Reviewer for Solid Mensuration

Plane Figure Naming Polygons

● It is the study of geometric figures that can be ● Polygons are named according to their number of
drawn on a two-dimensional surface called a sides. Generally, a polygon with n sides is called
plane. Figures that lie on a plane are called an n-gon.
two-dimensional figures or simply plane figures.
● The most common plane figure are the polygons
Polygon is a closed plane figure formed by line
segments.

Parts of a Polygon
Side or Edge - is one of the line segments that make
up the polygon
Adjacent Sides - are pairs of sides that share a
common endpoint.

Vertices - are the endpoints of each side of the polygon ● To form the name of polygons with 13 to 19
Adjacent Vertices are endpoints of a side sides, begin with the prefix of the unit digit,
Diagonal - is a line segment joining two non-adjacent followed by kai (Greek word for ‘and’) and the
vertices of the polygon. suffix for the tens digit.
Interior Angle - is the angle formed by two adjacent
sides inside the polygon

● To form the name of polygons 20 to 99 sides,

begin with the prefix of the tens digit, followed
Exterior Angle - is an angle that is adjacent to and by kai and the suffix for the unit digit.
supplementary to an interior angle of the polygon. ● For numbers from 100 to 999, form the name of
the polygon by starting with the prefix for the
hundreds digit taken from the ones digit, affix the
Types of Polygon word “hecta” then follow the rule on naming
polygons with 3 to 99 sides.
➢ Equiangular Polygon - all of its angles are
congruent
➢ Equilateral Polygon - all of its sides are equal.
➢ Regular Polygon - equiangular and equilateral.
➢ Irregular Polygon - neither equiangular nor
equilateral
➢ Convex Polygon - Every interior angle of a
convex polygon is less than 180°
➢ Concave Polygon - at least one interior angle that
measures more than 180°
 Diagonal
𝑛
d= 2
(n-3)
Where: n = number of sides.
Triangle
● A closed figure formed by joining three line
segments is called a TRIANGLE.
Example: Equilateral Triangles - each angle measures 60°
532 - Pentahectariacontakaidigon Isosceles Triangle - two congruent sides and two
36 - Triacontakaihexagon congruent angles.
Similar Polygons Scalene Triangle - no congruent angles
● The ratio of two quantities is the quotient of one Right Triangle - has > quantity divided by another quantity. Note, Oblique Triangle - A triangle with no right angle
however, that the two quantities must be of the ● Acute Triangle - three acute angles
same kind. ● Equiangular Triangle - Three congruent angles.
● A proportion is an expression of quality between ● Obtuse Triangle - One obtuse angle
two rations Congruent Triangle
● NOTE: Two polygons are similar if, interior ● Two triangles are congruent when they have the
angles = congruent & proportional sides. Similar same shape and size
polygons have the same shape but differ in size. Similar Triangles
● Are similar if, congruent and their corresponding
sides are proportional.
● Same shape but differ in size.
Parts of a Triangle

Note: A regular polygon of n sides can be subdivided

into n congruent isosceles triangles, whose base is a
side of the polygon. The common vertex of the
triangles is the center of the polygon

Properties of a Regular Polygon

Perimeter Central Angle Altitude - line segment drawn from a vertex
360°
P = ns θ= perpendicular to the opposite side.
𝑛
Orthocenter - The point intersection of the altitudes of
a triangle
Where: Where:
Median - Line segment connecting the midpoint of a
n = number of sides n = number of sides
side and the opposite vertex
s = length of each side
Centroid - Is the point of intersection of the medians of
a triangle
Sum of Interior Angles Area
Angle Bisector - Divides an angle of the triangle into
S.I.A = 180°(n – 2) A = ½ Pa
two congruent angles and has endpoints on a vertex
and the opposite side
Where: Where:
Incenter - Point of intersection of the angle bisectors of
n = number of sides P = Perimeter
a triangle
A = apothem
 Perpendicular Bisector - divides the side into two
congruent segments and is perpendicular to the side Rhombus
Circumcenter - Is the point of intersection of the
𝑑₂
perpendicular bisectors of the sides of a triangle. 2
= 2b2(1 – cosθ) P = 4b
Euler Line - is the line which contains the orthocenter,
centroid and circumcenter of a triangle 𝑑₁
= 2b2(1 + cosθ) A = ½ d1d2
2
Formulas
A = bh A = b2sinθ
Where:
a, b and c are the sides
of the triangle
𝑎+𝑏+𝑐
S=
2

