Analytic

Geometry
Introduction

Solid mensuration
• also known as SOLID GEOMETRY. It is the study of various solids, the
measure of its volume, area, height, length and many more.

PLANE GEOMETRY
• A large part of solid mensuration has to do with the computation of
surface areas and the volumes of solids. In this connection, it is
necessary to pass a plane through a solid to form a plane section,
find the area of this section, and multiply it by the length of a line.
Points, Lines, and Planes

POINT
▪ has no dimension
▪ has no length, no width, and no thickness
▪ only shows one exact location in space

LINES
▪ has no width, no thickness, only length
▪ has one dimension
Points, Lines, and Planes

PLANE
▪ has no thickness
▪ it contains an infinite number of points
and lines
▪ it extends indefinitely in all directions

SPACE
▪ Set of all points
Points, Lines, and Planes
Pairs of lines

▪ PARALLEL LINES are lines on a plane that do not intersect.

▪ INTERSECTING LINES are lines on a plane that met at a
point.
▪ Points are COLLINEAR when they lie on the same line
▪ Points are COPLANAR when they lie in the same plane.
▪ Three or more distinct lines that pass through a common
point are said to be CONCURRENT.
Polygon

Polygon

A polygon is a plane closed figure whose sides are line

segments that are non-collinear each intersects exactly
two other line segments at their endpoints.
 TRIANGLES

• Given the base and the Altitude

1
𝐴 = 𝑏ℎ
2

Where A = area of the triangle

b = length of the base
h = altitude
 TRIANGLES

• Given all sides

𝐴= 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)

Where A = area of the triangle

s = semi-perimeter
𝑃
𝑠= , where P = perimeter
2
a, b, c = length of the sides
 TRIANGLES

• Area of equilateral triangle given all sides

3 2
𝐴= 𝑎
4

Where A = area of the triangle

a = length of the side
 TRIANGLES

• Given two sides and an included angle

1
𝐴 = (𝑎𝑏)𝑠𝑖𝑛𝐶
2
1
𝐴 = (𝑎𝑐)𝑠𝑖𝑛𝐵
2
1
𝐴 = (𝑏𝑐)𝑠𝑖𝑛𝐴
2
Where A = area of the triangle
a, b, c = length of the side
A, B, C = angle of the triangle
 TRIANGLES

In ∆𝐷𝐿𝑆, 𝑚∠𝐷 = 30°, DL = 8 cm and the base DS = 11cm. Find the

altitude to DS and area of ∆𝐷𝐿𝑆.
 TRIANGLES

The altitude of an equilateral triangle is 12 cm. Find the length of

a side and the area of the triangle.
 TRIANGLES

Find the area of the triangular lot with sides 30 m, 27 m and 42 m.

 QUADRILATERALS

TYPES OF QUADRILATERALS
1. Parallelogram – a quadrilateral with two pairs of opposite sides parallel
a. Rectangle – a parallelogram with four right angles and congruent
diagonals.
b. Rhombus – a parallelogram with four congruent sides. Its diagonals
are perpendicular to each other.
c. Square – a rectangle with four congruent sides.
2. Trapezoid – a quadrilateral with only one pair of opposite sides parallel.
3. Trapezium – general quadrilateral, has no opposite sides parallel.
 QUADRILATERALS

Trapezoid
𝟏
𝑨 = 𝒉(𝒃𝟏 + 𝒃𝟐 )
𝟐
𝑷 = 𝒂 + 𝒃𝟏 + 𝒃 𝟐 + 𝒄

Where A = area of the trapezoid

P = perimeter
h = altitude
𝑏1 = length of the upper base
𝑏2 = length of the lower base
a and c = legs
 QUADRILATERALS
Isosceles Trapezoid Where A = area of the trapezoid
𝟏
𝑨 = 𝒉(𝒃𝟏 + 𝒃𝟐 ) P = Perimeter
𝟐
h = altitude
𝑷 = 𝒃𝟏 + 𝒃𝟐 + 𝟐𝒔
𝑏1 = length of the upper base
(𝒃 − 𝒃 ) 𝟐
𝒔= 𝟐
𝒉 +
𝟐 𝟏 𝑏2 = length of the lower base
𝟒 s = legs
(𝒃 + 𝒃 ) 𝟐 d = diagonal
𝟐 𝟏
𝒅= 𝒉𝟐 +
𝟒
 QUADRILATERALS
Parallelogram Where A = area of the parallelogram
𝑨 = 𝒃𝒉 P = Perimeter
𝟏 h = altitude θ
𝑨 = 𝒅𝟏 𝒅𝟐 𝒔𝒊𝒏θ
𝟐 a = side
𝑷=𝟐 𝒂+𝒃 b = base
𝑑1 = longer diagonal
𝑑1 = 𝒂𝟐 + 𝒃𝟐 + 𝟐𝒃 𝒂𝟐 − 𝒉𝟐
𝑑2 = shorter diagonal

