Subject: Applied Engineering Mathematics
Prepared by: Engr. Patrick Owen Alimuin
II – Plane Geometry
Plane Geometry Definitions
1. Altitude of a Triangle - A line drawn
perpendicular from any vertex of a triangle to the
opposite side (or its extension).
2. Angle -The opening between two straight lines
drawn from the same point.
3. Apothem - The radius of a polygon’s inscribed
circle.
4. Area - The numerical value representing the ratio
between a figure's surface area and the surface of
a unit square.
5. Center of Polygon - The common center of a
regular polygon’s inscribed and circumscribed
circles.
6. Circle - A closed plane curve where all points are
equidistant from a fixed point (the center).
7. Complementary Angles - Two angles whose
sum is 90∘ . Each is the complement of the other.
8. Concurrent Lines - Three or more lines meeting
at a single point.
9. Definition of 𝛑 - The constant ratio of a circle's
circumference to its diameter (π = 𝐶 / 𝐷).
Approximations: π = 3.1416 (to 4 decimal
places).
10. Diagonal - A line segment connecting two nonconsecutive vertices of a polygon.
11. Hypotenuse - The side opposite the right angle
in a right triangle, and the longest side.
12. Isosceles Triangle - A triangle with two sides of
equal length.
13. Locus - A set of points satisfying a specific
condition or fulfilling a given requirement.
14. Parallel Lines - Lines in the same plane that
never intersect, no matter how far extended.
15. Parallelogram - A quadrilateral whose opposite
sides are parallel.
16. Perpendicular - Two lines that intersect at a right
angle (90°).
17. Quadrilateral - A plane figure bounded by four
straight lines.
18. Rectangle - A parallelogram where all angles are
right angles (90∘ ).
19. Regular Polygon - A polygon with all sides and
angles equal.
20. Similar Polygons - Polygons whose
corresponding
angles
are
equal,
and
corresponding sides are proportional.
21. Supplementary Angles - Two angles whose sum
equals 180∘ .
22. Tangent - A straight line that touches a circle at
exactly one point without crossing it.
23. Trapezoid - A quadrilateral with exactly one pair
of parallel sides.
24. Triangle - A plane figure formed by three
straight lines.
25. Vertical Angles - Two angles formed when two
lines intersect. They share the same vertex and
are opposite each other.
TRIANGLE
Similar Triangles
1. Two triangles are similar if the angles of one
triangle are respectively equal to the angles of the
other, or if two angles of one triangle are equal to
two angles of the other.
2. Two triangles are similar if an angle of one equals
an angle of the other and the sides including these
angles are proportional.
3. Two triangles are similar if their sides are in the
same ratio.
4. Two triangles are similar if their sides are
respectively parallel or perpendicular to each
other.
QUADRILATERALS
Area of a Triangle
1. Given base 𝒃 and altitude 𝒉:
1
Area = 𝑏ℎ
2
2. Given two sides 𝒂 and 𝒃 and the included
𝒂𝒏𝒈𝒍𝒆 𝜽:
Area =
1
𝑎𝑏 sin 𝜃
2
3. Given three sides 𝒂, 𝒃, and 𝒄 (Hero's
Formula):
Area = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)
Rectangle
Area: Area = 𝑎 ⋅ 𝑏
Perimeter: 𝑃 = 2(𝑎 + 𝑏)
Diagonal: 𝑑 = √𝑎2 + 𝑏 2
Square
Area: Area = 𝑎2
Perimeter: 𝑃 = 4𝑎
Diagonal: 𝑑 = 𝑎√2
General Quadrilateral
where:
𝑠=
𝑎+𝑏+𝑐
2
4. Given three angles 𝑨, 𝑩, and 𝑪, and one
side 𝒂:
Area =
𝑎2 sin 𝐵 sin 𝐶
2 sin 𝐴
The area under specific conditions can be
calculated by finding one side using the sine law
and applying the formula for two sides and the
included angle.
