Basic Geometrical Ideas Ex 4.6: Circles and Their Parts

Understanding the Circle

Imagine you have a post fixed in the ground and you tie a rope to it. Now, hold the other end of the rope and walk around the post, keeping the rope tight. The path you trace on the ground is a perfect circle! In geometry, a circle is a collection of all the points in a plane that are at a fixed distance from a fixed point. That fixed point is called the center of the circle, and the fixed distance is called the radius. Every point on the edge of the circle is exactly the same distance away from the center. This simple rule creates a perfectly symmetrical shape. A circle also divides the plane into three parts: the interior (inside the circle), the circle itself (the boundary), and the exterior (outside the circle).

Key Parts of a Circle

Center

The fixed point in the middle of the circle from which all points on the boundary are at an equal distance.

Radius

A line segment that connects the center of the circle to any point on the circle. Plural is 'radii'.

Diameter

A line segment that passes through the center of the circle and has its endpoints on the circle. It is the longest chord of the circle.

Chord

Any line segment whose two endpoints lie on the circle. The diameter is a special type of chord.

Circumference

The total distance or length of the boundary of the circle.

Arc

A part of the circumference of the circle. Think of it as a curve between two points on the circle's edge.

Sector

The region inside a circle enclosed by two radii and the arc between them. It looks like a slice of pizza!

Segment

The region inside a circle enclosed by a chord and the arc it cuts off. It looks like the crust part of a pizza slice.

A Step-by-Step Guide to Identifying Parts of a Circle

1. Step 1: Find the Center — Look for the single point at the very middle of the shape. Let's call it point 'O'. This is the heart of the circle.
2. Step 2: Identify a Radius — Draw a straight line from the center 'O' to any point on the edge of the circle, say point 'P'. The line segment OP is a radius. You can draw countless radii in a circle, and they will all be the same length.
3. Step 3: Spot the Diameter — Draw a straight line that starts at one point on the circle's edge, passes through the center 'O', and ends on the opposite edge. Let's call this line segment 'AB'. AB is the diameter. Notice that it's made of two radii (AO and OB).
4. Step 4: Find a Chord — Draw any straight line connecting two different points on the circle's edge, say points 'X' and 'Y'. The line segment XY is a chord. Remember, the diameter 'AB' is also a chord, and it's the longest possible one!
5. Step 5: See the Arc, Sector, and Segment — The curved part of the circle's edge between points X and Y is an arc. Now, look at the area formed by two radii (like OA and OP) and the arc between A and P. This 'pizza slice' is a sector. Finally, the area enclosed by the chord XY and the arc XY is a segment.

Exam Tips: Key Relationships in a Circle

Pay close attention to these relationships as they often appear in questions!

• Diameter and Radius: The most important formula is Diameter = 2 × Radius. If you know one, you can always find the other. For example, if the radius is 5 cm, the diameter is 2 × 5 = 10 cm.
• Diameter as the Longest Chord: Always remember that the diameter is the longest possible chord you can draw in a circle because it passes through the center.
• All Radii are Equal: Any line you draw from the center to the edge of the same circle will have the same length. This is the fundamental property of a circle.

Practice Questions with Solutions

• Q: In a circle with center O, points A, B, C, D, E, F are on the circle. OA is a radius, AB is a chord, CD is a diameter passing through O. The region formed by radii OE, OF and arc EF is a sector. The region formed by chord AB and arc AB is a segment. Name: a) A radius b) A diameter c) A chord (other than diameter) d) A sector e) A segment A: Step 1: Recall definitions: Radius connects center to circumference. Diameter is a chord passing through center. Chord connects two points on circumference. Sector is region by two radii and arc. Segment is region by chord and arc. Step 2: Identify from the description: a) Radius: OA b) Diameter: CD c) Chord (other than diameter): AB d) Sector: Sector EOF e) Segment: Segment formed by chord AB and arc AB Final answer: a) OA b) CD c) AB d) Sector EOF e) Segment AB
• Q: Draw a circle with center P and radius 3 cm. Mark the following: a) A point Q on the circle b) A point R in the interior of the circle c) A point S in the exterior of the circle A: Step 1: Draw a point P as the center. Step 2: Using a compass, open it to 3 cm. Place the compass needle on P and draw a circle. Step 3: Mark any point Q on the boundary of the circle. Step 4: Mark any point R inside the circle (distance from P < 3 cm). Step 5: Mark any point S outside the circle (distance from P > 3 cm). Final answer: A circle with center P, radius 3 cm, and points Q (on), R (interior), S (exterior) marked accordingly.
• Q: If the radius of a circle is 7.5 cm, what is its diameter? A: Step 1: Understand the relationship between radius and diameter. The diameter of a circle is twice its radius. Step 2: Apply the formula: Diameter = 2 × Radius. Step 3: Substitute the given radius: Diameter = 2 × 7.5 cm. Step 4: Calculate the product: Diameter = 15 cm. Final answer: The diameter of the circle is 15 cm.
• Q: A circle has its center at point O. Its radius is 6 cm. Point P is 5 cm away from O. Point Q is 6 cm away from O. Point R is 7 cm away from O. State whether each point lies inside, outside, or on the circle. A: Step 1: Understand the conditions for a point to be inside, outside, or on a circle. Step 2: If distance from center < radius, point is inside. For P: 5 cm < 6 cm, so P is inside. Step 3: If distance from center = radius, point is on the circle. For Q: 6 cm = 6 cm, so Q is on the circle. Step 4: If distance from center > radius, point is outside. For R: 7 cm > 6 cm, so R is outside. Final answer: P is inside the circle. Q is on the circle. R is outside the circle.

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