𝑑2 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒃 𝒂𝟐 − 𝒉𝟐
 QUADRILATERALS
Rectangle
𝑨 = 𝒃𝒉
𝑷=𝟐 𝒃+𝒉
𝒅= 𝒃𝟐 + 𝒉𝟐
Where A = area of the rectangle
P = Perimeter
h = altitude
b = base
d = diagonal
 QUADRILATERALS
Square
𝑨 = 𝒔𝟐
𝟏 𝟐
𝑨= 𝒅
𝟐
𝑷 = 𝟒𝒔
𝒅=𝒔 𝟐 Where A = area of the square
P = Perimeter
s = side
d = diagonal
 QUADRILATERALS
Rhombus
𝒅𝟏 𝒅𝟐
𝑨=
𝟐
𝑷 = 𝟒𝒔
𝑨 𝒅𝟏 𝒅𝟐
𝒉= = Where A = area of the rhombus
𝒔 𝟐
P = Perimeter
s = side
𝑑1 = longer diagonal
𝑑2 = shorter diagonal
 QUADRILATERALS

Isosceles trapezoid BCAT has bases of 12 and 20, BT=CA=5. What is the area
of the trapezoid?
 QUADRILATERALS

The diagonal measure of a television screen is 24 in. If the screen is a

square, what is the length of a side?
 QUADRILATERALS

How many feet of lace trim is needed to make borders for two rectangular
tablecloths each with width 4 ft. and length 6 ft.?
 QUADRILATERALS

A certain city block is in the form of a parallelogram. Two of its sides are each
421 ft. long; the other two sides are each 227 ft. in length. If the distance
between the first pair of sides is 126 ft., find the area of the land in the block,
and the length of the diagonals.
QUADRILATERALS
 QUADRILATERALS

Given a parallelogram whose diagonals are 10 and 22 and whose angle

opposite of the base is 115°, determine its area.
POLYGONS

𝑄 = Central angle
360°
Θ= , where 𝑛 = number of sides of the polygon
𝑛

Area and Perimeter of a polygon

𝟏
𝑨= 𝒂𝒃𝒏,
where a the altitude of the triangle or
𝟐
the APOTHEM of the polygon which is also the line
formed between center of the polygon and
perpendicular to its side. If the perimeter of the
polygon s given by 𝑷 = 𝒏𝒃. The area formula
𝟏
becomes 𝑨 = 𝟐 𝒂𝑷.
 POLYGONS

Other important formulas for polygon are listed below:

2
𝐸𝑎𝑐ℎ 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑝𝑜𝑙𝑦𝑔𝑜𝑛 = 1 − 180°
𝑛

360°
𝐸𝑎𝑐ℎ 𝑣𝑒𝑟𝑡𝑒𝑥 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑟 𝑝𝑜𝑙𝑦𝑔𝑜𝑛 =
𝑛

𝑆𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 = 𝑛 − 2 180° = 360°

 POLYGONS

Find the area of a regular hexagon if a side is 6 cm.

 POLYGONS

How many sides has a regular polygon if the measure of an

exterior angle N is 24°?
 POLYGONS

Determine the measure of an interior angle N of a regular

polygon with 5 sides.
 POLYGONS

Find the degree measure of a central angle of a regular polygon

with 15 sides.
 CIRCLE

Circle
𝝅𝒅 𝟐 𝑪𝒓
𝟐
𝑨 = 𝝅𝒓 = =
𝟒 𝟐
𝒅 = 𝟐𝒓
𝟐𝑨
𝑪 = 𝟐𝝅𝒓 = 𝝅𝒅 =
𝒓
Where A = area of the circle
C = circumference of the circle
r = radius of the circle
d = diameter of the circle
 CIRCLE