Trapezoid
Formulas for the Area of a Triangle A = ½ (a + b) h
A = ½ bh Median of a trapezoid
m = a+b/2
A = ½ absinθ A = mh

A = √s(s-a)(s-b)(s-c) Mensuration of Plane Figures

Quadrilateral ● Refers to the branch of mathematics that deals
● AKA tetragon or quadrangle, is a general term with the measurement of various proportions and
for a four sided polygon. characteristics of two-dimensional geometric
Classification of Quadrilaterals shapes or plane figures.
Square - special rectangle, which all the sides are equal Square
Parallelogram - quadrilateral in which the opposite
A = s² d=𝑎 2 P = 4a
sides are parallel
Rectangle
Rectangle - a parallelogram in which the interior angles
are all right angles A = bh d = 𝑎² + 𝑏² P = 2a + 2b
Rhombus - parallelogram; all sides are equal Right Triangle
Trapezoid - quadrilateral; one pair of parallel sides A = ½ bh

General Formulas for the area of Quadrilaterals

A = ½ e1e2sinθ

A = ¼ (a2 + c2 – b2 – d2)|tanθ|

A= (𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑) − 𝑎𝑏𝑐𝐶𝑂𝑆²[ 12 (𝐴 + 𝐶)]

Parallelogram

d2 = a2 + b2 – 2abcosθ P = 2a + 2b

A = bh A = absinθ
Trapezoid Area of an Annulus Region
It is the difference between
One pair of opposite the area of the two circles
sides are parallel that define its shape.

A = ½ (a+b)h A = Area of Outer Circle -

Circle Area of inner Circle
Circumference = 2πr = πD
π A = π (r22 - r22)
Area = πr² = 4 D²
Sector of a Circle Area by Approximation
● Is the figure bounded by two radii and an The area of any irregular shaped plane figure can be
included arc found approximately dividing it into an even number of
strips of panel by series of equidistant parallel

1
A= 3
𝑑[(𝑦0 + 𝑦𝑛) + 4(𝑦1 + 𝑦3 +.. + 𝑦𝑛−1) +
Segment of a Circle ..2(𝑦2 + 𝑦4 +.. + 𝑦𝑛−2)]
Note: The larger the value of n in general the greater is
Area of Segment = the accuracy of approximation.
Area of Sector - Area Circle
of Triangle ● Is a set of points, each of which is equidistant
from a fixed point called the center.
A= ½ rs - ½ ba Radius is the line joining the center of a circle to
any points on the circle
Elipse Arc is a portion of a circle that contains two
a comic section with an eccentricity that is less than 1. endpoints and all the points on the circle between
the endpoints.
A = πab Chord - line segment joining aunty points on the circle
P = 2π
1
(a²+b²) Central Angle - the vertex lies at the center of the
2
circle and which sides are the two radii.
Inscribed Angle - the vertex lies on the circle and
which two sides are chords of the circle.
Parabolic Segment
Is a region bounded by a parabola and a line
Spandrel Parabolic Segment
A = ⅓ bh A = ⅔ bh
Perimeter of parabola
1 𝑏² 4ℎ+ 𝑏²+16ℎ²
P= 2
𝑏² + 16ℎ² + 8ℎ
𝑙𝑛( 𝑏
)

