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# Subtended Angle|Definition & Meaning

## Definition

A **subtended** **angle** represents the **angle** cast by an **object** onto some viewpoint. Suppose a person is looking at a building. The top and bottom of the building form two **lines** to the person’s eyes, and the **angle** between these **lines** is the **angle** **subtended** by the building onto the person’s eyes. Thus, the **object** (building) subtends an **angle** onto the viewpoint (eyes).

## What Is a Subtended Angle?

A simple definition of **subtended** **angle** would be that it is an **angle** between two points. A more practical explanation is that if an **object** is placed at a **distance** from the **viewer**, then the **lines** that pass through the corners of that **object** will make an **angle** such that the measure of the **angle** depends on the **distance** between the **object** and the **viewer**.

If the shortest possible **distance** between the two increases, the **subtended** **angle** will tend to decrease, whereas it will increase if the **distance** keeps on decreasing.

In geometrical terms, such an **angle** can be formed by an **arc**, a certain **segment**, or any other curve that contains at least two corners or endpoints. The **line** segments that join the two endpoints of an **arc**, **segment**, or curve are sometimes called rays of an **angle**.

The **object** that is contained inside these rays of an **angle** is thought of as the subtension of that **subtended** **angle**. Additionally, we will occasionally argue that **angle** intercepts or encapsulates a curve.

The definition of **subtended** **angle** alters with the context of the position. For example, you can say an **angle** made by an **arc** is **subtended** even if the vertex of the **angle** lies in the center of the **circle**.

## What Is an Angle?

An **angle** is a measurement of a shape that is made by two **lines** going away from a shared point. This shared point is known as the vertex.

An **angle** is represented by a unique symbol, $\angle$, whereas the **angles** are measured in degrees or radians.

It consists of 4 building blocks,

#### Vertex

The common point of the edge of an **angle** where the two rays or straight **lines** meet. Usually, the vertex is represented by the symbol ‘O’, as shown in the figure.

#### Arms

The two bars that extend from a common or shared point are known as arms. They form the two sides of an **angle**. In the above figure, OA and OB form the arms of an **angle**.

#### Reference

A straight **line** that is used as a reference for drawing an **angle** is known as a reference side or reference **line**. Here OB is the reference **line**.

#### Terminal

The ray that is used as a rotating arm for measuring the **angle** is known as the terminal **line**. This terminal ray decides how much the **angle** is formed. Here OA is the terminal **line**.

## How To Find an Angle Subtended by an Arc?

Subtended **angles** are any of the several **angles** that may be created by **merging** the **arcs’** ends in a **circle**. These arcs may be used to generate a variety of **angles**, including **angles** that are inside a similar section, **angles** that are within a curve, **angles** that intersect with the ring, etc.

An **arc** is a portion of a **circle** that is extracted from its **circumference**. The **angle** of that **arc** would be formed when two straight **lines** pass through the endpoints of the **arc** and meet at a shared point. As shown in the following figure, an **arc** is located on the **circumference** of the **circle**, and two distinct **angles** are formed from two different common points.

Finding the **angle** that the **arc** subtends is based on a theorem that relates the **angles** formed from the center of the **circle** and at any other point on the **circle**‘s circumference.

The theorem says that the **angle** formed at the center of the **circle** is two times the **angle** that is formed anywhere on the circumference.

In the above figure, the same **arc** subtends two different **angles**, $\angle$AOB at the center and $\angle$ACB, at the tangent point C.

If we draw a perpendicular **line** from point C and point O, it will meet the other side of the **circle** at point D, thus making two triangles $\triangle$OAC and $\triangle$OBC.

In both triangles, one **angle** and one side are concurrent. Thus using the exterior **angle** **theorem**, we can prove that,

$\angle$AOD = 2 x $\angle$ACO

$\angle$DOB = 2 x $\angle$OCB

Adding the above equations gives:

$\angle$AOD + $\angle$DOB = 2 x ($\angle$ACO + $\angle$OCB)

$\implies$ $\angle$AOB = 2 x $\angle$ACB

This proves that any **subtended** **angle** on the circumference will be **one-half** of the **angle** formed at the center.

## Subtended Angles of Different Objects

### Moon

If a **person** looks up to a moon from the surface, the **angle** that is formed is 0.54$^{\circ}$. Since the moon’s orbit is not a perfect circular movement, the **angle** of **subtension** can vary a little but not too much.

This can be easily **visualized** if you hold your thumb out with your arm stretched, making a full round of the **moon**. This is because the thumb will subtend a larger **angle** than your eyes, even at a **distance** of about 380,000 km.

### Sun

On the other hand, the sun will subtend an **angle** of 0.52$^{\circ}$ to an **observer** on the **surface** of the earth. This strikes an alarming situation since it is almost the same as that of the moon. Now you know why during a solar **eclipse**, the sun is almost fully covered by the moon.

So this proves that the **object** that is closer to the eyes will subtend a larger **angle** than the **object** that is farther away.

Because both the sun and **moon** subtend almost at a similar **angle**, the tiny moon covers up the gigantic sun, despite a huge size **difference** between the two. The glowing ring formed during the eclipse is due to the glowing gases.

The sun is about 3 times away from the earth as **compared** to the **moon**.

## Solved Examples of Subtended Angles

### Example 1

Find x in the following figure.

### Solution

The theorem states that:

$\angle$BOC = 2 x $\angle$BAC ……. (1)

$\angle$BOC = 2x

$\angle$BOC + $\angle$BOA + $\angle$AOC = 360$^{\circ}$

$\angle$BOC + 120 + 90 = 360

$\angle$BOC + 210 = 360

$\angle$BOC = 360-210

$\angle$BOC = 150$^{\circ}$

Now finding the value of $\angle$BAC, which will **result** in the value of **x.**

From (1)

$\angle$BOC = 2 x $\angle$BAC

150 = 2$\angle$BAC

$\angle$BAC = 150 / 2

$\angle$BAC = 75$^{\circ}$

So the x comes out to be 75$^{\circ}$

### Example 2

Find the value of **x.**

## Solution

Since,

OA = OB = OC (all are **radii** of the **circle**)

$\angle$OCA = $\angle$OAC = 30$^{\circ}$

$\angle$OBA = $\angle$OAB = 25$^{\circ}$

$\angle$BOC = 2 $\angle$BAC

$\angle$BAC = $\angle$BOA + $\angle$OAC

= 25 + 30

= 55$^{\circ}$

So,

$\angle$BOC = 2 x 55$^{\circ}$ = 110$^{\circ}$

So y is equal to 110$^{\circ}$.

*All images are created using GeoGebra.*