1. Given diagonals 𝑑1 and 𝑑2 and included
angle 𝛉:
1
Area = 𝑑1 𝑑2 si𝑛𝜃
2
2. Given four sides 𝑎, 𝑏, 𝑐, 𝑑 and the sum of
two opposite angles:
Area = √(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑) − 𝑎𝑏𝑐𝑑 cos 2 𝜃
where:
𝑠=
For the given triangles △ 𝑨𝑩𝑪 and △ 𝑨𝑿𝒀:
Area of △ 𝐴𝐵𝐶 (𝐴𝐵)(𝐴𝐶)
=
Area of △ 𝐴𝑋𝑌 (𝐴𝑌)(𝐴𝑋)
𝑎+𝑏+𝑐+𝑑
,
2
𝜃=
∠𝐴 + ∠𝐶
2
𝐬𝐞𝐦𝐢 − 𝐩𝐞𝐫𝐢𝐦𝐞𝐭𝐞𝐫
𝑜𝑟
𝜃=
∠𝐵 + ∠𝐷
2
3. Given four sides 𝑎, 𝑏, 𝑐, 𝑑 and two opposite
angles 𝐵 and 𝐷:
Divide the quadrilateral into two triangles:
Area =
1
1
𝑎𝑏si n 𝐵 + 𝑐𝑑si n 𝐷
2
2
Parallelogram
1. Properties:
-Opposite sides are equal and parallel.
-Diagonals bisect each other.
2. Given diagonals 𝑑1 and 𝑑2 and included
angle 𝛉:
1
𝐴𝑟𝑒𝑎 = 𝑑1 𝑑2 si n 𝜃
2
3. Given two sides 𝑎 and 𝑏 and one angle 𝐴:
1. Properties:
-A trapezoid has one pair of parallel sides.
2. Given bases aa and bb, and height hh:
𝑎+𝑏
Area =
⋅ℎ
2
3. Given four sides 𝑎, 𝑏, 𝑐, 𝑑 and two opposite
angles 𝐵 and 𝐷:
Divide the quadrilateral into two triangles:
1
1
Area = 𝑎𝑏si n 𝐵 + 𝑐𝑑si n 𝐷
2
2
Cyclic Quadrilateral
𝐴𝑟𝑒𝑎 = 𝑎𝑏si n 𝐴
Rhombus
A cyclic quadrilateral is a quadrilateral whose
vertices lie on the circumference of a circle.
Properties
1. Opposite angles are supplementary:
∠𝐵 + ∠𝐷 = 180∘
∠𝐴 + ∠𝐶 = 180∘
1. Properties:
-A rhombus is a parallelogram with four equal
sides.
-Diagonals bisect each other at 90∘ .
2. Given diagonals 𝑑1 and 𝑑2 :
1
Area = 𝑑1 𝑑2
2
3. Given side 𝑎 and one angle 𝐴:
𝐴𝑟𝑒𝑎 = 𝑎2 si n 𝐴
Trapezoid
Area = √(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑)
where:
𝑠=
𝑎+𝑏+𝑐+𝑑
2
(semi-perimeter)
Ptolemy's Theorem
"For any cyclic quadrilateral, the product of the
diagonals equals the sum of the products of the
opposite sides":
𝑑1 𝑑2 = (𝑎)(𝑐) + (𝑏)(𝑑)
POLYGONS
Definition
1. Convex Polygon: A polygon where no extended
side passes inside the polygon.
2. Concave Polygon: A polygon where at least one
extended side passes inside the polygon.
The figure provided represents a convex polygon.
