 2 1 A  ac sin B 2 A

A

1 bh 2

or

Area of triangle with three sides given: (Heron’s Formula)

A  s s  a s  b s  c  where s – semiperimeter abc s 2 Area of triangle inscribed in a circle:

A

a r b

c

abc where r – radius of the circumscribing circle 4r

The center of the circle circumscribing a triangle is obtained by constructing the perpendicular bisectors of the sides of the triangle. These will intersect at a point which is known as the circumcenter.

If one side of a triangle inscribed in a circle is the diameter of the circle, then the triangle is a right triangle.

Area of triangle circumscribing a circle:

A  rs

a

where s – semiperimeter of the triangle

c

r

abc 2

s

The center of the inscribed circle is obtained by constructing the angular bisectors of the triangle. These will intersect at a point which is known as the incenter.

b

The median of a triangle is a line ing a vertex to the midpoint of the opposite side. The medians of a triangle intersect at a point known as the centroid of the triangle. Orthocenter – the point of intersection of the altitudes of a triangle.

QUADRILATERALS A quadrilateral (also known as quadrangle or tetragon) is a polygon with four sides. The sum of the interior angles of a quadrilateral is always equal to 360o.

Parallelogram (or Rhomboid): A parallelogram is a quadrilateral where both pairs of opposite sides are parallel.

d1

θ θ

A = bh

d2

h

A

1 d1d 2 sin  2

b The diagonals of a parallelogram bisect each other.

Rhombus: A rhombus is a parallelogram all of whose sides are equal.

d1

d2

A = bh

h

A

1 d1d 2 2

b The diagonals of a rhombus not only bisect each other but are also perpendicular to each other.

Trapezoid: A trapezoid is a quadrilateral which has only one pair of opposite sides that are parallel. The parallel sides of a trapezoid are called bases. a

A Median

1 a  b h 2

h Median =

ab 2

b POLYGONS A polygon is a closed plane figure with three or more angles. There are as many sides as angles in a polygon. Polygons are named according to the number of sides or vertices. Number of Sides 3 4 5 6 7 8 9 10 11 12 15 20 1000

Name Triangle Quadrilateral or Tetragon Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon Pentedecagon or Quindecagon Icosagon Chillagon

There are two basic types of polygons: 1. Convex polygon – one in which no side, when extended, will inside the polygon. 2. Concave polygon – a polygon having one interior angle greater than 180o. Sum of interior angles of a polygon of n sides:

S  180n  2  Sum of exterior angles of a polygon = 360

Interior angle

The diagonal of a polygon is a line that connects any two non-adjacent vertices. Number of diagonals of a polygon of n sides: D 

Exterior angle

n n  3 2

Similar polygons are polygons whose corresponding angles are equal and whose corresponding sides are proportional.

Regular Polygons: Polygons whose sides are equal are called equilateral polygons. Polygons whose interior angles are equal are called equiangular polygons. Polygons that are both equilateral and equiangular are called regular polygons. The center of a regular polygon is the common center of its inscribed and circumscribed circles. The apothem of a regular polygon is the perpendicular line drawn from the center of the inscribed circle to any one of its sides. It is also the radius of the inscribed circle. The area of a regular polygon can be found by considering one segment, which has the form of an isosceles triangle i.e.

A  nAsegment 



In general, for a polygon of n sides,

θ

A

nb 2 4 tan 

where b – length of a side  - central angle subtended by a side

b apothem



 360  2 2n

Circle: A circle is the locus of points which are at the same distance from a point within called the center. The diameter is a line ing through the center, terminating at both ends on the circle. The radius is a line drawn from the center to any point on the circle. d r

d2 A r  4 2

C  2 r   d

The chord of a circle is a line ing any two points on the circle. The product of the segments of two chords intersecting each other are equal, i.e.

a

ab  cd

d b

c

Inscribed angle is an angle whose vertex is a point on the circle, the sides of which are chords. Central angle is an angle whose vertex is at the center of the circle the sides of which are any two radii of the circle. For the same intercepted arc (arc AB), the value of the central angle is twice that of the inscribed angle.

θ

θ – inscribed angle 2θ – central angle

2θ

B

A Circular Sector: A circular sector is a part of a circle bounded by two radii and an arc.

s= rθ where θ is central angle in radians s – arc length r – radius

s r

A θ

1 1 rs  r 2 2 2

r

Circular Segment: A circular segment is that part of a circle bounded by an arc and a chord.

s

r

A  Asec tor  Atriangle  

θ r

1 2 1 r   r 2 sin  2 2

1 2 r   sin   where  - central angle in radians 2

SOLID GEOMETRY Definition of : Polyhedron – a solid bounded by planes. Regular polyhedron –a polyhedron whose faces are congruent regular polygons and whose polyhedral angles are equal. There are only five (5) regular polyhedrons: 1. Tetrahedron – one having four (4) triangular faces. 2. Hexahedron (or cube) – one having six (6) square faces. 3. Octahedron – one having eight (8) triangular faces. 4. Dodecahedron – one having twelve (12) pentagonal faces. 5. Icosahedron – one having twenty (20) triangular faces. Solids for which V = Bh: Cube: A cube is a polyhedron whose six faces are all squares.

a

Volume, V  Bh  a 2 a  a 3

a

Surface area, S  6a 2

d

Diagonal, D  a 3

a

Rectangular Parallelepiped: A rectangular parallelepiped is a polyhedron whose six faces are all rectangles. Volume, V  Bh  (ab)c  abc Surface area, S  2ab  2bc  2ca

d c

 2(ab  bc  ca)

b

Diagonal, D 

a2  b2  c2

a Prism: A prism is a polyhedron whose bases are equal polygons in parallel planes and whose sides are parallelograms. Volume, V  Bh

h

Lateral area, A  Ph where P - perimeter of base

B

Right Circular Cylinder: A cylinder is a solid bounded by a closed cylindrical surface and two parallel planes.

r

Volume, V  Bh   r 2 h  

h

d2 h 4

Lateral Area, A  2 rh   dh

B

Solids for which V =

1 Bh (Pointed Solids): 3

Right Circular Cone: A cone is the surface generated by a straight line (called the generator) ing through a fixed point (called the vertex) and moving along a fixed curve (called the directrix).

Volume, V  L

h

1 1 Bh   r 2 h 3 3

Lateral area, A   rL where L – slant height

L  r 2  h2 r

Pyramid: A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a common point called the vertex.

h

B

Volume,

1 V  Bh 3

Frustum of a Cone & Pyramid: The frustum of a pyramid/cone is a section of the pyramid/cone between the base and a section parallel to the base. A1

A1

h

h

A2

A2

V 



h A1  A2  A1 A2 3



Sphere A sphere is a solid bounded by a surface all points of which are equidistant from a point called the center. Great circle – the intersection of a sphere and a plane ing through the center. Small circle – the intersection of a sphere and a plane not ing through the center.

Volume, V 

r

4 3 r 3

Surface Area, S  4 r 2

Spherical Segment: A zone is that portion of the surface of a sphere included between two parallel planes. A spherical segment is a solid bounded by a zone and the plane of the zone’s base.

h

r

r

h

r

r

Azone  2 rh

V 

 h2 3r  h 3

Review Exercises in PLANE & SOLID GEOMETRY 1. An oblique equilateral parallelogram. a. square b. rectangle

c. rhombus

d. recession

2. A rectangle with equal sides. a. rectangle b. polygon

c. square

d. trapezoid

3. A plane closed curve, all points of which are at the same distance from a point within, called the center, is a a. hyperbola b. circle c. ellipse d. parabola 4. The area of a circle circumscribed about an equilateral triangle is 254.47 m2. What is the area of the triangle in m2? a. 105.24 b. 121.06 c. 83.15 d. 96.12 5. A statement the truth of which follows with little or no proof from a theorem. a. axiom b. hypothesis c. conclusion d. corollary 6. A plane figure with closed broken lines. a. sector b. arc

c. polygon

d. circle

7. A polygon having fifteen sides. a. undecagon b. dodecagon

c. pentedecagon

d. icosagon

8. In triangle ABC, angle C = 70o, A = 45o, AB = 40 m. What is the length of the median drawn from vertex A to side BC? a. 36.8 m b. 37.1 m c. 36.3 m d. 37.4 m 9. The study of the properties of figures of three dimensions. a. physics b. plane geometry c. solid geometry

d. trigonometry

10. The sum of the sides of a polygon. a. perimeter b. hexagon

d. circumference

c. square

11. A hexahedron is a solid having _____ faces. a. 4 b. 6 c. 8

d. 20

12. The sides of a triangle are 8 cm, 10 cm and 14 cm. Determine the radius of the inscribed circle. a. 2.45 cm b. 3.02 cm c. 2.72 cm d. 3.26 cm 13. In plane geometry, the apothem of a polygon is what of the inscribed circle? a. area b. diameter c. arc d. radius 14. Each interior angle of a regular dodecagon is equal to a. 135o b. 150o c. 125o d. 105o

15. A triangle inscribed in a circle of radius 10 cm have two of its sides equal to 12 cm and 18 cm respectively. It is required to find the altitude upon the third side of the triangle. a. 10.6 cm b. 20.2 cm c. 10.8 cm d. 20 cm 16. A five-pointed star is known as a. pentagon b. pentatron

c. pentagram

d. quintagram

17. The median of a triangle is the line connecting the vertex to the midpoint of the opposite side. For a given triangle, these medians intersect at a point which is called the a. orthocenter b. circumcenter c. centroid d. incenter 18. The diagonals of a parallelogram are 18 cm and 30 cm respectively. One side of the parallelogram is 12 cm. Find the area of the parallelogram in cm2. a. 216 b. 224 c. 232 d. 208 19. The altitudes of a triangle intersect at a point known as a. orthocenter b. circumcenter c. centroid

d. incenter

20. The central angle subtended by the arc of a circle equal in length to the radius of the circle is a. 1 quarter circle b. 1 radian c. π radians d. 60o 21. The area of a rhombus is 168 m2. If one of its diagonals is 12 m, find the length of the sides of a rhombus in m. a. 13.89 b. 14.32 c. 16.84 d. 15.23 22. The area bounded by two concentric circles is called a. ring b. disk c. annulus

d. sector

23. It is a polyhedron where two faces are equal polygons in parallel planes and the other faces are parallelograms. a. tetrahedron b. prism c. frustum d. prismatoid 24. Find the area of a trapezoid having a median of length 32 m and an altitude of 6 m in m2. a. 192 b. 186 c. 178 d. 169 25. In geometry, prisms are named according to their a. diagonals b. sides c. areas

d. bases

26. Polygons are classified according to the number of a. vertices b. sides c. diagonals

d. angles

27. A polygon has one of its sides equal to 20 cm and having an apothem of 14 cm. Compute the area of the circle circumscribing the polygon in cm2. a. 300π b. 248π c. 296π d. 286π

28. All circles having the same center but with unequal radii are called a. encircles b. tangent circles c. concyclic circles d. concentric circles 29. A regular polygon with 54 diagonals is a a. bidecagon b. pentedecagon

c. icosahedron

d. dodecagon

30. What is the ratio of the area of a hexagon to that of a pentagon inscribed in a circle of radius “r”? a. 1.09 b. 1.82 c. 1.32 d. 1.42 31. A line segment ing two points on a circle is called a. arc b. tangent c. sector

d. chord

32. In a regular polygon, the perpendicular line drawn from the center of the inscribed circle to any one of the sides is called a. apothem b. median c. radius d. altitude 33. How many sides has an equiangular polygon if each exterior angle is 30o? a. 15 b. 10 c. 12 d. 14 34. Two geometric figures are said to be _______ when they have the same size and shape. a. identical b. symmetrical c. similar d. congruent 35. A line drawn from one angle of a triangle dividing its opposite side equally into two. a. bisector b. apothem c. median d. neutral axis 36. Find the length of the side of a regular pentagon inscribed in a circle of radius 10 cm. a. 11. 76 cm b. 9.47 cm c. 12.22 cm d. 10.89 cm 37. A polygon is ______ if no side, when extended, will through the interior of the polygon. a. convex b. equilateral c. isoperimetric d. congruent 38. The section of the sphere cut by a plane through its center is called a. small circle b. incircle c. big circle d. great circle 39. The perimeter of a circular sector is 9 units and its radius is 3 units. What is the area of the sector? a. 4 sq. units b. 9/2 sq. units c. 11/2 sq. units d. 27/2 sq. units 40. A part of a line included between two points on the line is called a. tangent b. secant c. sector

d. segment

41. Points which lie on the same plane are said to be a. collinear b. coplanar c. concurrent

d. congruent

42. The area of a circle is 89.42 in2. What is the circumference in inches?

a. 35.33 in.

b. 32.25 in.

c. 33.52 in.

d. 35.55 in.

