# Mutua’s Formula: the Land Surveying Area Solver

**Topics:**Cartesian coordinate system, Hypotenuse, Triangle

**Pages:**16 (1578 words)

**Published:**August 24, 2013

MUTUA’S FORMULA: THE LAND SURVEYING AREA SOLVER

SAMUEL FELIX MUTUA, MUTUA NICHOLAS MUTHAMA, MUTUA NICHOLAS MUNG’ITHYA Abstract. Land surveying has been a challenge for many years both in developed and developing countries. For instance the triangular survey consumes a lot of time as several sides of the field need to be measured. Solutions to this have been reached by the discovery of M utua’s formula of finding the area of a triangular shape.

1.0 Introduction In future the time spent on land surveying will be reduced by half, based on the current work to present a new formula of finding the area of triangular land. We detail the derivation of this new formula as follows: 1.1 A brief description of previous work. The book named, “Journey through Genius” motivates us very much. Severally, over the past we have tried to come up with new formulae which could ease the complex arithmetic algorithms encountered while executing results for just simple arithmetic problems which mathematicians commonly face in their life. In June (2012) we tried to extend the reduction formula for evaluating the integrals of + and m (is a real number other than zero (0)) by the form x n e mx dx where n expressing it in terms of factorials. This seemed not to bear fruits as the modification made had no much contribution (as advised by peers).

Our first attempt triggered off further efforts; further research was done to solve future surveying problems. It was not until on June 2013 while studying on curvilinear coordinates that we were prompted with an idea which enabled us to discover that the area of a triangle could be found by a relation connecting area to length of one side and trigonometric ratios of two angles emanating from the line considered. The knowledge of Vector analysis, equation of straight lines and calculus all combined proved to be helpful in our formula derivation. 2.0 NEW FORMULA OF FINDING THE AREA OF A TRIANGLE 2.1 PRELIMINARIES _______________ Key words and phrases: Mutua’s formula, Land surveying © 2013 Science Asia 1/8

2/8 SAMUEL FELIX MUTUA, MUTUA NICHOLAS MUTHAMA, MUTUA NICHOLAS MUNG’ITHYA

Consider the triangle ordinates x 1 , y 1 ,

A1 A 2 A 3

on the first quadrant of Cartesian plane with co-

x 2 , y 2 , x 3 , y 3 respectively. If the vertices A1 x 1 , y 1 , A2 x 2 , y 2 , A3

x 3 , y 3 are arranged in anticlockwise manner as shown in the figure below.

y-axis A3 x 3 , y 3

A2 x 2 , y 2

A1 x 1 , y 1

x-axis

0

Figure 1: Triangle

A1 A 2 A 3

on the first quadrant of Cartesian plane

A 1 MPA x3

3

Area of A 1 A 2 A 3 Area of trapezium 1 2 1 2 1 2 1 2

Area of trapezium

A 3 PNA

2

Area of trapezium

A MNA

1

2

x 3

x 1 y 1 y 3

1 2

x 2

y 2

y3

1 2

x 2

x 1 y 1 y 2

x 3 y 1 x 2 y 3

1 1

x1 y1 x 3 y 3 x1 y 3 x 2 y 2 x 2 y 3 x 3 y 2 x 3 y 3 x 2 y1 x 2 y 2 x1 y1 x 2 y 2 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1 __________ 1 x3 y3 A3

__________

__________

(1 )

x1 y1

x2 y2

This gives a positive determinant if the points A 1 , A 2 , manner.

are taken in anticlockwise

Using this result we consider a general case of triangle length A 1 A 2 a units

A1 A 2 A 3

and given that

, angles A 3 A 1 A 2

1 , A1 A 2 A 3 2

as shown below.

3/8 MUTUA’S FORMULA

A3

A1 If we let line origin i.e.

y axis

a

A1 A 2

A2

x

lie on the positive

A 2 will

-axis of the Cartesian plane with point which can be illustrated as:

A1

at the

A1 0 , 0

then

be at

A 2 a ,0

A3 x 3 , y 3

1

A1 0 , 0

a

2

A 2 a ,0

x

-axis

The gradient of the line

y tan 1 x

A1 A 3...

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