Applied geometry

Lines , Angles, And Polygons

Point: A point is the most basic geometric figure. We represent it graphically by a dot and denote it by a

capital letter such as P •

Lines

A line i.e. a straight line, is a one dimensional geometric figure consisting of points extending infinitely

far in both directions. We denote it by small letter such as 𝑙. We also denote a line by giving two points

that lie on the line, for example 𝐴𝐵 , as in the fig.

• 𝑙

𝐴 •

𝑩

Two distinct 𝑙1 lines, and 𝑙2 , will either intersect in a point or never intersect. If 𝑙1 lines, and 𝑙2 never

intersect we say that they are parallel lines and we write 𝑙1 ⃦𝑙2

If and intersect in a point, then the point and the two lines form four angles. If the four angles are

equal, that is, if they are the same size or have the same measure, we say that the lines are

perpendicular and we write 𝑙1 ⊥ 𝑙2

parallel lines perpendicular lines

Line Segment, Ray

A line segment is a straight line having a beginning and an end.

𝐴 • •𝐵

It ha s a definite length and two definite points .In figure 𝐴𝐵 is a line segment.

Ray

A ray is a part of a line which extends endlessly in one direction only.

𝐴 𝐵 .In figure 𝐴𝐵 is a ray

Angles

An angle consists of a point and two rays extending from that point. (Whenever two rays meet at a

point, angle is formed)

The point is called vertex of the angle and the two rays are called the sides of the angle.

Figure of Angle

Side B

vertex A C

Right angles

The angles formed when two perpendicular lines intersect are called right angles. The measure of a

right angle is 90°

Right angle

Acute angle, Obtuse angle, Complementary & Supplementary angles

An angle whose measure is less than90° is called an acute angle. An angle whose measure is greater

than 90° is called an obtuse angle. If the sum of the measures of two angles is 90°, we call the angles

complementary. When the sum of the measures is 180°, we call the angles supplementary.

D

acute angle obtuse angle B B

C

A D C A

In the above figure angle ACB and angle BCD are complementary and supplementary angles

Polygon

A two-dimensional geometric figure is called a plane figure. A plane figure that consists of line segments

and angles and completely enclose a region is called a polygon.

We name polygons according to the number of sides (or angles) the polygon has.

Name of Polygons

Name of polygon Number of sides (or angles)

triangle 3

quadrilateral 4

pentagon 5

hexagon 6

heptagon 7

octagon 8

Figures

Triangle, Quadrilateral, Pentagon, Hexagon

Regular Polygon

A polygon with equal sides and equal angles is called regular polygon.

Specific name of Triangles & Quadrilaterals

Name Figure Description

Right triangle a triangle with a right angle

isosceles triangle a triangle with two equal sides

parallelogram a quadrilateral with opposite sides

parallel and equal

Rectangle a parallelogram with four right angles

Trapezoid a quadrilateral with one pair of

opposite sides parallel

Pythagorean Theorem

The opposite side of the right angle is the longest side of the right triangle and is called the hypotenuse.

The other two sides are called the legs of the right triangle. According to this theorem hypotenuse

square =sum of the square other two sides. That means if c be the hypotenuse and a, b be other two

sides, then 𝑐2 = 𝑎2 + 𝑏2

b c

a

Congruency and Similarity

Congruent polygons are polygons that have exactly the same size and shape.

In congruent corresponding sides and angles are same.

For example, C F

A B D E

In triangle ΔABC and ΔDEF, if AB = DE, AC = DF, BC = EF and , , , then

ΔABC and ΔDEF are congruent.

Similarity

Similar polygons have the same shape but are not necessarily the same size. Corresponding angles of

similar polygons are equal and corresponding sides are in proportion, that is, the ratios of the

corresponding sides are equal.

A B G E

D C H F

In the above parallelograms. The corresponding angles are equal. But the corresponding sides are not

equal. And the ratios of the corresponding sides are the same.

3

2=

9

6

DA EB FC

Measures associated with polygons

Perimeter: The perimeter of a polygon is the sum of the lengths of the sides of the polygon.

Example-1: Find the perimeter P of the polygon. All angles are right angles.

4 9

5

6 8

B C

Solution: The unknown side is BC.

BC = 4+6+9=19

Therefore, the perimeter, P=19+8+9+5+6+5+4+8=64

Example-2: Find the number of yards of fencing needed to enclose a rectangular swimming pool whose

dimensions are 24ft × 50ft.Assume that the distance between the pool and the fence is to be 10 ft and

that the fence is to be rectangular.

Solution 70ft

50ft

24ft 44ft

10

We know,

Perimeter of the fence is, 𝑃 = 2𝑙 + 2𝑤 = 2 70 + 2 44 = 228 𝑓𝑡

Now, the number of yards of fencing are 228𝑓𝑡 = (228/3)𝑦𝑟𝑑 = 76𝑦𝑟𝑑

Area

Name Geometric figure formula of Area

Triangle height(h) 𝐴 =1

2𝑏𝑕

Base(b)

Rectangle width(w) 𝐴 = 𝑙𝑤

Length(l)

Parallelogram 𝐴 = 𝑏𝑕

base (b)

base(b1)

Trapezoid 𝐴 = 1/2 𝑏1 + 𝑏2 𝑕

base(b2)

side(a)

Square side(a) 𝐴 = 𝑎2

Examples

Ex-1: Find the area of the triangle with height 8.32 in. and base 11.5 in.

Solution: We know, Area of triangle is

Area of triangle is 𝐴 =1

2𝑏𝑕

Here, base b=11.5 in., height h= 8.32 in.

𝐴 = 0.5 11.5 8.32 𝑖𝑛2 = 47.8 𝑖𝑛2

The answer is given to three significant digits.

Ex-2: Find the height of a triangle with base 5.0 ft and area 19 ft2 .

Solution: Given, A=19 ft2 ,b=5.0 ft ,

We know,

𝐴 =1

2𝑏𝑕 ⇒ 𝑕 =

2𝐴

𝑏

So, 𝑕 =2×19

5= 7.6 𝑓𝑡

General Result

If two polygons are similar, then the ratio of their perimeters is equal to the ratio of any two

corresponding sides. The ratio of the areas of similar polygons is equal to the ratio of the squares of any

two corresponding sides.

Circle

A circle consists of all the points that are given distance from a fixed point. The fixed point is called the

center of the circle. The distance between the center and the points on the circle is called the radius of

the circle. The line segment that connects two points on a circle and passes through the center is called

the diameter of the circle. The line segment joining any two points on a circle is called a chord of the

circle. A line that touches the circle at exactly one point is called a tangent of the circle.

Tangent line

diameter center

Inscribed

A circle is said to be inscribed in a polygon if all the sides of the polygon are tangents of the circle. We

also say that the polygon is circumscribed about the circle. Circle O is inscribed in quadrilateral ABCD.

Or, quadrilateral ABCD is circumscribed about the center.

A B

D C

Circumscribed

A circle is said to be circumscribed about a polygon if the polygon is contained within the circle and each

vertex of the polygon lies on the circle, that is, if the sides of the polygon are chords of the circle. We

also say that the polygon is inscribed in the circle. In figure, circle O is circumscribed about quadrilateral

ABCD. A B

D C

Circumference

The circumference of a circle is the distance around the circle, that is, the length of the curved line that

forms the circle. The formula for the circumference C of a circle is 𝐶 = 2𝜋𝑟 = 𝜋𝑑 ,where r is the radius

of the circle. (d=2r )

The area of a circle is 𝐴 = 𝜋𝑟2

Where r is the radius of the circle.

Example

Find the circumference of the circle whose radius is .50 in.(𝜋 = 3.14)

Solution: 𝐶 = 2𝜋𝑟 = 2 3.14 0.50 𝑖𝑛 = 3.14𝑖𝑛.

Example

Find the area of a circle whose diameter is 5.0 𝑓𝑡

Solution: We know, 𝐴 = 𝜋𝑟2 = 3.14 2.5 2 = 19.625𝑓𝑡2

Measures associated with solids

A three dimensional geometric figure is called a solid. Some specific solids that we will consider are

prisms, circular cylinders, circular cones, and spheres.

A prism is a solid bounded by polygons, two of which are identical and parallel, polygons, and the

remainder of which are parallelogram. The identical and parallel polygons are called the bases of the

prism and the parallelograms are called the sides of the prism. The height of the prism is the

perpendicular between the parallel bases. If the sides of a prism are perpendicular to the bases, then

the sides are rectangles and we call the prism a right prism.

Figure of Prism

Triangular bases Prism

base

side side

side

base

Area of the prism

The surface area of a right prism is the sum of the areas of the bases and the sides. The units involved in

surface area are square units.

32121 RRRbbs AAAAAA

Example

Find the surface area of the right prism for the following figure.

6 cm

4 cm 3 cm

Solution:

The width of the invisible rectangular surface = 42 + 32 = 5

So the area of the invisible rectangular surface= 5 × 6 𝑐𝑚2 = 30𝑐𝑚2

Therefore, the area of the rectangular surfaces = 30 + 24 + 18 𝑐𝑚2 = 72𝑐𝑚2

Also the area of the triangular surfaces= 2 1

2× 4 × 3 = 12𝑐𝑚2

∴ The surface area = 72 + 12 𝑐𝑚2 = 84𝑐𝑚2

Circular cylinder

A circular cylinder (or just cylinder) is a solid consisting of two identical and parallel circles as bases and a

curved surface joining the bases. If the curved surface is perpendicular to the bases we call the cylinder

is a right cylinder. The height of a cylinder is the perpendicular distance between the bases.

Figure

Cylinder

h

Area of Right Circular Cylinder

The surface area of a right circular cylinder is given by the formula , bA where is the

circular base, bC is the circumference of the base, and is the height of the cylinder. Also 2rAb ,

∴

Example

Find the surface area of a right cylinder with radius m25.1 and height m50.3 14.3 .

Solution:

Given, 50.3h mr 25.1 rhrAs 22 2 50.325.114.3225.114.322 2875.37

23.37 m

Circular Cone

A circular cone is a solid consisting of a circular base and a curved surface that comes to a point called

the vertex of the cone. The height of the cone is the perpendicular distance from the vertex to the base.

If the perpendicular distance from the vertex to the base meets the base at the center of the circle, then

the cone is a right circular cone.

Figure

Cone

h

Right circular Cone

Example: Find the surface area of a right circular cone with radius cm5.12 and height cm16 .

Solution: We know,

Surface area of a right circular cone is 222 hrrrAs

222165.125.1214.35.1214.3

5547.1287 2cm

hCAA bbs 2

rCb 2

rhrAs 22 2

Sphere

The surface area of a sphere is given by𝐴𝑆 = 𝜋𝑑2 = 𝜋 2𝑟 2 = 4𝜋𝑟2

Example: Find the surface area of a sphere with diameter 5𝑓𝑡

Solution: 𝐴𝑆 = 𝜋𝑑2 = 3.14 5 2 = 78.5𝑓𝑡2

Volume of a solid

Right prism

Right circular cylinder,

Right circular cone

Sphere

Examples

Find the Volume of the following right prism

6ft

4 ft 3 ft

Solution:

We know,

Now

hAV b

hrV 2

hrV 2

3

1

3

3

4rV

hAV b

26435.02

1ftbhAb

33666 fthAV b

Example of cylinder

Example: Find the volume of right circular cylinder with radius 5 ft and height 12 ft.

Solution:

Given, r=5, h= 12

So,

=3.14(5)2 (12)=942 ft3

Example :Find the volume of right circular cone with radius 6.5 and height 8 ft .

Solution: Given, r=6.5 ft, h=8 ft

We know,

=1

3 3.14 6.5 2 8 = 353.8 𝑓𝑡3

Example: Find the volume of the sphere with radius 9.2 ft .

Solution:

Given, r=9.2ft

We know, ∴ 𝑉 =4

3 3.14 9.2 3 = 3260𝑓𝑡3

hrV 2

3

3

4rV

3

3

4rV

Exercise -3

1. A rectangular strip of steel is 9.00 inches wide and 6.50 feet long. Find the area of the strip in square

feet.

2. A rectangular platform is 24 feet long and 14 feet wide. Find the area of the platform

3. A school shop 10.0 meters wide and 14.0 meters long is to be built. Allow 5.0 square meters for each

workstation. How many workstations can be provided?

4. The bottom of a rectangular carton is to have an area of 2400 square centimeters. The length is to be

one and one-half times the width. Compute the length and width dimensions.

5. The area of a trapezoid is 376.58 square centimeters. The height is 16.25 centimeters, and one base is

35.56 centimeters. Find the other base.

6. Find the area of the sheet metal piece in the following Figure

7. The plot of land shown in the following Figure has an area of 6350 square meters. Find the distance x.

8. The section of land shown in the following Figure is to be graded and paved. The cost is $10.35 per

square yard. What is the total cost of grading and paving the section? [≈ $6960]

9. Find the area of an isosceles triangle with base 6 cm and equal sides of length 5 cm

10. Find the circumference of a circle that is inscribed in a square with side 12 cm .

11. Find the area of a circle that is circumscribed about a regular hexagon with 4 in sides.

12. Find the volume of a cylinder with a base area of 30.0 square centimeters and a height of 6.0

centimeters.

13. A spherical gas storage tank, which has a 96.4-foot diameter, is to be painted. Compute the surface

area of the tank to the nearest hundred square feet. Compute, to the nearest gallon, the amount of paint

required. One gallon of paint covers 530 square feet. [≈ 29,200𝑠𝑞𝑓𝑡, 55𝑔𝑎𝑙]

14. A spherical storage tank has an 80.0-foot diameter. The storage tank will be repainted, and the cost of

preparation, priming, and applying a finish coat of paint is estimated at $0.20 per square foot. Compute

the total cost of repainting the tank.

15. A stainless steel ball bearing contains balls that are each 1.80 centimeters in diameter. Find the

volume of a ball. Find the weight of a ball to the nearest gram. Stainless steel weighs 7.88 grams per

cubic centimeter.

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