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- Definition
- Representation
- Properties of Quadrant of Circle
- Steps to Make a Quadrant of a Circle
- Constructing a Quadrant of a Circle Practically
- Steps to Make a Circle From Its Quadrant
- Constructing a Circle From Its Quadrant Practically
- Area of Quadrant
- The Perimeter of the Quadrant
- An Example of Identifying Quadrants of Circles

**Quadrant (Circle)|Definition & Meaning**

**Definition**

If **Circle** is **decomposed** into** 4** equal **parts** then **one out of that four parts** is known as the **quadrant of the circle** or in other words,** ¼ portion of the circle** is referred to as a quadrant. There are **90-degree sectors** in each quadrant. All of the parts **contribute equally** in the area so when these parts or sectors a **joined** together a **circle is formed.**

Consider a **circle** having **two radii** **r1** and **r2** that are at **right angles** to each other (90 degrees). The **arc connecting** the two **radii** will be taken as a **quadrant** of a circle.

**Representation**

Here is the **representation** of the quadrant of a circle, the **red portion** constitutes the **quadrant** which is typically a **one-fourth portion** of a whole circle.

If we **bisect** the **circle** into **four equal parts,** then we get the following quadrants.

**Quadrant 1**

** ****ABC** is the **first quadrant** of the circle as shown in the figure below.

**Quadrant 2**

** ****AED** is the **second quadrant** of the circle as shown in the figure below.

**Quadrant 3**

** ****AFG** is the **third quadrant** of the circle as shown in the figure below.

**Quadrant 4**

** ****AKL** is the **fourth quadrant** of the circle as shown in the figure below.

**Properties of Quadrant of Circle**

- The quadrant
**must**be a**sector**of**90 degrees**. - The
**radii**that form the quadrant**must**be at the**right angle**to**each other**. - The
**area of one quadrant**must**not overlap**with any of the**other**three quadrants to maintain the 4 equal quadrants (the circle has four quadrants).

**Steps to Make a Quadrant of a Circle**

**Step 1**

** **From the center of the circle **sketch a line** to **any point** on the **circle**.

**Step 2**

** **Sketch another **line** that must be at a **right angle** to the above like drawn (90-degree).

**Step 3**

** **The **arc joining** the above **two lines** **including** the **center** and **radii** will be **referred** to as a **quadrant of the circle.**

**Constructing a Quadrant of a Circle Practically**

**Step 1**

** **Consider below the circle having a **center A**. We will **take** a **point** in a circle to let us suppose **B** now we will **join** the points **A** and **B** as shown. We can call the line segment AB as radius or** radii 1** of the circle.

**Step 2**

** **Consider another point **suppose C** on the circle but here we will **apply** the **property** of the quadrant, the **supposed point** must be in such a manner that it is at the** right angle** to the previous **point A.** **Join** given **point C** with the center of** circle A**, we get a line segment **AC,** take AC as the **second** **radii** of the circle that is the **right angle** to the line segment **AB.**

**Step 3**

** **The **arc joining** the **radii AB** and **AC** **along** with the two **radii** constitute a **quadrant of the circle** as shown in the figure.

**Steps to Make a Circle From Its Quadrant**

**Step 1**

** **Sketch two **line segments** of **equal** length.

**Step 2**

**Join** the above two **lines together** in such a way that they are at a **right angle** to each other and share a s**ingle common point.**

**Step 3**

** ****Connect** the two **uncommon points** of the line segment with an arc .

**Step 4**

** ****Repeat** the above step **three** more **times** we have now **four quadrants** we **join** these **four quadrants** to a single common point that will eventually **make the circle.**

**Constructing a Circle From Its Quadrant Practically**

**Step 1**

** **Draw two line segments **AD** and **EF** of equal length as shown in the figure.

**Step 2**

** ****Join AD** and **EF** together in such a way they **meet** at **single point A** as shown in the figure we have now segments **AD** and **AE** and the **angle** between them **is 90** degrees.

**Step 3**

** ****Connect** the points **D and E** of segments **AD** and **AE** **with** an **arc** saying** “C”** as shown in the figure.

**Step 4**

** **On **repeating** the above steps **three** more **times** we got **four quadrants AED, HIG, JKL, and NMO** we will now **join** these quadrants **together** in such a way that they share a single common **point A.** Finally, a **circle** is **constructed** and **each** quadrant is **one-fourth portion** of the whole circle.

**Area of Quadrant**

The **area** of the **quadrant of a circle** is generally equal to **one-fourth times** the **area of the circle**. Calculating a quadrant’s area **requires** knowing a **circle’s area**. We can **calculate** a circle’s area of **quadrant** by using the** formula given** below since a quadrant’s area is one-fourth of its total area.

**Area of circle** =$\pi r^{2}$

**Area of Quadrant of circle** =$\frac{1}{4} \times \pi r^{2} $

**The Perimeter of the Quadrant**

**Quadrants** of a circle **have** a **perimeter** that is **one by four times** the **circumference** and **two times** the **radius**. An **angle** between** two points** on a circle is also referred to as a **quadrant’s circumference**. The perimeter of a quadrant of a circle can be calculated by using the **given formula.**

**Perimeter** =$r(\frac{\pi}{2} +2)$

**An Example of Identifying Quadrants of Circles**

Consider the following figure to **demonstrate** **which** is the **quadrant** of a circle and **which** is **not**, and **explain** with **reasons.**

**Solution**

**In Figure A, **there are **two radii AD** and **AE**. We can see that they are **not right angles** to each other rather they are at somewhat **60 degrees** to each other, moreover **AED** is **not** **one-fourth portion** of the whole circle so we will conclude that **AED** is **not** a **quadrant** of the circle.

**In Figure B, **there are **two radii AF** and **AG**, we can visualize that these **two radii** are **bisecting** the **circle** into** two parts** moreover they are **180 degrees** so **AFG** is **not** a **quadrant** of the circle.

**In Figure C, **there are **two radii AC** and **AH**. Both of the radii are r**ight angle** or 90 degrees to each other moreover it is **constituting** exactly **one-fourth** of the **whole circle** so **ACH** is** referred** to as the **quadrant of the circle**. Generally, the figure is representing the** first quadrant** of a circle

**In Figure D, **there are **two radii AK** and **AJ** **both** are **perpendicular** to each other i.e. 90 degrees moreover they are** making one-fourth portion** of the given circle also this is generally the **third quadrant** of a circle so ** AKJ** is the quadrant.

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