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# Arc|Definition & Meaning

## Definition

An **arc** may be defined as a **piece** or segment of a **circle’s circumference.** In other words any **curved path** or **segment** which may be fitted on a circle is called an arc.

The concept of **arc** is a is a very **fundamental** idea commonly used in **geometry.** An arc is any curved surface or path. Arcs may be associated with all **geometrical shapes** containing some curved or arched **boundary. **

Commonly used arcs may belong to circles and **parabolas.** Both of these examples hail from a class of second degree **polynomials.** However, the general idea or arc is not limited or restricted to any particular standard **geometrical** shape. Its just a **curved path** (any curved path).

## Explanation of Arc With Visual Intuition

The **core idea** of an arc can be understood with the help of **examples.** Following figure shows a **circle** and highlights an example of arc.

Figure 1: Arc of a **Circle**

It can be seen that an arc of a circle is nothing but a piece or **segment of its circumference** which is shown in **blue** **color.** Following figure shows another example of an arc but here we use **parabola** for the demonstration.

Figure 2: Arc of a **Parabola**

In the above figure the **red** **solid line** shows the arc whereas the **blue dotted line** represents the total path of the **parabola.** The concept of arc is not limited to the **circles or parabolas.** In fact in its general form, an arc can be along **any curve.** Following figure shows an arc along a **higher degree polynomial.**

Figure 3: Arc of a **Higher Order Polynomial**

In the above figure the **red** **solid line** shows the arc whereas the **blue dotted line** represents the **higher order polynomial.** The striking similarity between above curve and the **sinusoidal function** is intentional. This is just to show that the same concept can be extended to **any mathematical function** that contains some **curved path.**

Now that we have understood the basic concept of an arc in reference to **geometry,** we can now delve into its **different types.** Lets consider the case of a **circle.** A circle may have **three types** of arcs. If the arc covers the less than half of the total circumference of the circle then its called a **minor** **arc.** If an arc covers more than fifty percent of total circumference of the circle, its called a **major arc.**

The borderline case where the arc traverses exactly 50 percent of half of the total circumference is commonly termed as a **semi circle.** A semicircle, understandably is just a **half circle** which is actually an arc.

In many geometry problems it may be necessary to find the **length of an arc.** Let us consider the simplest case of a **circle** as shown in the figure below:

Figure 4: Arch Length of a **Circular Arc**

The segment shown in blue is an **arc of length** equal to **L**. This arc makes an angle of **θ degrees **at the center of the circle. Now to find the **arc length,** we can use the following formula:

**L = r θ**

Where, **r** is the **radius** of the circle.

Arcs are very common in **daily life.** A **rainbow,** a curved **road,** a bowl, the path of a **rocket** etc. are all examples of the arc. The formula presented above, can be used to solve many types of **numerical** **problems.** Suppose a **rocket** is fired at some angle with the horizon. The path along which the rocket is moving is of a **parabolic shape.**

Now if you wanted to calculate the distance traveled by the rocket, you would need to calculate the length of the arc or the curved path along which the rocket had moved. Similarly if you were traveling in a **car** along a curved road, you would use the **same concept** of the arc lengths to evaluate the distance.

For **larger distances** such as the ones used in **astronomy,** the concept of arc length may be used to find the **size of an object.** This method is used to find the size of **Sun** and the **Moon.** In this case one simply needs to measure the angle that the body generates at the eye of the observed and the distance of the object from the observer.

Once you have these two things, using the above formula, size L of the object can easily be calculated. The **converse formula** can also be used where the size of the object is known and we need to find the distance to that object. This calculation may be used to find the **distance to objects** on earth surface since we know the sizes of most of the objects on Earth.

## Solving Numerical Problems Related to Arc Length

Lets now introduce the formal **method of calculating an arc length.**

– **First step** is to see what data is given. Normally two out of the three parameters **( L, r & θ )** are given.

– **Secondly** find the required quantity by using the formula **L = r θ** directly or after some manipulation ( i.e.** L / θ = r** or **θ = L / r )**

– It should be noted that the value of the angle used in this formula is in **radians.** So you may have to convert the angle from **degrees to radians.**

**(a)** Suppose a **car is moving along a curved road**. As it moves along this path, it covers a distance of **10 meters**. If the circular angle covered by the car is **180 degrees**, find the radius of the circular path.

**(b)** Suppose **a rocket was launched from the ground** and it followed a circular trajectory during its flight before hitting the ground. Given that the **maximum height** of the rocket was **1000 meters**, find the length of its trajectory.

**(c)** A** bicycle covered a distance of 0.5 meters.** If the radius of its wheel is **1 meters** how much did the wheel turn (in degrees).

### Solution

(a) Given that:

**L = 10 meters**

and

**θ = 180 degree = π radians**

Using the formula:

**r = L / θ = 10 / π = 3.18 meters**

(b) Given that:

**r = 1000 meters**

and

**θ = 180 degree = π radians**

Using the formula:

**L = r θ = 1000 x π = 3141.6 meters**

(c) Given that:

**L = 0.5 meters** and** r = 1 meter**

Using the formula:

**θ = L / r = 0.5 / 1 = 0.5 radians = 28.65 degrees**

*All mathematical drawings and images were created with GeoGebra.*