Classification of Polygons by Number of Sides
Number
Name
of Sides
2
Digon
3
Triangle (Trigon)
4
Quadrilateral (Tetragon)
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon (Enneagon)
10
Decagon
11
Undecagon (Hendecagon)
12
Dodecagon
13
Tridecagon (Triskaidecagon)
14
Tetradecagon (Tetrakaidecagon)
15
Pentadecagon (Pentakaidecagon)
16
Hexadecagon (Hexakaidecagon)
17
Heptadecagon (Heptakaidecagon)
18
Octadecagon (Octakaidecagon)
19
Enneadecagon (Enneakaidecagon)
20
Icosagon
30
Triacontagon
40
Tetracontagon
50
Pentacontagon
60
Hexacontagon
70
Heptacontagon
80
90
100
1,000
10,000
Octacontagon
Enneacontagon
Hectogon
Chiliagon
Myriagon
Theorems in Polygons
1. Interior Angle Sum
The sum of the interior angles of a convex
polygon with 𝑛 sides is:
(2𝑛 − 2) ⋅ 90∘
2. Exterior Angles
The sum of the exterior angles, one from each
vertex, is always 360∘ or 4 right angles.
3. Congruence
Homologous parts of congruent polygons are
equal.
Sum of Interior Angles
The sum of the interior angles (𝜃) of a polygon
with 𝑛 sides is:
Sum, Σθ = (𝑛 − 2) × 180°
Sum of Exterior Angles
The sum of the exterior angles (𝛽) of any polygon is
always:
β = 360∘
Number of Diagonals (DDD)
The number of diagonals in a polygon (line
segments joining two non-adjacent vertices) is
given by:
𝑛
𝐷 = (𝑛 − 3)
2
where 𝑛 is the number of sides.
REGULAR POLYGONS
CIRCLE
Equilateral Polygon: All sides are equal.
Equiangular Polygon: All interior angles are equal.
Regular Polygon: A polygon that is both equilateral
and equiangular.
Definitions and Properties:
Radius: A line segment from the center of the circle
to any point on its circumference.
Diameter: A line segment passing through the center
and touching two points on the circumference.
Chord: A line segment joining any two points on the
circle.
Secant: A line that intersects the circle at two points.
Tangent: A line that touches the circle at exactly one
point.
Arc: A portion of the circle's circumference.
The area of a regular polygon can be calculated by
dividing it into isosceles triangles.
Key Properties of a Regular Polygon
• 𝑥: Length of a side.
• 𝜃: Angle subtended by a side at the center.
• 𝑅: Radius of the circumscribing circle.
• 𝑟: Radius of the inscribed circle (also called
the apothem).
• 𝑛: Number of sides.
Regular Polygon Formulas
1. Central Angle (𝛉):
360∘
θ=
𝑛
2. Area (𝑨):
𝑛
𝑛
𝐴 = 𝑅 2 sin θ = (𝑥 ⋅ 𝑟)
2
2
where 𝑅 is the radius of the circumscribing circle, 𝑥
is the side length, and 𝑟 is the apothem.
3. Perimeter (PPP):
𝑃 = 𝑛 ⋅𝑥
4. Interior Angle (𝛃):
𝑛−2
β=
⋅ 180∘
𝑛
5. Exterior Angle (α\alphaα):
360∘
α=θ=
𝑛
Theorems on Circles
1. Unique Circle Through Three Points
-Through three points not in a straight line, one
and only one circle can be drawn.
2. Tangent and Radius
-A tangent to a circle is perpendicular to the
radius at the point of tangency, and conversely.
3. Tangents from an External Point
-Tangents drawn from an external point are
equal in length and make equal angles with the
line joining the point to the center of the circle.
4. Inscribed Angle
-An inscribed angle is measured as half the
intercepted arc:
1
α or α = 2θ
2
5. Right Triangle in a Semicircle
-An angle inscribed in a semicircle is a right
angle. If a right triangle is inscribed in a circle,
the hypotenuse is the diameter of the circle.
8. Angle Formed by a Tangent and a Chord
-The angle included by a tangent and a chord
drawn from the point of contact is measured by
half the intercepted arc:
1
α= θ
2
θ=
9. Angle Formed by Secants and Tangents
-An angle formed by two secants, two tangents,
or a tangent and a secant drawn to a circle from
an external point is measured by half the
difference of the intercepted arcs.