Concentric Circles
If circles of different radii have a
common center.
Tangent Line - a line in the same plane ● Diameter as Perpendicular Bisector - diameter
as the circle is a tangent line of the that bisects a chord is perpendicular to the chord
circle if it intersects the circle at exactly and vice versa. Consequently, the perpendicular
one point on the circle. bisector of a chord is the diameter of the circle
which must pass through the center
Secant Line - a line is called a ● Central Angles of Equal Circles - the same or
secant line if it intersects the circle equal circles have the same ratio as their
at two points on the circle intercepted arcs
● Angles Subtended on the Same Arc - Angles form
two points on the circle are equal to the other
Cyclic Quadrilateral - is a four sided angle, in the same, formed from those two points.
figure inscribed in a circle with each
vertex (corner) of the quadrilateral
touching the circumference of the circle

Theorems on Circles
● Two Chords intersect at a point - the product of
the segments of one chord is equal to the product
of the segments of the other chord. ● Lines of Center of Tangent Circles - The line of
● Two Secant Lines Intersecting at an Exterior centers of two tangent circles passes through the
Point - the product of lengths of the entire secant point of tangency.
line and its external segment is equal to the ● Central Angle and its Intercepted Arc - measure
product if the lengths of the lengths of the other of central angle is the angular measure of its
secant line and its external segment. intercepted arc.

● Tangent and Secant Lines Intersecting at ● Inscribed Angle - And inscribed angle is
Exterior Point - the product of the lengths of the measured by one-half of its intercepted arc.
secant line and its external segment is equal to ● Angle in a Semi-Circle - formed by constructing
the square of the length of the tangent line. lines from the ends of the diameter of a circle to
● A Radius Perpendicular to a Tangent Line - point on the circle is a right angle.
Every tangent line of a circle is perpendicular to
the radius of the circle drawn through the point
of tangency.
 ● Cyclic Quadrilateral - The sum of the products of Radius of the Circle Circumscribing a Triangle
opposite sides of a cyclic quadrilateral is equal Formula:
to the product of the diagonals. 𝑎𝑏𝑐
r= or
● Two Intersecting Tangents - The lengths of two 4𝐴
tangents from the points of tangency on the circle 𝑎𝑏𝑐
r=
to their point of intersection are the same. 4 𝑠(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)
Where:
A = Area of a Triangle
𝑎+𝑏+𝑐
s= 2
Radius of the Circle Inscribed in a Triangle
Formula:

𝐴
● Angles Formed by Intersecting Secant Lines -
r= 𝑠 or
The lengths of two tangents from the points of
(𝑠−𝑎)(𝑠−𝑏)(𝑠−𝑐)
tangency on the circle to their point of r=
𝑠
intersection are the same.
- if two secants intersect at an exterior point of a
Area of a Cyclic Quadrilateral
circle the measure of the angle formed by the
Formula:
secants is one-half the difference between the
angular measurements of the intercepted arcs

Note:
∠P + ∠R = 180°
∠Q + ∠S = 180°
∠P + ∠Q + ∠R + ∠S=
360°

Inscribed and Circumscribed Polygons A = (𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑)

Inscribed - a polygon inscribed in a circle if all of its
vertices lie on the circle. LINES / PLANES / ANGLES: LSN#3
Plane - A surface such that a straight line joining any
S² = r² + r² - 2r(r) cosθ two any two points lies wholly in the surface.
where : Collinear Points - Are three or more points that lie on
r= radius of the circle the same line.
θ= Central angel Coplanar Points - are points that lie in the same plane.
θ=
360° Non-Collinear Points - A point that does not lie on the
𝑛
same straight line as another point or set of points.
n= number of sides Coplanar Lines - Lines that lie in the same plane
Circumscribed Polygons - A
polygon is circumscribed about a circle if each of its A plane is determined by:
sides is tangent to the circle. - Three noncollinear points
- Two intersecting lines
- Two Parallel lines
Relative positions of a line and Plane - The angle that the line makes with its projection
on a plane is called the angle of inclination of a
line to a point.
Dihedral Angles - The
angle formed between two
intersecting planes.

