You will need

- The value of the radius of the circle, the Central angle of the arc, the value of the constant π (considered to be 3.14)

Instruction

1

To calculate the length of arcs p you need to use the formula:

p = (2*π*R*n)/360 or p = (π*R*n)/180, where

p - length of a circular arc;

R is the radius of the circle;

n is the size of the Central angle of the arc (in degrees).

Example: Given a circle with a radius of 4 cm and an arc angle of 90 degrees. To calculate

p = (2*3.14*4*90)/360 = 6.28 see Or:

p = (3.14*4*90)/180 = 6.28 see

p = (2*π*R*n)/360 or p = (π*R*n)/180, where

p - length of a circular arc;

R is the radius of the circle;

n is the size of the Central angle of the arc (in degrees).

Example: Given a circle with a radius of 4 cm and an arc angle of 90 degrees. To calculate

**the length of**its arc, it is necessary to use the above formula:p = (2*3.14*4*90)/360 = 6.28 see Or:

p = (3.14*4*90)/180 = 6.28 see

# Advice 2: How to find the arc length of a circle

Increasingly in everyday practice necessary to solve the problem that once the seeds flipped on the mathematics lessons, but over the years, something forgotten. Find the length of the

**arc****of a circle**is one of the tasks with which a person can face in life.You will need

- calculator, the value of the number π = 3,14 , the value of the radius r and Central angle α taken from the conditions of the problem.

Instruction

1

First you need to decide the main concepts. A circle is the set of all points plane that are at a given positive distance from some given point of the plane, called the center

**of the circle**(point O). Arc - part of**circumference is**located between two points A and b of this**circle**, where OA and OV radii of this**circle**. To distinguish between these**arcs**, each of them mark an intermediate point L and M. Thus, we obtain two**arcs**ALB and AMB.2

The arc

?, and if more straight angle, 360° - ?.

**of the circle**determined by a Central angle ?. Angle with vertex in the center**of the circle**is called its Central angle. If the Central angle is less than straight angle, then its degree measure is equal?, and if more straight angle, 360° - ?.

3

So, the arc

L = ?r?/180

where ? is a numeric constant equal 3,14.

Substituting in the formula values ?, r ? and using a calculator you can easily calculate

**of the circle**defined by the radius**of the circle**r and Central angle ?. Knowing these two values, it is easy to calculate*the length***of the arc**L by the formula:L = ?r?/180

where ? is a numeric constant equal 3,14.

Substituting in the formula values ?, r ? and using a calculator you can easily calculate

*the length***of the arc**L.# Advice 3: How to calculate arc length

The need to compute the length of arc may occur during different project works. Is the development of arched ceilings, the building of bridges and tunnels, laying of roads and Railways and much more. The initial conditions for the solution of this task can be very different. In order for the most optimal way to calculate the arc length, you need to know the radius of the circle and the Central angle.

You will need

- - a sheet of paper;
- a pair of compasses;
- - the range;
- - protractor;
- - a computer with AutoCAD;
- calculator.

Instruction

1

Construct a circle with a given radius. The principles of its construction in AutoCAD the same as on the sheet of paper. Having mastered the methods of constructing various geometric figures in the classical way, you will quickly understand how this is done on the computer. The difference is that in the ordinary construction with a compass you find the center of the circle at the point where you put the needle. In AutoCAD, look in the top menu click "arc" or "Arc". Choose build in the center, start point and angle and enter the desired parameters. Mark the center of the circle as O.

2

Using pencil and ruler or a computer mouse, swipe radius. If you draw on the sheet, then with a protractor lay aside a predetermined size of the angle. For this zero mark of the protractor will align with the dot On, check the desired angle and swipe through the resulting point of the second radius. Label the angle as α. You can call it as AOW, if appropriate letters to mark the points of intersection of the radii of the circle. You need to find the length of the arc AB.

3

If the size of the angle is specified in degrees, then arc length is equal to twice the product of the radius of the circle by a factor of π and the ratio of angle α to the full size of the Central angle of a circle. He is 360°. That is, it can be found by the formula L=2πRα/360°, where L is the desired arc length, R is the radius of the circle, and α is the size of an angle in degrees. The angle can be specified in radians. Then the arc length is equal to the product of the radius by the angle, i.e. L=RA. In this case, the rest of the formula has decreased in translating degrees to radian.

4

Designers often have to calculate the arc length, the value just estimated the height of the bridge or overlap and the span length. In this case, make a drawing. The span will be the chord and the height of the part radius. Swipe it from the top of the future arch is perpendicular to the chord and continue on to the intended center of the circle. The height divides the chord in half. The center will connect with the ends of the chord, thus obtaining a 2 radius. Calculate the radius by using the Pythagorean theorem, that is, R=√a2+(R-h)2.

5

Knowing the radius and the difference between him and the tall, by theorem of sines, find the magnitude of the half angle of the sector. Sine is the ratio of the opposite leg to the hypotenuse, i.e. sinα=a/R. In the table of sines to find the size of the angle and substitute it into the formula.

Note

The two points divide the circle into two arcs. The job can be specified, the length of which one need to find. In this case, you must calculate the greater angle, and subtracting from the full angle given is acute.