10. Intersecting Secants and Tangents
6. Angle Between Two Chords
-The angle formed by two chords intersecting
within the circle is measured by half the sum of
the intercepted arcs.
7. Segments of Intersecting Chords
-Intersecting Secants
If two secants intersect outside a circle:
(𝑂𝐴)(𝑂𝐵) = (𝑂𝐶)(𝑂𝐷)
The angle formed by the intersecting secants
is:
1
θ = (𝐴𝑟𝑐AC − 𝐴𝑟𝑐 BD )
2
Angle relationships:
∠𝐴𝐵𝐶 = ∠𝐴𝐷𝐶
If two chords intersect in a circle, the product of
the segments of one is equal to the product of
the segments of the other:
(𝐴𝐸)(𝐵𝐸) = (𝐶𝐸)(𝐷𝐸)
The angle formed is:
1
θ = (Arc AC + Arc BD)
2
∠𝐴𝐷𝐶 = ∠𝐴𝐵𝐶
∠𝐵𝐴𝐷 = ∠𝐵𝐶𝐷
and ∠𝐵𝐶𝐷 = ∠𝐵𝐴𝐷
-Intersecting Tangent and Secant
If a tangent and a secant intersect outside
the circle:
(𝑂𝐶)2 = (𝑂𝐴)(𝑂𝐵)
The angle formed is:
1
θ = (𝐴𝑟𝑐BC − 𝐴𝑟𝑐AC )
2
1
α = 𝐴𝑟𝑐AC
2
CIRCLE
Area of a Circle
Circumference:
Circumference = 2π𝑟
Area:
π
Area = π𝑟 2 = 𝐷2
4
Sector of a Circle
Length of Arc:
𝐶 = 𝑟θradians =
11. Perpendicular to the Diameter
-A perpendicular drawn from a point on the
circumference of a circle to its diameter forms a
mean proportional between the segments of the
diameter:
ℎ2 = 𝑎 ⋅ 𝑏
where:
ℎ: Height of the perpendicular
𝑎, 𝑏: Segments of the diameter.
π𝑟θ
180∘
Area of Sector:
1
π𝑟 2 θ 1
Area = 𝑟 2 θradians =
= 𝐶𝑟
2
360∘ 2
Note: 1 radian is the angle θ such that
𝐶 = 𝑟
Segment of a Circle
1. Area of a Segment (𝒘𝒉𝒆𝒏 𝛉 ≤ 𝟏𝟖𝟎∘ ):
12. Circumferences and Arcs of Circles
-The circumferences of two circles are
proportional to their radii.
-Arcs subtended by equal central angles in two
circles are proportional to their radii.
Area = 𝐴sector − 𝐴triangle
1
1
𝐴𝑟𝑒𝑎 = 𝑟 2 θ − 𝑟 2 sin 𝜃
2
2
1 2
Area = 𝑟 (θ − sin θ)
2
Simplified formula:
∘
2. Area of a Segment (𝒘𝒉𝒆𝒏 𝛂 = 𝟑𝟔𝟎 − 𝛉):
Areashaded =
2
(𝐴𝐶)(ℎ)
3
Where:
𝐴𝐶 is the base.
ℎ is the height of the segment.
Spandrel of a Parabolic Segment
Area = 𝐴sector + 𝐴triangle
1
Area = 𝑟 2 (α + sin θ)
2
Where θ and α are angles in radians.
The area of the spandrel is given by:
1
Area = 𝑏ℎ
3
Where:
𝑏 is the base of the spandrel.
ℎ is the height.
