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Practical Geometry - Construction of a Circle

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A circle is a collection of all points in a plane that are at a constant distance from a fixed point. The fixed point is called the center (OO), and the constant distance is called the radius (rr).

A circle showing the center O and the radius r.
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To construct a circle of a given radius (e.g., 4 cm4 \text{ cm}), open the compass using a ruler so that the pointer and the pencil lead are 4 cm4 \text{ cm} apart.

A ruler measuring 4cm to set the compass width.
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The center of the circle is marked with a sharp pencil as a point. The metal pointer of the compass is placed exactly on this point while the pencil traces the circumference.

An incomplete circle being drawn around a fixed center point.
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Concentric circles are circles that share the same center point but have different radii. To draw them, keep the compass pointer at the same spot but change the opening for each new circle.

Two concentric circles with the same center O.

📐Formulae

d=2×rd = 2 \times r

r=d2r = \frac{d}{2}

Diameter=2×Radius\text{Diameter} = 2 \times \text{Radius}

Radius=Diameter2\text{Radius} = \frac{\text{Diameter}}{2}

💡Examples

Problem 1:

Construct a circle of radius 3.5 cm3.5 \text{ cm} using a ruler and compass.

Solution:

Step 1: Open the compass and use a ruler to set the distance between the metal pointer and the pencil tip to exactly 3.5 cm3.5 \text{ cm}.\nStep 2: Mark a point OO on the paper to serve as the center of the circle.\nStep 3: Place the metal pointer of the compass on point OO.\nStep 4: Turn the compass slowly to draw the full curve, ensuring the pointer does not shift from OO. The resulting shape is the required circle with r=3.5 cmr = 3.5 \text{ cm}.

Explanation:

The construction relies on keeping the distance between the pointer and pencil constant to ensure every point on the boundary is exactly 3.5 cm3.5 \text{ cm} from the center OO.

Problem 2:

If the diameter of a circle is 10 cm10 \text{ cm}, find its radius and explain how to adjust the compass to draw it.

Solution:

Step 1: Use the formula r=d2r = \frac{d}{2}. Given d=10 cmd = 10 \text{ cm}, the radius is r=10 cm2=5 cmr = \frac{10 \text{ cm}}{2} = 5 \text{ cm}.\nStep 2: To draw this circle, place the metal tip of the compass at the 00 mark on a ruler and extend the pencil tip to the 5 cm5 \text{ cm} mark.\nStep 3: Fix a center point OO and rotate the compass to complete the circle.

Explanation:

Since a compass is set using the radius, we must first divide the diameter by 22 before we can begin the construction process.

Problem 3:

Construct two concentric circles with a common center CC and radii 2.5 cm2.5 \text{ cm} and 4 cm4 \text{ cm}.

Two concentric circles with radii 2.5 cm and 4 cm.

Solution:

  1. Mark a point and label it CC.
  2. Set the compass to a radius of 2.5 cm2.5 \text{ cm} using a ruler.
  3. Place the pointer on CC and draw the first circle.
  4. Without moving the center point, adjust the compass to 4 cm4 \text{ cm}.
  5. Draw the second circle around the first one.

Explanation:

Concentric circles are drawn by maintaining the same center CC while varying the distance between the compass needle and the pencil.

Problem 4:

Draw a circle and any two of its diameters. If you join the ends of these diameters, what is the shape formed?

A circle with two perpendicular diameters forming a square/rectangle inside.

Solution:

  1. Draw a circle with center OO.
  2. Draw two diameters ABAB and CDCD.
  3. Join AA to CC, CC to BB, BB to DD, and DD to AA.
  4. The resulting shape ACBDACBD is a rectangle.

Explanation:

In a circle, diameters are equal and bisect each other at the center. When we join the endpoints of two diameters, the diagonals of the resulting quadrilateral are equal and bisect each other, which is the property of a rectangle.