NCERT Solutions for Class 6 Maths Chapter 8: Playing with Constructions — Free PDF

Question: Draw a circle with radius $3.5$ cm.

Solution:

Step 1: Open the compass to $3.5$ cm by placing the metal tip at the $0$ mark of the ruler and adjusting the pencil tip to $3.5$ cm.

Step 2: Mark a point $O$ on the paper. This will be the centre.

Step 3: Place the metal tip of the compass on $O$.

Step 4: Rotate the compass $360°$ to draw the complete circle.

Answer: The circle has centre $O$ and radius $3.5$ cm. Every point on the circle is exactly $3.5$ cm from $O$.

Key terms:
- Radius — distance from centre to any point on the circle ($r = 3.5$ cm)
- Diameter — distance across the circle through the centre ($d = 2r = 7$ cm)
- Circumference — the perimeter of the circle

Question: Construct a line segment $\overline{AB}$ of length $6.4$ cm.

Solution:

Step 1: Draw a ray starting from point $A$.

Step 2: Place the ruler with the $0$ mark at $A$.

Step 3: Mark point $B$ at the $6.4$ cm mark on the ruler.

Step 4: Draw the segment $\overline{AB}$.

Answer: $\overline{AB} = 6.4$ cm.

Using a compass (alternative):
Step 1: Open the compass to $6.4$ cm using the ruler.
Step 2: Draw a ray from $A$.
Step 3: With the compass on $A$, draw an arc cutting the ray at $B$.
This method is more accurate for longer segments.

Question: Given a line segment $\overline{PQ}$, construct another line segment $\overline{RS}$ of the same length without using a ruler.

Solution:

Step 1: Place the compass tip on $P$ and pencil on $Q$, capturing the length $PQ$.

Step 2: Without changing the compass width, draw a ray from point $R$.

Step 3: Place the compass tip on $R$ and draw an arc cutting the ray at $S$.

Answer: $\overline{RS} = \overline{PQ}$.

This technique is essential because it copies a length exactly without needing to read a measurement.

Question: Construct the perpendicular bisector of a line segment $\overline{AB}$ of length $8$ cm.

Solution:

Step 1: Draw $\overline{AB} = 8$ cm.

Step 2: With $A$ as centre and radius more than $\frac{8}{2} = 4$ cm (say $5$ cm), draw arcs above and below $\overline{AB}$.

Step 3: With $B$ as centre and the same radius ($5$ cm), draw arcs above and below $\overline{AB}$, intersecting the first arcs at points $P$ and $Q$.

Step 4: Draw the line through $P$ and $Q$.

Answer: The line $PQ$ is the perpendicular bisector of $\overline{AB}$. It:
- Passes through the midpoint $M$ of $\overline{AB}$ (so $AM = MB = 4$ cm)
- Makes a $90°$ angle with $\overline{AB}$
- Every point on this line is equidistant from $A$ and $B$

Question: Construct the bisector of a given angle $\angle AOB = 60°$.

Solution:

Step 1: Draw $\angle AOB = 60°$.

Step 2: With $O$ as centre and any convenient radius, draw an arc cutting $\overrightarrow{OA}$ at $P$ and $\overrightarrow{OB}$ at $Q$.

Step 3: With $P$ as centre and radius $> \frac{PQ}{2}$, draw an arc.

Step 4: With $Q$ as centre and the same radius, draw an arc intersecting the previous arc at $R$.

Step 5: Draw ray $\overrightarrow{OR}$.

Answer: $\overrightarrow{OR}$ bisects $\angle AOB$, so $\angle AOR = \angle ROB = 30°$.

Note: This method works for any angle, not just $60°$.

Question: Construct an angle of $60°$ using only a ruler and compass.

Solution:

Step 1: Draw a ray $\overrightarrow{OA}$.

Step 2: With $O$ as centre and any radius $r$, draw an arc cutting $\overrightarrow{OA}$ at $P$.

Step 3: With $P$ as centre and the same radius $r$, draw an arc cutting the first arc at $Q$.

Step 4: Draw ray $\overrightarrow{OQ}$.

Answer: $\angle AOQ = 60°$.

Why it works: Triangle $OPQ$ is equilateral (all sides $= r$), so each angle is $60°$.

From this, you can construct:
- $30° = \frac{60°}{2}$ (bisect $60°$)
- $120° = 2 \times 60°$ (repeat the arc)
- $90°$ (bisect $120°$ and $60°$, or construct a perpendicular)

Question: Create a flower pattern by drawing $6$ circles of the same radius, each passing through the centre of the original circle.

Solution:

Step 1: Draw a circle with centre $O$ and radius $r$.

Step 2: Mark a point $A$ on the circle. With $A$ as centre and radius $r$, draw a circle. It passes through $O$ and cuts the original circle at two points.

Step 3: One of these intersection points becomes the centre for the next circle. Repeat this process, moving around the original circle.

Step 4: After $6$ circles, you return to the starting point, forming a petal pattern.

Answer: The resulting figure has $6$ overlapping regions that look like flower petals. This works because $6$ circles of radius $r$ fit exactly around a circle of radius $r$ (since the central angle for each is $60°$).

This is one of the most beautiful constructions in geometry and has appeared in art and architecture for centuries.

Question: Construct a regular hexagon inscribed in a circle of radius $4$ cm.

Solution:

Step 1: Draw a circle with centre $O$ and radius $4$ cm.

Step 2: Mark any point $A$ on the circle.

Step 3: With compass set to $4$ cm (the radius), place the tip at $A$ and mark point $B$ on the circle.

Step 4: Move to $B$ and mark point $C$. Continue for $D$, $E$, $F$.

Step 5: Connect $A$-$B$-$C$-$D$-$E$-$F$-$A$.

Answer: The hexagon has $6$ equal sides, each $= 4$ cm (equal to the radius).

Why it works: The side of a regular hexagon inscribed in a circle equals the radius of the circle. Each central angle is $\frac{360°}{6} = 60°$, and the triangle formed is equilateral.