Parabolic Segment
ELLIPSE
1. Area:
2
𝑏ℎ
3
2. Length of Curve (𝑨𝑩𝑪):
𝑏2
[𝑚 + ln(𝑚 + 𝑒)]
Length =
8ℎ
Area =
4ℎ
Where: 𝑚 = 𝑏 ,
Area:
𝑒 = √1 + 𝑚2
Shaded Area in a Parabolic Segment
Area = π𝑎𝑏
Where:
𝑎: Semi-major axis
𝑏: Semi-minor axis
Perimeter (Approximation):
Perimeter ≈ 2π√
The shaded area can be calculated as:
Areashaded = 𝐴𝐴𝐷𝐵 − 𝐴𝐴𝐷𝐸 + 𝐴𝐵𝐹𝐶 + 𝐴𝐶𝐹𝐸
𝑎2 + 𝑏 2
2
Relation of Axes and Foci:
𝑎2 = 𝑏 2 + 𝑐 2
Where:
𝑐: Distance from the center to the
focus
𝑐
Eccentricity (First Eccentricity): 𝑒 = 𝑎
𝑐
Second Eccentricity: 𝑒 ′ = 𝑏
𝑐
Angular Eccentricity: 𝛼 = 𝑎
3. Circles Escribed about a Triangle (Excircles)
A circle is escribed about a triangle if it is tangent to
one side and the extensions of the other two sides. A
triangle has three excircles.
Flatness:
Ellipse Flatness: 𝑓 =
𝑎−𝑏
Second Flatness: 𝑓 ′ =
𝑓
𝑎
𝑎−𝑏
𝑏
RADIUS OF CIRCLES
𝐴
1. Circle Circumscribed about a Triangle
(Circumcircle)
A circle is circumscribed about a triangle if it passes
through all the vertices of the triangle.
Radius (𝒓): 𝑟 =
𝑇
Where:
𝐴𝑇 : Area of the triangle
𝑎, 𝑏, 𝑐: Sides of the triangle
2. Circle Inscribed in a Triangle (Incircle)
A circle is inscribed in a triangle if it is tangent to
all three sides of the triangle.
𝐴
Radius (𝒓): 𝑟 = 𝑠𝑇
2
√(𝑎𝑏+𝑐𝑑)(𝑎𝑐+𝑏𝑑)(𝑎𝑑+𝑏𝑐)
4𝐴quad
𝐴quad : Area of the quadrilateral
Semi-perimeter: 𝑠 =
𝑎+𝑏+𝑐+𝑑
2
5. Circle Inscribed in a Quadrilateral
A circle is inscribed in a quadrilateral if it is tangent
to all its sides.
Radius (𝒓): 𝑟 =
𝑎+𝑏+𝑐
𝐴
𝑇
𝑟𝑐 = 𝑠−𝑐
Where:
Where:
𝑠: Semi-perimeter 𝑠 =
𝑇
𝑟𝑏 = 𝑠−𝑏
,
4. Circle Circumscribed about a Quadrilateral
A circle is circumscribed about a quadrilateral if it
passes through all the vertices.
𝑎𝑏𝑐
Radius (𝒓): 𝑟 = 4𝐴
𝐴
𝑇
Radii: 𝑟𝑎 = 𝑠−𝑎
,
𝐴quad
𝑠
Where:
𝐴quad = √𝑎𝑏𝑐𝑑,
𝑎+𝑏+𝑐+𝑑
𝑠=
2
AREA BY APPROXIMATION
The area of any irregular plane figure can be
approximated by dividing it into a number of strips
or panels using equidistant parallel chords (offsets)
ℎ1 , ℎ2 , … , ℎ𝑛 with the common distance between
chords being 𝑑.
Area by Trapezoidal Rule
Assumes each strip is a trapezoid. The area
is:
𝑑
Area = [ℎ1 + 2(ℎ2 + ℎ3 + ⋯ ) + ℎ𝑛 ]
2
Area by Simpson’s One-Third Rule
A more accurate method that considers
curved sides. It requires an odd number of
offsets (𝑛 must be odd).
𝑑
Area = [ℎ1 + 2 ∑ ℎeven + 4 ∑ ℎodd + ℎ𝑛 ]
3
AREA BY COORDINATES
For a planar shape (convex or concave) with
vertices (𝑥1, 𝑦1), (𝑥2, 𝑦2), … , (𝑥𝑛, 𝑦𝑛)
Area Formula: