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Regents Review #5
Area and Perimeter:
Area: The unique real number assigned to any polygon that indicates the number of nonoverlapping square units contained in the polygon’s interior.
Perimeter: The sum of the length of the sides of a polygon.
Area Formulas:
Parallelogram:
A = bh
Rectangle: A = bh
h
h
b
Square:
b
A = s2
A = bh
A=
1
d1 d 2
2
d2
s
h
d1
b
Rhombus:
A = bh
A=
h
1
d1 d 2
2
d2
b
Triangle:
A=
1
bh
2
Trapezoid: A =
b
A=
A = height x median
median
h
h
b
1
hb1  b2 
2
b1
h
Circle:
d1
h
b2
r2
C = 2r
r
d
C=
d
3-Dimensional Figures
A polyhedron is a three-dimensional figure whose surfaces are polygons.
 The faces of a polyhedron are the polygons that make up the figure

The edges are the intersections of the faces.

The vertices (points) are the intersections of the edges.
Prisms:
A prism is a polyhedron in which two of the faces, called the bases of the prism, are
congruent polygons in parallel planes.
All of the lateral sides of a right prism are rectangles.
A parallelepiped is a prism that has parallelograms as bases.
A rectangular parallelepiped is a parallelepiped that has rectangular bases and lateral
edges perpendicular to the bases. (rectangular solid)
base
altitude
Lateral
face
Lateral
edge
base
Pyramids:
A pyramid is a solid figure with a base that is a polygon and
lateral faces that are triangles.
The length of the altitude of a triangular lateral face of a
regular pyramid is the slant height of the pyramid.
Cylinders:
A cylinder is a solid formed by congruent parallel
curved bases and the surface that joins them.
Lateral surface
altitude
base
Cones:
A cone is like a pyramid, but it is circular.
slant height
altitude
radius
base
Spheres:
A sphere is the set of points equidistant from a
fixed point called the center.
The radius of a sphere is the length of the line
segment from the center of the sphere to any
point on the sphere.
A plane in can have no points in common, one
point in common, or infinitely many points in
common.
A great circle of a sphere is the intersection of a sphere and a
plane through the center of the sphere.
Lateral Area, Surface Area, and Volume:
The lateral area is the sum of the areas of the lateral faces.
The total surface area is the sum of the lateral area and the area(s) of the base(s). (the
sum of the areas of all the faces)
Volume is the space that a figure occupies.
PYRAMID:
1
LA = pl
2
SA = LA + B
1
V = Bh
3
PRISM:
LA = ph
SA = LA + 2B
V = Bh
CONE:
LA = πrl
SA = LA + B
1
V = Bh
3
CYLINDER:
LA = 2πrh
SA = LA + 2B
V = Bh
SPHERE:
SA = 4πr2
4
V = πr3
3
Points, Lines, & Planes
Parallel lines are coplanar lines that have no points in common, or have all points in common
and, therefore, coincide.
A transversal is a line that intersects two other coplanar lines in two different points.
Two lines are parallel if and only if
- alternate interior angles are congruent.
-
corresponding angles are congruent.
-
interior angles on the same side of the transversal are supplementary.
Alternate Interior Angles:
1
3 4
2
,
Corresponding Angles:
,
5 6
7 8
,
,
Interior Angles on the Same Side of the Transversal:
,
Vertical angles – two angles in which the sides of one angle are opposite rays to the sides
of the second angle. (In the diagram above:
,
,
,
)
Complementary angles – two angles, the sum of whose degree measure is 90
Supplementary angles – two angles, the sum of whose degree measure is 180
Perpendicular lines – two lines that intersect to form right angles.
Bisects – divides into two congruent parts.
- Line segment bisector – any line that intersects the segment at its midpoint.
-
Angle bisector – a ray whose endpoint is the vertex of the angle and that divides
that angle into two congruent angles.
Playfair’s Postulate: - in a plane, through a given point not on a given line, there exists one
and only one line parallel to the given line.
If, in a plane, a line intersects one of two parallel lines, it intersects the other.
In a plane, if a transversal is perpendicular to one of two parallel lines, it is perpendicular
to the other.
If two of three lines in the same plane are each parallel to the third line, then they are
parallel to each other.
Rules for Points, Lines, and Planes:
*Use your paper as a plane and your pencil as a line*
 A set of three non-collinear points determine a plane.
 A plane containing any two points contains all of the points on the line determine by
those two points.
 There is exactly one plane containing a line and a point not on the line.
 If two lines intersect, then there is exactly one plane containing
E
them.
Skew Lines – lines in space that are neither parallel nor intersecting
Ex) Skew lines – HB and GE, HB and AF, DC and GE, DC and AF
Parallel lines – FC and AB, AF and BC, EF and HB, DC and GA





D
G
H
F
C
A
B
If a line not in a plane intersects the plane, then it intersects in
exactly one point.
A line is perpendicular to a plane if and only if is perpendicular to each line in the
plane through the intersection of the line
If a line is perpendicular to each of two intersecting lines at their point of
intersection, then the line is perpendicular to the plane determined by these lines.
Through a given point on a plane, there is only one line perpendicular to the given
plane.
If two planes intersect, they intersect in exactly one line.
Dihedral angle - the union of two half-planes with a common edge. (The angle formed by
the intersecting planes)
Perpendicular Planes - two planes that intersect to form a right dihedral angle
 If a plane contains a line perpendicular to another plane, then the planes are
perpendicular
Parallel Planes - planes that have no points in common.
 If a plane intersects two parallel planes, then the intersection is two parallel lines.
 If two planes are perpendicular to the same line, then they are parallel.
Part I:
______1) Which of the following is an expression for the volume
of the prism in the figure?
(1) 4t + 2
(2) 3t3 + 6t2
(3) 3t3 + 3t + 2
(4) 14t2 + 16t
______2) Which of the following statements is false?
(1) A line parallel to the intersection of two planes is parallel to each of the
planes.
(2) Two planes parallel to the same line are parallel to each other.
(3) The shortest segment from a point to a plane is the perpendicular from
that point.
(4) Parallel segments drawn to a plane from points in a line parallel to the
plane are congruent.
______3) If two parallel planes are intersected by a third plane, how many lines of
intersection are there?
(1) one
(2) three
(3) two
(4) infinitely many
______4) The length, width, and height of a rectangular prism are each tripled. What will
happen to the volume of the prism?
(1) The volume of the prism will be three times the original volume.
(2) The volume of the prism will be nine times the original volume.
(3) The volume of the prism will be eighteen times the original volume.
(4) The volume of the prism will be twenty-seven times the original volume.
______5) If point P lies outside of plane n, how many planes parallel to plane n can be
drawn through point P ?
(1) none
(2) one
(3) two
(4) infinitely many
_____6) What is the volume of this rectangular oblique prism?
(1) 64 in3
(2) 90 in3
(3) 72 in3
(4) 124 in3
______7) Through a point outside a given plane, how many planes can be drawn
perpendicular to the given plane?
(1) none
(2) one
(3) two
(4) infinitely many
______8) The base of a regular pyramid is an octagon with an area of 45 square inches. If
the volume of the pyramid is 75 cubic inches, what is the height of the pyramid?
(1) 1.67 inches
(2) 10 inches
(3) 5 inches
(4) 30 inches
______9) Point P is on line k and line k is in plane q. How many lines in plane q are
perpendicular to k at P ?
(1) none
(2) one
(3) two
(4) infinitely many
______10) The surface area of a sphere is 324π cubic meters. What is the volume of the
sphere in terms of π?
(1) 9π m3
(2) 972π m3
(3) 524π m3
(4) 1,082π m3
______11) How many planes can pass through a given point?
(1) none
(2) two
(3) one
(4) infinitely many
______12) A regular square pyramid has an altitude of 15 centimeters, and a volume of
8,000 cubic centimeters. What is the length of one side of the square base, to the nearest
centimeter?
(1) 23 centimeters
(2) 40 centimeters
(3) 36 centimeters
(4) 45 centimeters
______13) The wheel of a steamroller is in the shape of a cylinder. The length of the
wheel is 6 feet, and the diameter of the wheel is 4 feet. To the nearest hundredth, how
many square feet will one revolution of the wheel cover?
(1) 24 ft2
(2) 75.40 ft2
(3) 48.56 ft2
(4) 150.72 ft2
______14) How many lines are parallel to a line through a point not on the line?
(1) none
(2) two
(3) one
(4) infinitely many
______15) Which of the following is not a true statement about a sphere?
(1) The radius of a great circle is also the radius of the sphere.
(2) The intersection of a plane and a sphere may be a circle.
(3) The intersection of a plane and sphere may be a point.
(4) Two planes that intersect a sphere equidistant from its center always
create parallel circles.
______16) If a line is perpendicular to a plane, how many planes containing this line are
perpendicular to the plane?
(1) none
(2) two
(3) one
(4) infinitely many
______17) If point P lies outside plane m, how many lines parallel to plane m can be drawn
through point P ?
(1) none
(2) two
(3) one
(4) infinitely many
E
D
______18) In the following figure, a line skew to line AG is:
G
(1) BH
(2) CF
H
(3) AB
(4) DC
F
C
A
B
______19) The diameter of a cylinder is doubled, and the height is cut in half. What will
happen to the volume of the cylinder?
(1) The volume will stay the same.
(2) The volume will be half the original volume.
(3) The volume will be twice the original volume.
(4) The volume will be four times the original volume.
______20) What is the value of x?
(1) 23
(2) 113
(3) 67
(4) The value of x cannot be determined
by the given information.
______21) Which statement is not true concerning the following diagram?
(1) Point B is the midpoint of AD
(2) ABE and ABC are vertical angles
(3) CBD and DBE form a linear pair
(4) By the angle addition postulate,
mABC + mCBD = mABD.
Part II:
22) The lateral area of a right circular cylinder is 60π cm2. If the height of the cylinder is
10 cm, find the radius of the cylinder.
23) Find the lateral area of a regular pyramid with a regular hexagonal base of sides each
68 cm and slant height 489 cm.
24) In the diagram below, m1 = (2x + 12)° and m2 = (3x + 18)°. If AB || CD, find the
degree measure of 1.
25) The radius of the base of a right circular cylinder is 6 ft and its altitude is 14 ft. Find
the total surface area in terms of π.
26) Express in terms of π the volume of a sphere whose radius is 6.
27) The base of a right prism is a rhombus whose diagonals are 12 cm and 16 cm. The
altitude of the prism is 20cm. Find its lateral area.
28) If the volume of a pyramid is 546 cm3 and the altitude is 97.7 cm, find the area of the
base to the nearest tenth of a square centimeter.
29) In the diagram below, mEGB = (6x – 10)° and mGHD = (4x + 20)°. If AB || CD find
the mCHG.
30) Find the radius of a sphere whose volume is 36π.
31) If the altitude of a triangular prism is 81 in, and the right triangular base has legs 10 in
and 24 in, find the volume.
32) If m || n and t is a transversal, find the degree measure of 1 if 1 = (5x + 10)° and
2 = (x + 110)°
33) Find the volume of a right circular cone if the measure of the radius is 4 and its height
is 9. Leave your answer in terms of π.
34) Find the radius of a sphere whose surface area is 100π.
35) Find the lateral area of a right circular cone whose radius is 4 inches and whose slant
height is 5 inches. Round to the nearest tenth of a square inch.
36) A cross section of a railroad embankment is in the shape of an isosceles trapezoid
whose bases are 20 feet and 36 feet. The congruent legs are 17 feet each. If the railroad
embankment is 300 feet long, how many cubic feet of earth have been used?