Solid Mensuration Reviewer

MODULE 1 Broken line is a set of line segments
connected end to end.
1.1 Points, Lines and Angles
A point in geometry is a location, which has no

size, no length, no depth and shown by a dot.
Midpoint is a point halfway between two

endpoints of a line segment.
A line is a one-dimensional figure having no
thickness and extending infinitely in both
directions.
A ray is a straight line which is limited from

one end and infinite from another end. A plane is a flat surface that has length and
width but no thickness.
A segment or line segment is a part of a

straight line which is limited from both ends.
An angle is a space formed by two rays called

Parallel lines are straight lines which lie on sides sharing a common endpoint called
the same plane and do not intersect each vertex.
other.
Intersecting lines are two or more different Most common angle measurement:
lines that meet at the same point.
Degree (◦) is defined as the unit of angle
measurement wherein one complete
revolution is divided into 360 parts.
Radian (rad) is defined as the unit of angle

measurement wherein one complete
Transversal line is a line that cuts across two revolution is equal to 2π.
or more lines.
Points on the same line are said to be

collinear.
Forms of Angles:
Acute angle is an angle whose measure is Supplementary angles are two angles whose
less than 90°. sum is 180°.
Right angle is an angle that measures 90°. Interior and exterior angles are the angles
formed by cutting two parallel lines with a
transversal line.
Obtuse angle is an angle whose measure is

more than 90° and less than 180°.
Alternate interior angles are a pair of
nonadjacent interior angles on opposite
sides of the transversal line. Alternate interior
angles are congruent.
Corresponding angles have the same

position with respect to their lines and the
Straight angle is an angle that measures 180°. transversal line. Corresponding angles are
congruent.
Adjacent angles are two angles which have

the same vertex and a common side
between them. The sum of two adjacent
Reflex angle is an angle whose measure is angles is 180°.
more than 180° but less than 360°.
Vertical angles are two nonadjacent angles
formed by two intersecting lines. Vertical
angles are congruent.
Dihedral angle is the amount of divergence

of two intersecting planes.
Circular angle is an angle that measures 360°.
Terms: Two angles having the same number of

angle measurements are said to be
Complementary angles are two angles
Congruent angles.
whose sum is 90°.
MODULE 2
Polygon is a two-dimensional closed figure

bounded by straight line segments.
Parts of Polygon:
Side or edge - is one of the line segments that

make up the polygon.
Vertex - is a point where the sides meet.
Diagonal - is a line connecting two

nonadjacent vertices.
Polygons are named or classified according to
their number of sides. A polygon with n sides is
Interior angle - is the angle formed by two called an n-gon.
adjacent sides inside the polygon.
n (sides) Polygon
Exterior angle – is the angle formed by two
adjacent sides outside the polygon. 3 triangle, trigon
4 quadrilateral, tetragon
5 pentagon
Apothem (of a regular polygon) is the
6 hexagon
segment connecting the center of a polygon 7 heptagon
and the midpoint of a side. It is also a 8 octagon
perpendicular bisector of the side. 9 nonagon, enneagon
10 decagon
Central angle (of a regular polygon) is the 11 undecagon, hendecagon
12 dodecagon
angle subtended by a side about the center.
13 tridecagon
14 tetradecagon
15 pentadecagon
Types of Polygons: 16 hexadecagon
17 heptadecagon
Simple and Complex Polygons 18 octadecagon
A simple polygon has only one boundary and 19 enneadecagon
20 icosagon
the sides do not cross each other, otherwise it
30 triacontagon
is a complex polygon. 40 tetracontagon
50 pentacontagon
60 hexacontagon
70 heptacontagon
80 octacontagon
90 enneacontagon
100 hectagon
1,000 chiliagon
10,000 myriagon
106 megagon
10100 googolgon
Concave and Convex Polygons
A convex polygon has no internal angle more
than 180° and if there are any internal angles, To construct a polygon’s name, combine the
then it is a concave polygon. corresponding prefix and suffix as follows:
Sides Prefix
20 icosa
30 triacon
40 tetracon
50 pentacon
60 hexacon
70 heptacon
80 octacon
90 enneacon
Sides Suffix Sum of interior angles (Is )
1 henagon
2 digon
3 trigon
4 tetragon
5 pentagon
6 hexagon
7 heptagon
8 octagon
9 enneagon
Formulas in Polygons:
Perimeter of a regular polygon (P)
P = sn
where P is the length around the boundary of

a closed two-dimensional region,
s is the measure of one side,
n is the number of sides
Area of a regular polygon (A)
where A is the amount of material that would

be needed to cover a surface completely,
P is the perimeter, a is the apothem,
n is the number of sides
A regular polygon can be subdivided into

congruent isosceles triangles whose common
vertex lies at the center of the polygon. Thus,
the area of a regular polygon is equal to the
product of the area of one triangle and the
number of sides of the polygon.
Number of diagonals in a regular polygon (d)
Number of triangles formed by diagonals

drawn through the same vertex (t)
Central angle in a regular polygon (ᶱc)
Each Interior Angle of a regular polygon (ᶱI )

1. Find the number of sides of each of the
two polygons if the total number of
sides of the polygon is 15, and the sum
of the number of diagonals of polygons
is 36.
Given:
total number of sides = 15

sum of diagonals = 36
Find:
a. number of sides of each of two

polygon
Solution:
3. Find the area of a regular dodecagon
whose sides measure 8 units. Determine
Let x be the first polygon
the number of distinct diagonals that
Let y be the second polygon
can be drawn from each vertex and the
sum of its interior angle.
Given:
number of sides=12
length of sides= 8
Find:
area of dodecagon
number of diagonals
sum of interior angles
x=9|x=6
2. Each interior angle of a regular polygon

measures 135. Find the sum of the
interior angles, each interior angle, and
the number of sides.
Given:
interior angle=135
Find:
sum of interior angle
each interior angle
number of sides
4. The sum of the interior angles of a 5. What is the measure of an interior angle
regular polygon is 1260 degrees. Find of a regular pentacontakaitrigon?
the area of the polygon if its perimeter is
45 centimeters. Given:
pentacontakaitrigon = 50
Given:
sum off interior angles =1260∘ Find:
perimeter = 45 cm interior angle
Find:
Area

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MODULE 1 Broken line is a set of line segments

connected end to end.

1.1 Points, Lines and Angles

A point in geometry is a location, which has no

Midpoint is a point halfway between two

A ray is a straight line which is limited from

A segment or line segment is a part of a

An angle is a space formed by two rays called

Radian (rad) is defined as the unit of angle

Points on the same line are said to be

Obtuse angle is an angle whose measure is

Corresponding angles have the same

Adjacent angles are two angles which have

Dihedral angle is the amount of divergence

Terms: Two angles having the same number of

Polygon is a two-dimensional closed figure

Side or edge - is one of the line segments that

Vertex - is a point where the sides meet.

Diagonal - is a line connecting two

Perimeter of a regular polygon (P)

where P is the length around the boundary of

Area of a regular polygon (A)

where A is the amount of material that would

A regular polygon can be subdivided into

Number of diagonals in a regular polygon (d)

Number of triangles formed by diagonals

Central angle in a regular polygon (ᶱc)

Each Interior Angle of a regular polygon (ᶱI )

total number of sides = 15

a. number of sides of each of two

2. Each interior angle of a regular polygon

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