PLANE AND SOLID

GEOMETRY

Created by: Mikko P. San Jose

BSED-MT 1-1D
ANGLES - Vertex
- Diagonals
- It is the amount of turn of the two straight lines
that shares common end point. Types of Polygon
- Equiangular – has equal angles
Parts - Equilateral – has equal sides
- Regular – all sides and angles are equal
- Vertex – it is the point where the two lines
- Convex – all interior is less than 180deg
meets - Concave – one or more interior angle is greater
- Arms – these are the lines that makes up the than 180deg
angle.
- Degrees – it the measure of the angle

Naming angles Undefined and Defined terms in

Geometry
- angle symbol with the three letters of the angle
- angle symbol with the vertex ‘s letter - Undefined terms – these are the basic concepts
- angle symbol with the letter between the angle that are not formally defined using other terms.
They are used to define other term.
Examples:
- Points – a location is apace with no size or
dimensions and not measurable
Common Angle Measures
- Line – has an infinite length, no dimension. An
- Acute – less then 90deg arrow head is places at the both ends of the line
- Right – exactly 90deg to show that it is continuously expanding
- Plane – a flat two-dimensional surface that
- Obtuse – greater then 90deg
extends infinitely in all sides
- Straight – 180deg
- Reflex – greater than 180deg
- Full rotation - 360deg Naming plane

Polygons
- Greek word Poly (many) Gon (angles)
- Closed shape made of straight lines.
- Has at least 3 sides and 3 angles.
- Sides connect only at endpoints (called
vertices).
- Lies on a flat (2D) surface.
- The number of sides = number of angles.

Parts of the Polygon

- Using the capital single letter name of the plane
- Sides (P)
- Interior angle - Using the three non-collinear points (points that
- Exterior angle will not create a straight line when connected)
- In naming co-planar, you can use any 4 points
that lies on the same plane - Its all four sides are equal
- Note: if the two plane intersect, they will create - It is a both rectangle and rhombs
a line. - Opposite sides are parallel
- Has equal diagonals
- It has 4 rotational symmetry
- Defined terms – these are the concepts that can
Formulas for the Square:
be clearly describe using the undefined terms.
- Perimeter = 4(sides)
- Area = side squared
Examples: - Diagonals = side square root of 2
- Line segment - Sides if area is known = square root of area
- Ray - Sides if diagonals are known = diagonal over
- Opposite ray square root of 2
- Angle - Radius of the in-circle (circle inside the
- Parallel lines square) = sides over 2
- Perpendicular lines - Radius of circumcircle (circle that outside the
- Intersecting lines square that touches its four corners) = sides
- Midpoint over square root of 2
- Circle
Common notations Rectangle
- Dot (point)
- Single line at the top of the two letters (line
segment)
- Single line with an arrow head at the right side
at the top of the letter (ray)
- Single line with a two arrow head at the both
ends (line)
- Zigzag (broken lines)

- All the opposite sides are equal

- Al angles are right angles
Plane and Solid Geometry: Plane - Diagonals are equal
Objects - It has two lines of symmetry

Square
Formulas for the Rectangle:
- Perimeter = 2 (length + width)
- Area = length x width
- Diagonals = square root of length squared +
width squared
- Length is diagonals and width are known =
square root of diagonal squared + width
squared
- Width is diagonals and length are known = - Scalene – no equal sides
square root of diagonal squared + length
squared
- Length if area and width are known = area
over width
- Width is are and length are known = area
over length

Types of triangle by Angle:

- Acute – all angles are less than 90deg

Triangle - Right – one angle is exactly 90deg
- Obtuse – one angle is more than 90deg

Formulas for the triangle:

- Area = ½ x base x height

- Area using Heron’s formula (if all sides are
given) (first find the semi-perimeter or s = a
+b+ c over 2) = square root of s(s-a)(s-b)(s-c)
- Perimeter = a+b+c
Types of triangle by sides:
- Pythagorean theorem (for right tringles only)
- Equilateral – all sides are equal = c squared = a squared + b squared

Trapezoid

- Isosceles – two sides are equal

Parts of Trapezoid:

- Top and bottom base

- Legs
- Mid segment
- Height/altitude
- Base angle

Types od trapezoid
- Isosceles trapezoid – has a pair of non-parallel Circle
and parallel sides

- Scalene trapezoid – has no congruent angles

and sides

Parts of the Circle:

- Chords – line segment whose end point lies on

the circle
- Radius – line segment that connects the center
to point in the circle
- Diameter – a chord that contains the circle
- Right trapezoid – has one pair of right angles - Arc – a part of a circle’s circumference
- Semicircle – arc that is half of the circle
- Minor arc – arc that is less than the semicircle
- Major arc – arc that is greater than the
semicircle
- Tangent – a line that passes at exactly 1point
- Secant - a line that passes at exactly two points

Formulas for the Trapezoid: Angles is Circle

- Area = ½ X (base1 + base2) X height • Central angle – angle inside the circle
- Perimeter = base1 + base2 + side1 + side2 that the vertex is at the center of the
- Mid-segment (median) = base1 + base2 over 2 circle
- It is equal to the arc length in degrees
- Central angle = 2 X inscribe angle
• Inscribe angle – angle inside the circle
that the vertex is on the circle
- Inscribe angle = ½ x central angle
Lines and segments - Geometric solid are the figures or shapes that
has a three-dimensional structure, it has
- sector – enclosed by
volume. They have length, width, and height.
radii and arc
- Solid geometry deals with the study of 3D
- Segment – bounded
shapes and figure.
by chord and arc
Common parts of a solid shapes:

Theorems in Inscribe Angles

- The degree measure of an inscribe angle s half

the measure of its intercepted arc
- If the two inscribe angle of the circle intercept
the same arc, they are congruent
- If an inscribe angle intercept a semicircle, then - Face – the side of the solid figure
it is a right angle - Edge – it is the formed when two faces meet.
- If quadrilateral is inscribe in a circle, then its - Vertices – Is is the point where three of more
opposite angles are supplementary edges meets

Two Types of Geometric Solid:

Formulas for the Circle: - Polyhedron – comes from the Greek words
Poly (many) and edron (face), polyhedrons are
- Area of the Sector = Pi radius squared x (angle the solid figures that has no curved surface. All
theta over 360deg) surface are flat
- Area of the segment = ½ x radius squared x - Non-Polyhedron – these are the solid figures
(pi angle theta over 180 – sin theta) that has curved surface like cylinder
- Circumference = pi diameter or 2 pi radius
- Area of the Circle = pi radius squared
- Arc Length = angle theta over 360deg x (2 pi
radius) Cube

Plane and Solid Geometry: Solid

Objects
- A cube is a perfectly balanced 3 shape made
entirely of 6 identical faces, with equal edges
and right angles.
- Cube has 6 faces, 12 edges, and 8 vertices
 Formulas for Rectangular Prism:

Formulas for the Cube: - Lateral Surface Area = 2 (length + width)

height
- Lateral surface area (it is the front, back, left ,
- Total Surface Area = 2 (LW + LH + WH) (L is
and right. Excluded the top and bottom squares)
length, W is width, and H is height)
= 4 (length of one side of the cube) squared
- Volume = length x Width x height
- Sides if the LSA is given = square root of LSA
over 4 (note: the LSA over 4 are both under the
radical)
- Total surface area (all sides are included) = 6 Pyramid
(length of one side of the cube) squared - A pyramid is a three-dimensional object that
- Sides if the TSA is given = square root of TSA has a polygonal base and triangular faces
over 6 (note: the TSA and 6 are both under the
radical)
- Volume = s cube or length of one side of the Basic Properties of the Pyramid
cube cube - Apex
- Sides if the Volume is given = cube root of the - Base
volume - Edge (base edge, lateral edges)
- Face diagonal = side length of the cube square - Vertices
root of 2 - Height
- Space diagonal (it is the longest diagonal in - Lateral Faces
the cube) = side length of the cube square root - Slant Height
of 3
Parts of the Pyramid:

- Apex
Rectangular Prism - Base
- Faces (triangular and the base)
- Lateral faces
- Edge
- Vertices (includes the apex)

Types of Pyramid:
- It is a 3D object that has six rectangular faces. - Triangular Pyramid – It is the type of pyramid
Its opposite sides are congruent and all angles that has a triangular faces and base. It is made
are right angle. of 4 triangle. There are three types of
triangular pyramid: regular, irregular, and
Parts of the Rectangular Prism: right- angled
• Regular pyramid – symmetrical and
- Face has all equilateral faces and base.
- Edge
- Vertex
 triangle can also had Isosceles triangle as its
lateral face not only equilateral triangle.

• Irregular – faces are not equal, might

be made up of Scalene or Isosceles

- Hexagonal Pyramid - it is also called

heptahedron, it is a 3D object that has a
hexagonal base. It faces are Isosceles triangles.

• Right Triangle - it has a right triangle

base

- Heptagonal Pyramid – it is a 3D shape that

has heptagon as its base. It has equilateral as its
lateral faces. There are types of heptagonal
pyramid: right heptagonal and the oblique

- Square Pyramid – it is the pyramid that has a

square base, and has 4 triangular faces heptagonal
There are types of square pyramid: right • Right heptagonal pyramid – the apex
square, oblique square, and equilateral is directly above the center
square • Oblique heptagonal pyramid - the
• Right square pyramid – the apex is apex is not aligned the center of the
places exactly above the center of the base.
square base
• Oblique square pyramid – the apex is - Octagonal Pyramid - it is a 3D shape that has
not place above the center of the quare an octagonal base, and triangular lateral faces.
base
• Equilateral square pyramid - all of
the triangular faces have equal edges

- Pentagonal Pyramid - it is the type of

pyramid that has a pentagon base, pentagonal
 - Right – axis is perpendicular to the plane of its
2 bases
- Oblique – axis is not perpendicular
- Elliptic – bases are in elliptical shape instead of
circular
- Right Circular Hollow – two right circular
cylinder bounded one inside another

Formulas for the Pyramid: Parts of a Cylinder:

- Total surface area (only is base is a regular - Bases
polygon and all of the lateral faces are - Curved surface
congruent triangle) = area of the base + ½ x - Axis
perimeter of the base x slant height of the - Height
pyramid or SA = A+ ½ P x l - Radius
- Lateral Surface Are (LA) = ½ x perimeter of
the base x slant height
- Volume of the Pyramid = 1/3 x area of the
base x height of the pyramid
Formulas for the Cylinder:

- Lateral surface Area (LSA) = 2pi x radius x

Cylinder height
- Total Surface Area (TSA) = 2 pi x radius
- A cylinder is a 3D shape that has two identical
squared + 2pi x radius x height
circle at its both ends that are connected by a
- Volume:
rolled-up rectangle around it. The word
• Volume of the hollow cylinder = pi x
cylinder comes from the Greek word Kylindros
height (radius 1squared – radius 2
which means roll or roller
squared)
• Volume of the not hollow cylinder = pi
radius squared x height

Cone
- A cone is a 3D shape with a circular base
and a pointed top called the apex,
connected by a curved surface.

Types of Cylinder
Types of Cone:

- Right Circular Cone - A cone with a circular

base and the apex directly above the center of
the base.
Most common and symmetrical.
- Oblique Cone - A cone with a circular base,
but the apex is off-center, making it slanted.
- Elliptical Cone - A cone with an oval
(elliptical) base instead of a circular one. Can
be right or oblique.
- Truncated Cone (also called a frustum) - A
cone that is cut horizontally near the top,
removing the apex. Has two circular surfaces
(top and bottom).

Parts of the Cone:

- Apex or Vertex
- Base
- Axis
- Radius
- Height
- Slant Height

Formulas for the Cone :

- Slant height = square root of radius squared +

height squared
- Lateral Surface Area = pi x radius x slant
height
- Total Surface Area = pi x radius (radius =
slant height)
- Volume = 1/3 x pi x radius squared x height

Note: include the number o f sidean and mote tht

are presented in ma'am’s ppt