# Advice 4: How to calculate the length of the curve

When calculating any length, remember that it is a finite quantity, that is just a number. If you mean the arc length

**of the curve**, such problem is solved by using the definite integral (planar case) or a curvilinear integral of the first kind (the arc length). The arc AB is denoted IAV.Instruction

1

The first case (flat). Let IAV is set to a flat

**curve**y = f(x). The argument of the function will change in the range from a to b and it is continuously differentiable this segment. Find**the length**L of the arc IAV (Fig. 1A). To solve this problem, break the reporting period into elementary segments ∆xi, i=1,2,...,n. As a result, the IAV will be divided into elementary arcs ∆Ui, plots the graph of y=f(x) at each of the elementary segments. Find**the length**∆Li of an elementary arc approximately, replacing it with the corresponding chord. It is possible to increase in to replace the differentials and use the Pythagorean theorem. After the issuance of the square root of the differential dx will get the result shown in figure 1b.2

The second case (arc IAV set parametrically). x=x(t), y=y(t), tє[α,β]. Function x(t) and y(t) have continuous derivatives on the interval this interval. Find their differentials. dx=f’(t)dt, dy=f’(t)dt. Substitute these differentials into the formula to calculate the length of the arc in the first case. Remove dt from the square root under the integral, put x(α)=a, x(β)=b and come to the formula for calculating arc length in this case (see Fig. 2A).

3

The third event. Arc IAV graph of a function given in polar coordinates ρ=ρ(φ) of the Polar angle φ during the passage of the arc changes from α to β. The function ρ(φ)) has a continuous derivative on the interval of consideration. In such a situation, the easiest way to use the data obtained in the previous step. Choose φ as a parameter and substitute in equations relating polar and Cartesian coordinates x=ρcosφ y=ρsinφ. Differentiate the formula and substitute the squares of the derivatives in the expression in Fig. 2A. After a bit of identical transformations that are based mainly on the use of trigonometric identities (cos)^2+(sinφ)^2=1 we obtain the formula for arc length in polar coordinates (see Fig.2b).

4

The fourth case (spatial curve, defined parametrically). x=x(t), y=y(t), z=z(t), tє[α,β]. Strictly speaking, we should apply the line integral of the first kind (the arc length). Curvilinear integrals calculated by translating them in ordinary certain. As a result, the practical answer remains the same as case two, the only difference is that under the root will appear in an extension term – the square of the derivative z’(t) (see Fig. 2C).

# Advice 5: How to measure the length of the arc

An arc is a portion of the circumference. The circle is the locus of points equidistant from one point called center. In everyday situations, when the error is not important and difficult to measure,

**the length****of the arc**is sometimes measured with the aid of a soft material such as yarn which follows the shape**of an arc**, and then rectified and measured. For serious measurements, this method is unacceptable.You will need

- line;
- the compass.

Instruction

1

*Find the radius*

**of the arc**of a circle. To do this, take a compass and draw a new circle at three points. Point it is advisable to choose are located far enough away from each other, therefore advisable to take the extreme point

**of the arc**and a point approximately in the center. Every two circles must intersect at two points. Swipe through these two points line. Where these two lines intersect is the center

**of the arc**of a circle. The radius is the distance from the center to any point on the circle.

2

Guide segments from the center to the extreme points of the

**arc**. They form an angle, called Central. If possible, measure it. Length**of an arc**in m of degree equal to the product of PI, radius,**arc**and m degrees divided by 180 degrees. pm=π*r*m/180.3

It may be that to measure the angle nothing. In this case, the output angle of the triangle, if possible, or use the formula of Huygens.

4

*Connect the extreme points*

**of the arc**A and B. C Find the midpoint of the segment AB. Note the arc at the middle of m. It lies on the perpendicular drawn to AB through C.

5

Calculate

**the length****of the arc**according to the formula of Huygens, having measured the required values: p≈2k+1/3*(2k-d). Here k=AM d=AB. The formula of Huygens estimated and has an error.# Advice 6: How to find square meter

To calculate the square

**meter**is not difficult. Need a mathematical formula for rectangles studying in second grade. Difficulties may arise with the calculation of the area of nonstandard shapes. For example, if we are talking about the Pentagon or a more complex configuration.You will need

- measure the sides and angles of a figure, paper, pencil, ruler, protractor.

Instruction

1

Draw desired shape on paper. Or draw a plan of the territory, which area are going to count. This will help for further calculations.

2

Divide the original shape into simple parts: rectangles, triangles or sectors of a circle. Calculate the area of the resulting parts. For rectangles, multiply the lengths of the sides S = a·b.

3

Determine the area of a triangle in any convenient way. In the General case, it can be calculated by several formulas. If there is a triangle with angles α, β, γ and opposite them the sides a, b, c, then its area S is defined as: S = a·b·sin(γ)/2 = a·c·sin(β)/2 = b·c·sin(α)/2. In other words, pick the angle whose sine is going to be the easiest to calculate, multiply the product of two adjacent sides and divide in half.

4

Use another method: S = a2·sin(β)·sin(γ)/(2·sin(β + γ). In addition, there is Heron's formula: S = √(p·(p – a)·(p – b)·(p – c)) where R is properiter triangle (p = (a + b + c)/2) and √ ( ... ) is the symbol for a square root. There are other ways. If you have a rectangular or equilateral triangle, the calculation is simplified. In the first case, use the length of the two sides adjacent to the angle 90°: S = a·b/2. In the second measure first the height of the isosceles triangle is lowered to its base. And use the formula S = h·c/2 where h is the height and length of the base.

5

Calculate the area of sector of a circle included in the figure. To do this, find the product of half the arc length of the sector to the radius of the circle. The most difficult aspect of this task is getting the correct radius value for the selected initial shape of the sector.

6

Fold the resulting square to the final result.

7

Use the triangulation method to compute the area of complex shapes like pentagons. Divide your source code into triangles. Calculate their areas and add the results.