A circle is INSCRIBED in a polygon if the circle is tangent to all the

sides of the polygon as shown below.
 CIRCLE

A circle is CIRCUMSCRIBED about a regular polygon if the circle

contains all the vertices of the polygon.
 CIRCLE

Find the area and the circumference of a circle whose radius is 12.
 CIRCLE

The side of a square is 8 inches long. Find the circumference and

area of its inscribed and circumscribed circle.
 CIRCLE

1. Find the area of the annulus if the diameter of

the iron washer is 6 cm and the diameter of the
hole is 3 cm.
(ANNULUS is a Latin word which means RING. The
shaded portion of the figure is an annulus, whose
area is the difference between the areas of the
larger and smaller circles)
CIRCLE
 ARCS, SECTORS AND
SEGMENTS ON CIRCLES

SECTOR of a circle is the region bounded by

two radii of the circle and their intercepted
arc. Sector PQR is bounded by radii QP, QR,
and PR.
𝜽
𝑨𝒔𝒆𝒄𝒕𝒐𝒓 = (𝝅𝒓𝟐 )
𝟑𝟔𝟎°
𝜽
𝑺(𝒂𝒓𝒄 𝒍𝒆𝒏𝒈𝒕𝒉) = (𝟐𝝅𝒓)
𝟑𝟔𝟎°
 ARCS, SECTORS AND
SEGMENTS ON CIRCLES

A SEGMENT of a circle is a region

bounded by an arc and a chord of the
circle.

𝑨𝒔𝒆𝒈𝒎𝒆𝒏𝒕 = 𝑨𝒔𝒆𝒄𝒕𝒐𝒓 − 𝑨𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆

 ARCS, SECTORS AND
SEGMENTS ON CIRCLES

The radius of a circle is 18. How long is an arc of 120° sector?

 ARCS, SECTORS AND
SEGMENTS ON CIRCLES

Find the area of a segment of a circle with radius 8 in. and arc AB
measuring 45°.
 ARCS, SECTORS AND
SEGMENTS ON CIRCLES

AOD is a sector of a circle, center O, radius 10 cm. ∠AOD = 50°.

Given that AB ⊥ OD find: (a) perimeter of the whole figure and (b)
area of the shaded figure.
ARCS, SECTORS AND
SEGMENTS ON CIRCLES
ARCS, SECTORS AND
SEGMENTS ON CIRCLES
ARCS, SECTORS AND
SEGMENTS ON CIRCLES
 SIMILAR FIGURES

Two polygons are SIMILAR if their corresponding angles are

equal and their corresponding sides are proportional.
Two segments are PROPORTIONAL if there exists a positive
number k such there the length is k times the length of the other
segments.

R
 SIMILAR FIGURES

Find the common factor 𝑘 and the ratio of the length of the segments if
𝐿1 = 10 𝑎𝑛𝑑 𝐿2 = 5.
 AREA OF SIMILAR FIGURES

The ratio of the areas of two similar triangles is the square of the ratio
of any two corresponding sides.
2
𝑎∆𝐴𝐵𝐶 𝐴𝐵 𝐴𝐵 𝐴𝐵
= ∙ =
𝑎∆𝑋𝑌𝑍 𝑋𝑌 𝑋𝑌 𝑋𝑌
 AREA OF SIMILAR FIGURES

Find the ratio of the perimeter

and the area.
 AREA OF SIMILAR FIGURES

Find the ratio of the perimeter

and the area.
 AREA OF SIMILAR FIGURES

Let AC=5, XZ=10 and the area of ΔABC=6. Find the area of ΔXYZ.
AREA OF SIMILAR FIGURES
 AREA OF SIMILAR FIGURES

If area of ΔABC=4 and area of ΔXYZ=9, and YZ=5, find BC.

 CUBE
A CUBE is a prism whose six faces are all squares. Where:
SIDES of the square are EDGES of the cube. The V = volume
angles of a cube are right angles. S = lateral
area
T = total area
Diagonal: 𝐷=𝑎 3 D = diagonal
Total Surface Area: 𝑇 = 6𝑎2 B = base
Lateral Area: 𝑆 = 4𝑎2 H = height
3 3 (𝑑𝑓 )3 2 A = edge
Volume: 𝑉 = 𝑎3 = 𝐵ℎ = 𝐷 =
9 4
 CUBE

A cube has edge 8 cm, find its total surface area, diagonal and
volume.
 CUBE

Find the edge of a cube whose volume is three times the volume
of another cube whose edge is 4 cm. Find the ratio of total
surface area.
 CUBE

The diagonal D of a cube is 10 cm. Find its total surface area

and volume.
 RECTANGULAR PARALLELEPIPED

A PARALLELEPIPED is a prism whose bases are parallelograms.

A RIGHT PARALLELEPIPED is a parallelepiped whose lateral edges are
perpendicular to the base.
A RECTANGULAR PARALLELEPIPED or simply RECTANGULAR SOLID is a
prism whose six faces are all rectangles. Hence, it is a right
parallelepiped. The length l, width w, and height h are its dimensions.
 RECTANGULAR PARALLELEPIPED

Total Surface Area

𝐴 = 2 𝐿𝑊 + 𝐿𝐻 + 𝐻𝑊
Volume
𝑉 = 𝐵ℎ = 𝐿𝑊𝐻
Diagonal of Solid

𝐷= 𝐿2 + 𝑊 2 + 𝐻2
 RECTANGULAR PARALLELEPIPED

If water weighs 62.4 lbs per cu. ft., what is the weight of water filling a
can 12 in. by 12 in. by 24 in.?
 RECTANGULAR PARALLELEPIPED

A concrete wall 1 ft. thick, 15 ft. long and 4 ft. high is made using the proportion 1:2:3 (one
part cement, two parts sand and three parts gravel by volume). Find the total volume of
cement, sand, and gravel used to make the wall.
 RECTANGULAR PARALLELEPIPED

The proportion of the dimensions of a block of metal is 5:3:1. If the

volume is 1000 𝑐𝑚3 , find its total surface area.
RECTANGULAR PARALLELEPIPED
 PRISM

A PRISM is a polyhedron of which two faces are equal polygons in

parallel planes, and the other faces are parallelograms.
Prisms are named according to the shape of their bases.

A RIGHT PRISM is a prism whose edges are perpendicular to the

bases. Otherwise, it is OBLIQUE PRISM . The lateral faces of a right prism are
rectangles.
 PRISM

PROPERTIES
1. The BASE (B) are the equal parallel polygons.
2. The ALTITUDE of a prism is the distance between the two bases; it is
usually denoted by h and is measured along a line perpendicular to the
bases.
3. The LATERAL FACES of a prism are the parallelograms included between
the bases.
4. The LATERAL EDGES (e) of a prism are the intersection of the two
adjacent lateral faces; these edges are equal and parallel.
5. The DIAGONAL OF PRISM is the line segment joining any two vertices not
in the same face (noncoplanar). DIAGONAL OF A FACE is a line
segment joining any two nonadjacent vertices within a face.
 PRISM

PROPERTIES
6. A RIGHT SECTION (K) of a prism is a section formed by passing a
plane cutting all the lateral edges of a prism at right angles.
7. A REGULAR PRISM is a right prism whose bases are regular
polygons.
8. The LATERAL SURFACE (S) AREA of a prism is the sum of the areas of
its lateral faces.
9. The TOTAL SURFACE AREA of a prism is the sum of its lateral surface
area and the areas of its bases.
 PRISM

Lateral Surface Area Volume of Prism

𝑆 = 𝑒𝑃𝐾 𝑉 = 𝐾𝑒
Where: S = lateral surface area Where: V = volume
e = lateral edge K = area of right section
PK = perimeter of right e = lateral edge
section
Volume
𝑉 = 𝐵ℎ
Where: V = volume
B = are of base
h = altitude
 PRISM

An irrigation canal is 300 m long and 3 m deep. It is 5 m wide at the top and
3 m at the bottom. Find the volume excavated to make the canal.
 PRISM

The trough shown in the figure has triangular ends which lie in parallel
planes. The top of the trough is a horizontal rectangle 84 cm by 51 cm and
the depth of the trough is 41 cm.
a. How many cubic meters of water will it hold?
b. How many liters does it contain when the depth of the water is 25
cm? What is the wetted area at this condition?
PRISM
PRISM
PRISM
PRISM
 PRISM