Right Dihedral Angle - The

1. A line in a plane if two points of the line lie in
angle formed when a plane
the plane
intersects another plane and
2. If a line and Plane do not intersect, the line is
makes the adjacent dihedral
parallel to the plane
angles equal.
3. A line not on the plane and not parallel to it can
intersect the plane at exactly one point
Polyhedral Angles - The angle formed by three or more
4. If a line is perpendicular to a plane, then it is
planes which meet at a common point
perpendicular to every line in the plane passing
through the point of intersection of the line and
➢ common point is called the vertex
the plane. The plane is also said to be
➢ The intersecting planes are the
perpendicular to the line.
faces of the polyhedral angle.
5. A line that intersects a plane but is not
➢ The lines of intersection of these
perpendicular to the plane is said to be an oblique
faces are called edges.
line to plane M.
➢ A plane which cuts all the faces of polyhedral
Relative Positions of Two Planes
angles is called a section.
1. If three noncollinear points of a plane, the two
➢ Face angle is the angle at the vertex and formed
planes coincide.
by any two adjacent edges.
2. If two planes intersect, the intersection is a
Solids
straight line.
a three-dimensional figure bounded by surfaces or plane figures
3. If two Planes do not meet, the planes are parallel.
● Volume of a Solid - The amount of space it
Other Facts about Parallel and lines and Planes
occupies. Unit of measure is in cubic³.
● Two Lines are perpendicular to the same plane
● Surface Area
are parallel
● Lateral Area
● Two planes perpendicular to the same line are
● Total Surface Area
parallel
The Cavalieri’s Principle
● A line perpendicular to one of two parallel planes
Given any two solids included between parallel
is also perpendicular to the second plane.
horizontal planes; If every right section has the same area
● Two lines which are cut by three parallel planes
in both solids, then the volume of the solids are equal.
are divided proportionally

The Volume Addition Theorem

- A solid region may be divided into non -
overlapping smaller regions so that the sum of
the volumes of these smaller regions is equal to
Additional Notes the volume of the solid.
- The projection of a straight line on a plane not
perpendicular to the line is a straight line.
INSCRIBED SOLIDS
A solid of maximum volume placed inside another solid
fixed volume, with their edges or surfaces touching each
other.
Cylinder Inscribed in a Sphere

The diameter of the sphere is equal to the diagonal of the Properties of Similar Polyhedra
rectangle formed by the height and diameter of the - Corresponding linear dimensions of any two
2 2 similar solids have the same ratio
cylinder. 𝐷 = (2𝑟) + ℎ
- The ratio of the area of similar plane figures or
A sphere Inscribed in a Cube
similar surfaces is equal to the square of the ratio
The diameter of the sphere is equal to the length of one
of any two corresponding dimensions.
edge of the cube
- The ratio of the volumes of two similar solids is
A Sphere inscribed in a Cylinder
equal to the cube of the ratio of any two
The cylinder and the sphere have the same diameter
corresponding dimensions.
A rectangular Solid Inscribed in a Sphere
Facts about a regular Polyhedrons
The diameter of the sphere is equal to the length of the
● Regular Polyhedrons of the same
diagonal of the rectangular solid.
number of faces are similar
2 2 2
𝐷 = 𝐿 +𝑊 +ℎ ● e = ½ np where: e is # of edges, p
POLYHEDRA is # of polygons enclosing the
Polyhedron - a solid which is bounded by polygons polyhedron, and n is # of sides in
joined at their edges. each polygon
● Faces = The bounding polygons are the faces of ● v = e-p+2 where: v = vertex
the polyhedron. Total Surface Area
● Vertices - The intersection of the edges are the 𝑛𝑝𝑠
2

vertices 𝑇𝑆𝐴 = 180°

4𝑡𝑎𝑛( 𝑛
)
● Edges - the intersection of these faces are the
edges Volume of a Regular Polyhedron
180°
● Lateral Faces - The surfaces on the sides of a −1 𝑐𝑜𝑠( 𝑓
)
polyhedron 𝑑 = 2𝑠𝑖𝑛 ( 180° )
𝑠𝑖𝑛( 𝑛
)
● Lateral Edges - Intersection of Lateral Faces 3 𝑑
● Diagonal - A line joining any two vertices not in 𝑛𝑝 𝑡𝑎𝑛( 2 )
the same face. 𝑉 = 2 180°
24𝑡𝑎𝑛 ( 𝑛
)
Section
The plane figure formed by the intersection of a plane Where: d = dihedral angle between two adjacent faces
cutting all the lateral edges of a polyhedron is called V = volume
section PRISMS
Note: If every section of a polyhedron is a convex polygon, the A polyhedron with two congruent bases that lie on
polyhedron is a convex polyhedron. parallel planes, and whose every section that is parallel to
Polyhedrons a base has the same area as that of the base.
A regular polyhedron or platonic solid is a polyhedron Types of Prisms
whose faces are congruent regular polygons and whose ● Right Prism – a prism whose lateral faces or
polyhedral angles are equal. lateral edges are perpendicular to the two bases.
● Regular Prism - a right prism whose bases are
regular polygons
 ● Oblique prism - prism whose lateral faces or Surface Area of a Rectangular Solid
lateral edges are not perpendicular to its bases. LSA = 2lh +2wh
Its lateral faces are parallelograms. Total Surface Area
Types of Sections TSA = 2lw + 2lh + 2wh
● Right Section - is a section made by a plane Volume of a Rectangular Solid
perpendicular to one of the lateral edges. V = lwh
● Oblique Section - is made by a plane oblique to CUBE
one of the lateral edges. ● A special type of prism whose faces and edges
Surface Area are all congruent
Lateral Surface Area ● Is a regular hexahedron
𝐿𝑆𝐴 = 𝑃𝑒 Formulas:
where : 𝑑 = 𝑠 3 𝐿𝑆𝐴 = 4𝑠
2

P - perimeter of a right section 2 3

𝑇𝑆𝐴 = 6𝑠 𝑉 = 𝑠
e - length of lateral Edge
CYLINDER
Total Surface Area
A solid bounded by a closed cylindrical surface and two
TSA = LSA + 2B
parallel planes cutting all the elements of the surface.
where: B = area of one base
Cylindrical Surface
Volume of Prism
- the surface generated by a straight line which
V = Bh V = Re
moves along a fixed curve, and which remains
R = Bsinθ
parallel to a fixed line not on the curve.
Where: B is area of base
Parts of Cylinder
h is height of prism
➢ The two parallel planes are called the bases of
R is area of the right section
the cylinder
e is length of lateral edge
➢ Cylindrical Surface is called the lateral face.
PARALLELEPIPED
➢ The distance between the two bases of a cylinder
A prism whose bases and lateral faces are parallelograms.
is the altitude.
Types of Parallelepiped
➢ A section perpendicular to all elements is a right
● Right parallelepiped - has lateral edges
section of a cylinder.
perpendicular to the bases
➢ Circular Cylinder - a cylinder whose bases are
● Oblique parallelepiped - has lateral edges that are
circles
inclined with the bases.
➢ Height of cylinder h - perpendicular distance
● Rectangular parallelepiped - Commonly known
between the circular bases of a cylinder.
as a rectangular solid. All the faces and bases of
➢ Radius of the cylinder r - radius of the base
a rectangular solid are rectangles.
➢ Right Circular Cylinder - if a line segment drawn
from the center of the bottom base to the center
of the top base is perpendicular to each of the
base.
➢ Otherwise, the cylinder is said to be oblique

Surface Area of a Right Circular Cylinder

LSA = 2πr² + 2πrh
TSA = 2πr² + 2πrh = 2πr(r+h)
Diagonal Length of a Rectangular Solid = C(r+h)
d = diagonal
2 2 2
d= 𝑙 +𝑤 +ℎ