43. A circle having an area of 452 m2 is cut into two segments by a chord which is 6 m from the center of the circle. Find the area of the bigger circular segment in m2. a. 363.68 b. 378.42 c. 383.64 d. 354.89 44. A quadrilateral whose opposite sides are equal in length is generally a a. square b. rectangle c. rhombus d. parallelogram 45. A triangle ABC is inscribed in a circle of radius R having its center at O. If the angle OBA is 48o, find the angle ACB. a. 60o b. 42o c. 75o d. 55o 46. A circular piece of cardboard with a diameter of one meter will be made into a conical hat 40 cm. high by cutting a sector off and ing the edges to form a cone. Determine the central angle of the sector removed. a. 160o b. 144o c. 125o d. 177o 47. A quadrilateral of which only two sides are parallel is a a. parallelogram b. trapezoid c. rhombus

d. rectangle

48. A conical vessel has a height of 24 cm. and a base diameter of 12 cm. It holds water to a depth of 18 cm. above its vertex. Find the volume of its content in cm3. a. 381.70 b. 298.40 c. 412.60 d. 188.40 49. A circular cylinder is circumscribed about a right prism having a square base one meter on an edge. The volume of the cylinder is 6.283 m3. Find its altitude in meters. a. 4 b. 4.5 c. 5 d. 5.5 50. A regular triangular pyramid has an altitude of 9 m. and a volume of 187.06 m3. If the base is an equilateral triangle, find the base edge in meters. a. 18 b. 14 c. 12 d. 16 51. If the radius of a right circular cylinder is tripled, by what percent is the volume of the cylinder increased maintaining the same height? a. 200% b. 300% c. 400% d. 800% 52. The base of a right prism is an isosceles triangle having two sides equal to 80 cm. each and the third side equal to 60 cm. If the height of the right prism is 200 cm, compute the volume of the circumscribing cylinder in cm3. a. 1,168,977 b. 1,619,877 c. 1,698,717 d. 1,169,877 53. A right prism having a hexagonal base with 8 cm. for each side has a lateral edge of 12 cm. Find the lateral area in cm2. a. 756 b. 675 c. 765 d. 576 54. What is the volume of a right circular cylinder inscribed in a cube of edge 10 cm.? a. 200π cm3 b. 230π cm3 c. 250π cm3 d. 300π cm3 55. A plane is ed parallel to the base and 8 cm. from the vertex of a pyramid whose

base is 45 cm2 and whose altitude is 12 cm. What is the volume of that part of the pyramid after the top had been removed? a. 139.63 cm3 b. 120.27 cm3 c. 126.67 cm3 d. 200.16 cm3 56. A water tank, open at the top, consists of a right circular cylinder at the top and a right circular cone at the bottom. If the altitude of the cylinder is three times its radius and the altitude of the cone is two times the same radius, find the number of square meters of sheet metal required to construct a tank having a capacity of 38,000 liters. a. 57.36 b. 49.28 c. 64.52 d. 51.29 57. The area of a zone of a spherical segment is 180π m2. If the radius of the sphere is 15 m., compute the volume of the spherical segment in m3. a. 864π b. 846π c. 648π d. 468π 58. The diameter of two spheres are in the ratio of 2:3 and the sum of their volumes is 1260 m3. Find the volume of the larger sphere in m3. a. 827 b. 972 c. 729 d. 592 59. The section of the sphere cut by a plane through its center is called a. small circle b. incircle c. big circle d. great circle 60. Lines that through a common point are said to be a. collinear b. coplanar c. concurrent

d. congruent

61. Points which lie on the same plane are said to be a. collinear b. coplanar c. concurrent

d. congruent

62. In two intersecting lines, the angles opposite to each other are known as a. opposite angles b. vertical angles c. horizontal angles d. inscribed angles 63. A normal to a given plane is a. perpendicular to the plane c. parallel to the plane

b. lying on the plane d. oblique to the plane

64. When two planes intersect with each other, the amount of divergence between the two planes is expressed by measuring the a. dihedral angle b. plane angle c. polyhedral angle d. reflex angle 65. In Plane Geometry, two circular arcs that together make up a full circle are called a. coterminal arcs b. conjugate arcs c. half arcs d. congruent arcs 66. Polygons with all interior angles less than 180o are called a. concave polygon b. convex polygon c. acute polygon d. supplemental polygon 67. One-fourth of a great circle is termed a. cone b. quadrant

c. circle

d. sphere

ANSWER KEY: : 1. c 2. c 3. b 4. a 5. d 6. c 7. c 8. c 9. c 10.a 51. d 52. d 53. d 54. c 55. c

11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

b a d b c c c a a b

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

d c b a d b c d d a

31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

d a c d c a a d b d

56. 57. 58. 59. 60.

a d b d c

61. 62. 63. 64. 65.

b b a a b

66. b 67. b

41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

b c a d b b b a a c