Find the volume of a piece of wood (prism) whose ends are

equilateral triangles which was cut from a cylindrical piece of wood
with radius 10 cm and 40 cm long.
 PRISM

A lead pencil whose ends are regular hexagons was cut from a
cylindrical piece of wood, with the least waste. If the original piece
was 8” long and ½” in diameter, find the volume of the pencil.
 CYLINDER
A cylinder is a solid bounded by a closed cylindrical surface and two
parallel planes cutting the elements of the surface. The surfaces
between the parallel planes is called the lateral surface of the
cylinder.
 CYLINDER

Lateral area
𝑳 = 𝑪 ∙ 𝒉 = 𝟐𝝅𝒓𝒉
Total surface area
𝑻 = 𝑳 + 𝟐𝑩 = 𝟐𝝅𝒓𝒉 + 𝟐𝝅𝒓𝟐 = 𝟐𝝅𝒓 𝒉 + 𝒓
Volume
𝑽 = 𝑩𝒉 = 𝝅𝒓𝟐 𝒉
 CYLINDER
Find the volume of a cylinder whose diameter is 10 in. and height is 3
in.
 CYLINDER
A milk can in the shape of a right circular cylinder has an inside
diameter 14 in. and is 3 ft high. If 1 gallon = 231 cu. in., find the
capacity of the milk can in gallons.
 CYLINDER
Find the number of square feet of metal tin needed to make the sides
of 30,000 orange juice cans if each can is 6 inches tall and 3 inches in
diameter.
 PYRAMID
A pyramid is a polyhedron of which one
face is any number of sides and the other faces
are triangles which have a common vertex.
The base of the pyramid is the face
formed by the polygon. The triangular faces are
called the lateral faces. The intersections of the
lateral faces are called the lateral edges. The
common vertex of the lateral faces is the vertex
or apex of the pyramid. The segment joining the
vertex perpendicular to the base is the altitude
of the pyramid.
 PYRAMID

Lateral area
𝐿 = 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑟𝑒𝑎𝑠 𝑜𝑓 𝑓𝑎𝑐𝑒𝑠
Volume

1
𝑉 = 𝐵ℎ
3
 PYRAMID
Find the volume of the pyramid whose base is a trapezoid with bases
4 cm and 7 cm and altitude 6 cm, and the altitude of pyramid is 8
cm.
 PYRAMID
In the corner of a cellar is a pyramid heap of coal. The base of the
heap is an isosceles right triangle whose hypotenuse is 20 ft. and the
altitude of the heap is 7 ft. If there are 35 cubic feet in a ton of coal,
how many tons are there in this heap?
PYRAMID
 REGULAR PYRAMID
A REGULAR PYRAMID is a pyramid whose base is a
regular polygon, the center of which coincides
with the foot of the perpendicular dropped from
the vertex.

PROPERTIES
1. Lateral edges are equal
2. Lateral faces are congruent isosceles triangle
3. Slant height of the pyramid is the altitude of
the lateral face
4. If a regular pyramid is cut by a plane parallel
to its base then the pyramid cut off is a regular
pyramid.
 REGULAR PYRAMID
Lateral area
1
𝐿 = 𝑃𝑙
2
Where: P = perimeter of the base
l = length of the slant height

Volume
1
𝑉 = 𝐵ℎ
3
 REGULAR PYRAMID
Find the lateral area and the volume of a regular square pyramid, if
the area of the base is 100 sq. m, and the length of lateral edges is 18
m long.
REGULAR PYRAMID
REGULAR PYRAMID
 Frustum
Frustum of a pyramid (cone) is a
portion of pyramid (cone) included
between the base and the section
parallel to the base not passing
through the vertex.

𝜋ℎ 2
𝑉= (𝑅 + 𝑅𝑟 + 𝑟 2 )
3

ℎ
𝑉 = (𝑆1 + 𝑆2 + 𝑆1 𝑆2 )
3
 Frustum
A cone 12 cm high is cut 8 cm from the vertex to form a frustum with
a volume of 190cu.cm. Find the radius of the cone.
Frustum
 Frustum
A block of granite is in the form of the frustum of a regular square
pyramid whose upper and lower base edges are 3 ft. and 7 ft.,
respectively. If the height is 4 ft, find the volume of the pyramid.
Analytic Geometry

END
OF
DISCUSSION