Performing Shop Computations (Intermediate)

Information Sheet 2.1: Calculation of Triangle

2. An airplane is flying at an altitude of 700 m when the co pilot spots a ship in distress at an
angle of depression of 37.6°. How far is it from the plane to the ship?

Given:
Altitude
Angle of depression

Required:
Distance from the ship to airplane

Solution:
opp 700 m

sin 37.6° = --------- = ---------------
hyp distance

700 m
Distance = ---------------

sin 37.6°

Distance = 1,147.26 m

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 5

Information Sheet 2.1: Calculation of Triangle

2.1.4: Pythagorean Theorem
The Pythagorean Theorem is the most important mathematical principles. Phytagorean
theorem states that in a right triangle, the square of the hypotenuse, which is the longest side
of a right triangle, is equal to the sum of the squares of the other two sides of the triangle. This
theorem is only applicable to right triangles.

The triangle in the figure is a right triangle. Side c is the hypotenuse. Sides a and b called legs.

Using the above figures, Pythagorean Theorem can be represented by the equation
c² = a² + b²

Note: We always need to know the measurements of two sides of a triangle in order to use
the basic formula.

Rule: To find the length of the hypotenuse when the length of the altutude and base are
known, use the formula:
c² = a² + b²

Rule: To find the length of one of the legs (altitude or base) of a right triangle when the length
of the hypotenuse and the other length are known, use one of the formulas:

b² = c² - b² or a² = c² - b²

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.1: Calculation of Triangle
Example:

Given a right triangle as shown below, calculate the missing side.

Given: a = 76 mm c=?
b = 102 mm

Required: c = ?

Solution:

Using c² = a² + b²

c² = 76² + 102²

c² = 5776 + 10404
c = √ 16180

c = 127.2 mm

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 7

Worksheet 2.1.1: Calculation of Triangle
Learning outcomes:
1 Perform calculations involving right triangles.
Learning Activity:
1.2 Performing calculations involving right triangles.

Solve the following problem:

1. The angle of elevation of the sun is 28°10’, and the shadow of a flagpole on horizontal
ground is 97.3 ft. long. How tall is the flagpole?

2. An observer on top of a 213 ft. building finds that the angle of depression of an open
manhole is 38°50’. How far is the manhole from the base of the building if they are in the same
horizontal plane?

3. A flagstaff 7.5m high stands on the top of a house. From a point on the ground on which the
house stand, the angle of elevation of the top and bottom of the flagstaff of observed to be 60°
and 45° respectively. Find the height of the house?

4. A and B are both directly east of a point immediately below a balloon. They are 1735 ft.
apart and find the angles of elevation of the balloon to be 27°32’ and 58°41’. How high is the
balloon if the two observers are in the same horizontal plane?

5. In diving, a submarine makes an angle of 62°20’ with the vertical. How far must it travel to
be 385 ft. below the surface?

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Worksheet 2.1.2: Calculation of Triangle

Learning outcomes:
1 Perform calculations involving right triangles.
Learning Activity:
1.2 Performing calculations involving right triangles.

Solve the following problem:

1) Calculate x in each figure b)
a)

4.5 cm x 11 cm 6 cm
10cm x

2) Calculate y in each figure : b)
a)

5 y y
cm

7cm

10c
m
3)

a) A 9 m post is stood up so that it meets the ground at right angles. A wire is strung
from the top of the post to a peg on the ground 7 m from the base of the post. How
long (to 1 decimal place) must the wire be?

b) A 4 m ladder is placed 1 m from the base of the building. How far up the building
does the ladder reach?

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Worksheet 2.1.2: Calculation of Triangle

c) A, B, C are corners of a rectangular field. AC is a diagonal. If it takes 10 steps to go
from A to B and 9 steps to go from B to C, how many steps could you save by
walking directly from A to C?

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Information Sheet 2.2: Calculation of Triangle
Learning outcomes:
1 Perform calculations involving non-right triangles
Learning Activity:
1.1 Performing calculations involving non-right triangles

Objectives:
 Define oblique triangle
 Discuss sine law and cosine law
 Identify and explain the different cases of sine and cosine function.

2.2.1 Oblique Triangle
An oblique triangle is a triangle which does not contain a right angle or 90°. It contains
either three acute angles, or two acute angles and one obtuse angle.

Consider the given triangles below:

A triangle is uniquely determined when three parts, not angles, are known. Thus any triangle
problem may fall under any one of the following cases:

Case I Given one side and two angles
Case II Given two sides and the angle opposite one of them
Case III Given two sides and the included angle
Case IV Given three sides

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
2.2.2 Law of Sine (Case I and Case II)
In any triangle, the sides are proportional to the sine of the opposite angles

abc
---------- = ---------- = ----------

sin A sin B sin C

Proof:

Let ∆ABC be an oblique triangle, where A, B and C are all acute angles. Let CD AB and
denote CD = h.

Considering the right triangle ACD above, using the definition of sine function,

h eqn. (1)
sin A = -----; h = b sin A

b

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Information Sheet 2.2: Calculation of Triangle
while in ∆BCD

h eqn. (2)
sin B = -----; h = a sin B

a

Combining equations (1) and (2)

h=h
a sin B = b sin A

ab
---------- = ----------

sin A sin B

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Information Sheet 2.2: Calculation of Triangle

Considering the right triangle ∆ACB above, using the definition of sine function at A:

aa
sin A = -----; b = ---------- eqn. (3)

b sin A

while in C:

cc
sin C = -----; b = ---------- eqn. (4)

b sin C

Combining equations (3) and (4)

b=b

ac
---------- = ----------

sin A sin C

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Information Sheet 2.2: Calculation of Triangle
Note:

ab ac

---------- = ---------- and ---------- = ----------

sin A sin B sin A sin C

Therefore:

ab c

---------- = ---------- = ---------- General formula for Sine Law

sin A sin B sin C

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
2.2.3 Case I – Given one side and two angles
Given: a, B and C
Required: A, b and c
Solution:

1. To find A, use A = 180 – (B + C)

ab a sin B
2. To find b, use --------- = ----------or b = ----------

sin A sin B sin A

ac a sin C
or c = ----------
3. To find c, use ---------- = ----------
sin A
sin A sin C

Example 1. Solve ∆ABC

Given: c = 20, A = 30°, B = 65°

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle

Required: C, a and b 85°
Solution:

For C: C = 180 – (30° + 65°) =

c sin A 20 sin 30°

For a: a = ---------- = --------------- = 10.04

sin C sin 85°

c sin B 20 sin 65° 18.20
For b: b = ---------- = --------------- =

sin C sin 85°

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MEE722207 (Intermediate) 7

Information Sheet 2.2: Calculation of Triangle

Example 2.
A and B are two points on opposite side of a street. From A, a line AC = 270 ft. is laid off and
the angles CAB = 120° and ACB = 45° are measured. Find the length of AB.

Given:
AC = b = 270 ft.
A = 120°
C = 45°

Required:
Length of AB
AB = c

Solution:
B = 180 – (120° + 45°) = 15°

b sin C 270 sin 45°

c = ---------- = ---------------

sin B sin 15°

c = 737.65 ft.

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
2.2.4 Case II – Given two sides and an angle opposite one of them

Given:
b, c and B

Required:
C, A and a

Solution:

Ab c

1. Find C (the angle opposite to the other given side) from ---------- = ---------- = ----------

sin A sin B sin C

i. If sin C > 1 no angle is determined.
ii. If sin C = 1, C = 90°, a right triangle is determined.

iii. If sin C < 1, two angles are determined.
a.) an acute angle C and
b.) an obtuse angle C = 180 – C

Thus, there may be one or more triangles determined.

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
Example 1.

Given:
a = 3984 b = 6593 and A = 48° 20’

Required:
B, C and c

Solution:

ab
For B: ---------- = ----------

sin A sin B

3984 6593

--------------- = ----------

sin 48° 20’ sin B

6593 sin 48° 20’
sin B = -------------------------

3984

sin B = 1.24 > 1

therefore: no solution.

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Information Sheet 2.2: Calculation of Triangle
Example 2.

Given: c = 128 C = 62°
b = 80

Required:
B, A and a

Solution:
bc

For B: ---------- = ----------
sin B sin C

b sin C 80 sin 62°

B = sin – 1 --------------- = sin – 1 ---------------

c 128

B = 32.76°

For A: A = 180 – (32.76° + 62°)

A = 85.23°

b sin A
For a: a = ---------------

sin B

80 sin 85° 14’
a = --------------------

sin 32.76°

a = 147.33

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Information Sheet 2.2: Calculation of Triangle
2.2.5 Law of Cosine (Case III and Case IV)

In any triangle, the square of any side is equal to the sum of the squares of the other two sides
minus twice the product of the sides and the cosine of their included angle, i.e..

A a² = b² + c² – 2bc cos

General Formula

B b² = a² + c² – 2ac cos of

Cosine Law

C c² = a² + b² – 2ab cos

Proof:
Consider ∆BCD:

a² = h² + (b – x)² eqn. (1)
a² = h² + b² – 2bx + x²

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Information Sheet 2.2: Calculation of Triangle
Consider ∆ABD

c² = h² + x² eqn. (2)
h² = c² – x²

Substitute eqn. (2) in eqn. (1)

a² = c² – x² + b² – 2bx + x² eqn. (3)

x eqn. (4)
cos A = -----

c

x = c cos A

Substituting eqn. (4) in eqn. (3)
a² = c² + b² – 2bc cos A

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
2.2.6 Case III – Given two sides and the included angle.

Given:
a, b, and C

Required:
c, A, and B

Solution:

1. To find c, use c² = a² + b² – 2ab cos C

2. To find A, use A = cos – 1 b² + c² – a²
2bc

3. To find B, use B = cos – 1 a² + c² – b²
2ac

4. To check, use A + B + C = 180°

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
Example 1.

Given: b = 230 C = 32° 20’
a = 150

Required:
c, A and B

Solution:
For c: c² = a² + b² – 2ab cos C
c² = 150² + 230² – 2(150)(230) cos 32° 20’
c² = 17098.39
c = 130.76

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MEE722207 (Intermediate) 15

Information Sheet 2.2: Calculation of Triangle

For A: (230)² + (130.76)² – (150)²
A = cos – 1 (2) (230) (130.76)

A = cos – 1 0.79

A = 42.02°

For B:
B = 180° – (A + C) = 180° – (42.02° + 32.33°)

B = 109° 50’

Example 2.

An electric transmission line is planned to go directly over a swamp. The power line will be
supported by tower at points A and B. A surveyor measures the distance from B to C as 573
m, the distance from A to C as 347 m, and BCA as 106.63°. What is the distance from tower
A to tower B?

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MEE722207 (Intermediate) 16

Information Sheet 2.2: Calculation of Triangle

Given: AC = b = 347 m BCA = C = 106.63°
BC = a = 573 m

Required:
distance from tower A to tower B

AB = c

Solution:
c² = a² + b² – 2ab cos C

c² = (573)² + (347)² – 2(573)(347) cos 106.63°

c² = 490077.21

c = 700.06 m

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Information Sheet 2.2: Calculation of Triangle
2.2.7 Case IV – Given three sides

Let a, b, and c be given. Apply the law of cosines for each of the angles. To find the angles,
use:

b² + c² – a² a² + c² – b² a² + b² – c²
cos A = --------------- cos B = --------------- cos C = ---------------

2bc 2ac 2ab

Example 1.

Solve the triangle with the parts given:

Given: b = 34.5 c = 52.8
a = 26.4

Required:
A, B and C

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 2.2: Calculation of Triangle
Solution:

For A: b² + c² – a² (34.5)² + (52.8)² – (26.4)²
cos A = --------------- = -----------------------------------

2bc 2 (34.5) (52.8)

Cos A = 0.90
A = cos – 1 0.90
A = 25.84°

For B: a² + c² – b² (26.4)² + (52.8)² – (34.5)²
cos B = --------------- = -----------------------------------

2ac 2 (26.4) (52.8)

Cos B = 0.82
B = cos – 1 0.82
B = 34.92°

For C: a² + b² – c² (26.4)² + (34.5)² – (52.8)²
cos C = --------------- = -----------------------------------

2ab 2 (34.5) (26.4)

Cos C = – 0.49
C = cos – 1 (– 0.49)
C = 119.34°

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MEE722207 (Intermediate) 19

Information Sheet 2.2: Calculation of Triangle
Example 2.
An isosceles triangle has sides that measure 24 cm, 24 cm, and 18 cm. Find the measure of
each angle.
Given:

a = 24 cm b = 24 cm c = 18 cm

Required:
A, B, and C

Solution:

For A: b² + c² – a² (24)² + (18)² – (24)² = 0.375
cos A = --------------- = -----------------------------

2bc 2 (24) (18)

A = cos – 1 0.375
A = 67.98°

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 20

Information Sheet 2.2: Calculation of Triangle

For B: a² + c² – b² (24)² + (18)² – (24)² = 0.375
cos B = --------------- = ----------------------------

2ac 2 (24) (18)

B = cos – 1 0.375
B = 67.98°

For C: a² + b² – c² (24)² + (24)² – (18)² = 0.718
cos C = --------------- = ---------------------------

2ab 2 (24) (24)

C = cos – 1 0.718
C = 44.05°

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MEE722207 (Intermediate) 21

Worksheet 2.2.1: Calculation of Triangle
Learning outcomes:
1 Perform calculations involving non-right triangles.
Learning Activity:
1.2 Performing calculations involving non-right triangles.

Solve the following problem:

1. Calculate the unknown side, a, in each figure:

a) b)
81º 11cm 9cm 76º

62º 38º
a a

c) d)

a 7cm 42º
4cm 128
31º
º 27º a

2. In the diagram, angle PRQ = 48º, angle PQR = 65º and PQ = 6cm. Calculate the length
of PR.

P

6cm

65º
48º
RQ

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Worksheet 2.2.2: Calculation of Triangle

Learning outcomes:
1 Perform calculations involving non-right triangles.
Learning Activity:
1.2 Performing calculations involving non-right triangles.

Solve the following problem:
1. Find the unknown side, a, in each figure. Give answers correct to 3 significant figures.

11cm a 4.5cm
34 a
13cm

10cm 3cm
25
108
a 13cm

2. Calculate A in each figure. Give answers correct to the nearest degree
a. b.

A 16cm

BC

9cm 12cm 7cm
C A 4cm
B

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Information Sheet 3.1: Taper Calculation

Learning outcomes:
1 Calculate Taper
Learning Activity:
1.1 Performing calculations involving tapers

Objectives:
 Learn the units used to express taper.
 Identify the common tapers used commercially and their standard sizes.
 Learn how to calculate the taper of the commercial tapers.
 Learn how to calculate inclination.
 Learn how to calculate taper ratio.
 Learn how to calculate setting angle.
 Lear how to calculate tailstock offset

Machinists frequently need to apply their mathematical skills to tapers. Tapered shanks
(shank with one end larger in diameter than the other) have been used on tools for a long time
in machine trades. They have been used for tools such as twist drills, mandrels, end mills,
lathe centers, and chucks. Tapered shanks are used because of two very important
characteristics. First, they permit a very tight grip for driving tools, yet the grip can be easily
broken when removal is desired. Second, they provide an automatic, accurate alignment of
the tool.

3.1.1: Taper and Units Used to Express Taper

The difference in diameters, or width, of a piece at its ends is called taper. If a round piece of
work is 2 inches in diameter at one end and 1 ½ inch in diameter at the other end, the taper
would be ½ inch. If a flat key is 1 inch wide at one end and ¾ inch wide at the other end, the
taper would be ¼ inch.

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 1

Information Sheet 3.1: Taper Calculation

Taper per foot is the difference in diameters, or widths, at the ends of a piece, expressed in
inches per foot length. If the diameters of a piece of work 1 foot long are 3 inches and 2 ½
inches, the piece would have a taper of ½ inch per foot. In some shops the piece maybe said
to have a ½ inch taper per foot. In both cases we have expressed the taper in inches per foot.
Sometimes the unit of length to measure taper is the inch. In this case it would be spoken of as
the taper per inch. For example, if a piece of flatwork 6 inches long is ¾ inch wide at one end
and ½ inch at the other end, the difference in width over 6 inches would be ¼ inch. The
difference in width over 1 inch would be 1/6 of ¼, or 1/24 inch. Therefore, the taper per inch
would be 1/24 inch.
It should be understood that taper per foot is 12 times the taper per inch, or conversely, taper
per inch is 1/12 the taper per foot. For example, a round piece of work 1 foot long is 3 inches in
diameter at its large end and 2 inches in diameter at its small end. 1 inch per foot is the taper
per foot, and 1/12 inch per inch is the taper per inch. The taper per foot is 12 times the taper
per inch, and conversely, the taper per inch is 1/12 the taper per foot.

This leads to three important facts:
1. Multiply taper per inch by 12 to obtain taper per foot.
2. Divide taper per foot by 12 to obtain taper per inch.
3. Multiply taper per inch by length in inches to obtain taper in inches for the length in
question.

3.1.2: Common tapers used commercially
1. Morse Standard
2. Brown & Sharpe Standard
3. ¾ inch per foot Standard
4. Taper-pin Standard
5. Jarno series

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 3.1: Taper Calculation

Morse Standard taper for each size

No. 1 0.598 in. per foot

No. 2 0.599 in. per foot

No. 3 0.602 in. per foot

No. 4 0.623 in. per foot

No.4 ½ 0.623 in. per foot

No. 5 0.631 in. per foot

The Brown & Sharpe taper is ½ in. per foot

The ¾ in. per foot series has a taper of ¾ in. per foot in all cases

The taper-pin Standard has a taper of ¼ in. per foot

The Jarno series of tapers has a taper of 0.6 in. per foot

With the Jarno tapers, the number of the taper is the key to all the dimensions. For example,
a No. 6 taper is 6/10 in. in diameter at the small end, 6/8 in. in diameter at the large end, and
6/2 in. in length. Consequently, the number of the taper indicates the number of tenths of an
inch in diameter at the small end, the number of eighths of an inch of diameter at the large
end, and the number of halves of an inch in length.

The Brown & Sharpe taper is used in milling-machine spindles, and the Morse taper is used on
lathes and drilling-machine spindles.

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
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Information Sheet 3.1: Taper Calculation
3.1.3: Computing the Taper

Example:
1. The taper per inch of a Brown & Sharpe taper is 0.04167 in. What is the taper in 6 in.? in

12 in.?

Taper in 6 in. = 0.04167 × 6 = 0.250 in.
Taper in 12 in. = 0.04167 × 12 = 0.500 in.

2. A Morse No. 4 taper has a taper of 0.623 in. per foot. What is the taper per inch?
Taper per inch = 0.623 ÷ 12 = 0.0519 in.

3. A piece of round work is 8 in. long, 3 in. and 2-9/16 in. in diameter at the large and small
respectively. What is the taper?

Taper = 3 – 2-9/16 = 7/16 in.

7/16
Taper per inch = ---------- = 0.0547 in.

8

Taper per foot = 0.0547 × 12 = 0.656 in.

Since taper is the difference in diameters at the ends, the taper is the difference between 3 in.
and 2-9/16 in., or 7/16 in.

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 4

Information Sheet 3.1: Taper Calculation

To find the taper per inch of length, divide the taper by the length, 8 in. Thus, 7/16 divided by
the length, 8 in., equals 0.0547 in., which is taper per inch.
To obtain taper per foot, multiply the taper per inch by 12. Thus, 0.0547 × 12 equals 0.656 in.,
which is the taper per foot. In other words the taper per foot is 12 times the taper per inch.

4. What is the large diameter of a round piece of work 10 in. long if the small diameter is 1.02
in. and the taper per foot is 0.75 in?

Taper per inch = 0.75 ÷ 12 = 0.0625 in.
Taper per 10 in. = 0.0625 × 10 = 0.625 in.
Large dia. = small dia. + 0.625 = 1.645 in.

Since the taper in 10 in. of length, which is 0.625 in., represents the difference in diameters at
the ends, the large diameter equals the small diameter plus the difference in diameters (the
taper in 10 in.). If we asked to find the small diameter, we would subtract the taper from the
large diameter.

5. Find the length of the tapered piece whose diameters at the ends are 3.21 in. and 2.03 in.
The taper is a Morse No. 3.

Taper = 3.21 – 2.03 = 1.18 in.

Length = 1.18 ÷ 0.05017 = 23.5 in.

The taper per foot of Morse No. 3 is 0.602 in. Dividing 0.602 in. by 12 gives us 0.05017 in.,
which is the taper per inch. Since taper per inch is the difference in diameters at 1 in., and
since the difference in diameters of the ends of this particular piece equals 1.18 in., we find the
length by dividing the taper over the entire length by the taper over 1 in. If the difference in
diameters over one inch equals 0.05017 in., the number of inches of length required to

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 5

Information Sheet 3.1: Taper Calculation

produce a difference in diameters of 1.18 in. is the quotient of the taper divided by the taper
per inch.

6. Find the length, diameter at large end, and diameter at small end of a No. 16 Jarno taper.

taper No. 16

Length = ---------------- = ----------- = 8 in.

22

taper No. 16

Large dia. = ---------------- = ---------- = 2 in.

88

taper No. 16

Small dia. = ---------------- = ---------- = 1.6 in.

10 10

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Information Sheet 3.1: Taper Calculation
3.1.4: Calculation of Inclination

A change of height in relation to the length of a workpiece.

Ex. An inclination ratio of 1:10 means that the height of the workpiece changes by 1 mm
per 10 mm length.

The formula for calculating the inclination ratio is:

C = H:L

H H–h
C = -------- or ----------

LL
Where:

C = inclination ratio
H, h = height
L = length of inclination

H = h + ( CL )
= 15 + [( 1 × 50 ) ÷ 10 ]
= 20 mm

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 7

Information Sheet 3.1: Taper Calculation

If there is no inclination ratio given, the size “H” can still be calculated provided the angle of the
inclination is given.

For the workpiece the way of calculating the total height will be:

Given: base = 50 mm angle (α) = 5º40'; small height = 15 mm

Finding the size of h’ finding the size of H

h’ = tan α × L H = h + h'

= tan 5º 40’ × 50 = 15 + 5

= 0.0992 × 50 = 20 mm

= 4.96  5 mm

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 8

Information Sheet 3.1: Taper Calculation
3.1.5: Calculation of Taper Ratio

A change in diameters (cone) or lengths of sides (frustum of pyramid) in relation to the length.

A taper ratio of 1:5 means that the diameter of a round workpiece changes by 1 mm per 5 mm
length.

The formula for calculating taper ratio is:

C = D:L

D D–d
C = -------- or ----------

L L

Where:
C = taper ratio
D, d = diameters
L = length of taper

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 9

Information Sheet 3.1: Taper Calculation

d = D – ( CL )
d = 28  [( 1 × 40 ) ÷ 5 ]
d = 20 mm

Example:
A pivot in the form of a truncated cone with a length of 80
mm has a cone ratio of 1:8. The smaller diameter is 30 mm.
What is the larger diameter in mm?
find D

given that C = 1:8
L = 80 mm
d = 30 mm

Solution:

D–d
C = ------------

L
D = d + ( CL )
D = 30 + [ ( 1 × 80 ) ÷ 8 ]
D = 30 + 10
D = 40 mm

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 10

Information Sheet 3.1: Taper Calculation

3.1.6: Calculation of Setting Angle for Lathe Machine
Short and abrupt tapers are turned by the use of the compound rest of the lathe machine. The
compound rest is a tool rest which can be swiveled clockwise or counterclockwise. It is
graduated in degrees.

For setting the lathe we do need the size of the setting angle. This angle is half the size of the
total half angle.

The setting angle /2 must be calculated, when not already given in the drawing. This can be
done similarly to the method of calculating the inclination. Again we have to consider length
(L) as the base and h1 as the perpendicular of a rectangular triangle.

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 11

Information Sheet 3.1: Taper Calculation

40

Step1: When not already given we calculate the small diameter “d” by using the
taper ratio.

d = D  (8 × 1 mm) = 28 mm  8 mm = 20 mm

Step 2: Now we have to calculate the size of the perpendicular.

2 × h1 = D  d; h’ = (D  d)/2

h’ = (28  20) / 2 = 4 mm

Step 3: By using trigonometric we find /2:

Tan /2 = [perpendicular (h1)] / base (L)

= 4 mm / 40 mm = 0.1

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 12

Information Sheet 3.1: Taper Calculation

Step 4: The value of tan 0.1 will be looked up in the trigonometric table or
scientific calculator.

Tan 0.1 = 5  40'  = 5  40'

3.1.7: Turning Taper by Offsetting the Tailstock

Turning tapers by offsetting the tailstock is used when the taper is small and the work is not
very long.

This figure illustrates taper turning with the tailstock offset.

The formula for finding the amount of offset to turn a taper is as follows:

Offset = taper per inch × total length of work
-----------------------------------------------------

2

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 13

Information Sheet 3.1: Taper Calculation
Example:

1. Compute the offset to turn the taper shown in the figure.

taper per inch × total length of work
Offset = ----------------------------------------------------

2

Taper = 2 ½ – 2 = ½ in.

Taper per inch = ½ ÷ 14 = 0.03571 in.

Offset 0.03571 × 14
= ---------------------- = 0.25 in.

2

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 14

Information Sheet 3.1: Taper Calculation
2. Compute the tailstock offset to turn the taper shown in the figure.

Offset = taper per inch × total length of work
-----------------------------------------------------

2

Taper = 1.73 – 1.03 = 0.70 in.

Taper per inch = 0.70 ÷ 9 = 0.07778 in.

0.07778 × 15
Offset = ----------------------- = 0.5834 in.

2

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 15

Information Sheet 3.1: Taper Calculation
3. Calculate the tailstock offset required to turn No. 3 Morse tapers in the figure.

The taper of a No. 3 Morse is 0.602 in per foot.

Taper per inch = 0.602 ÷ 12
= 0.05017 in.

0.05017 × 14
Offset for 4-in. length = ----------------------- = 0.3512 in.

2

0.05017 × 14
Offset for 5-in. length = ----------------------- = 0.3512 in.

2

Code No. Performing Shop Computations Date: Developed Date: Revised Page #
MEE722207 (Intermediate) 16

Worksheet 3.1.1: Taper Calculation
Learning outcomes:
1 Calculate taper
Learning Activity:
1.2 Performing calculations involving tapers.

Solve the following problem:

1. A taper has a length of 150 mm and a diameter of 75 mm. What is the taper ratio?

2. A 60 mm long slide valve has a taper ratio 1:5. What is the diameter?

3. A shaft end with a diameter of 60 mm receives a cone with a cone-ratio of 1:5. How long
is the cone?

4. A taper has a diameter of 50 mm and is 150 mm long. What is the taper ratio?

5. The taper ratio of a 120 mm long mandrel is 1:80, and the larger diameter is 65 mm.
Calculate the smaller diameter.

6. A conical slide valve with a cone ratio of 1:6 has diameters of 70 and 75 mm. Calculate
the cone length and setting angle.

7. What is the taper per inch of a Morse No. 1 taper?
8. The taper-pin series has a taper of ¼ in. per foot. What is the taper per inch?
9. What is the taper per inch in the ¾ in. per foot series?
10. A No. 4 Morse taper has a length of 4 ¼ in. The small diameter is 1.020 in. Find the

large diameter.

Code No. Servicing Starting System Date: Developed Date: Revised Page #
ALT723307
Nov. 28, 2003 Mar 01, 2006 1