Std 5 Mathematics · Chapter 7

7. Circles

Solutions to Problem Sets 28 to 31, with diagrams — radius, chord and diameter, interior and exterior, circumference and arcs. Maharashtra State Board (Balbharati), Grade Five Mathematics, Part One.

Key Ideas

Radius: a line joining the centre to any point on the circle. All radii of a circle are equal.

Chord: a line joining any two points on the circle.

Diameter: a chord that passes through the centre. Diameter = 2 × radius, and radius = diameter ÷ 2.

Interior / exterior: points inside the circle are in its interior; points outside are in its exterior. Points on the circle itself are neither.

Circumference: the length of the circle all the way round. Arc: a part of the circle between two points on it.

Radius, Chord and Diameter

AThe centre of the circle is O. Find the radii, chords and diameters in the diagram and write their names in the box provided.

Type of line	Names in the diagram
Radius	OM, OT, OS, OP
Diameter	MT, SP
Chord	NM; and also MT and SP (every diameter is a chord)

MT and SP pass through the centre O, so they are diameters. NM does not pass through O, so it is a chord but not a diameter. The halves of each diameter (OM, OT, OS, OP) are radii.

Problem Set 28

1Draw circles with the radii given below: (1) 2 cm (2) 4 cm (3) 3 cm

Step 1: Place the metal tip of the compass on the 0 mark of a ruler and open the compass until the pencil point is on the required mark (2 cm, 4 cm or 3 cm).

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

Step 3: Keep the metal tip steady on O and turn the pencil once all the way round.

Step 4: Mark any point A on the circle and join OA. OA is the radius; write its length on it.

(1) Radius OA = 2 cm

(2) Radius OA = 4 cm

(3) Radius OA = 3 cm

The figures are drawn to the same scale as one another; in your notebook, measure the radius with a ruler to make it exactly 2 cm, 4 cm and 3 cm.

2Draw a circle of any radius. Show one diameter, one radius and one chord on that circle.

Circle with centre O: AB is a diameter, OC is a radius and DE is a chord.

Diameter: draw a line through the centre O from one side of the circle to the other and name its ends A and B.
Radius: join O to any point C on the circle.
Chord: join any two points D and E on the circle (not through O).

Relationship between Radius and Diameter

BAnswer the questions about the circle with centre O and diameter AB.

Which are the radii in the circle? OA and OB.
How many radii make up diameter AB? Two radii, OA and OB.
If one radius is 3 cm, what is the diameter? 3 cm + 3 cm = 6 cm.
How long is the diameter compared to the radius? The diameter is twice the radius.
Will another diameter CD be the same length as AB? Yes. All the diameters of a circle are of the same length.

CTest 1: Measure the diameters and radii of the circles given and verify the relationship between their lengths.

	Circle 1	Circle 2	Circle 3
Radius	1 cm	2 cm	3 cm
Diameter	2 cm	4 cm	6 cm

In every circle, the diameter is twice the radius: 2 × 1 = 2, 2 × 2 = 4, 2 × 3 = 6. This verifies the relationship.

Problem Set 29

1If the radius of a circle is 5 cm, what will its diameter be?

Diameter = 2 × radius = 2 × 5 = 10 cm

2If the diameter of a circle is 6 cm, what will its radius be?

Radius = diameter ÷ 2 = 6 ÷ 2 = 3 cm

3Complete the following table by filling in the blanks.

	(1)	(2)	(3)	(4)
Radius	4 cm	8 cm	9 cm	11 cm
Diameter	8 cm	16 cm	18 cm	22 cm

(1) 2 × 4 = 8 (2) 16 ÷ 2 = 8 (3) 2 × 9 = 18 (4) 22 ÷ 2 = 11

Problem Set 30

QIn the table below, write the names of the points in the interior and exterior of the circle and those on the circle.

Points in the interior of the circle	Points in the exterior of the circle	Points on the circle
Y, T, R, X	W, M, Z	P, N

Look closely at M: its dot lies just outside the circle, so M is in the exterior. P and N lie exactly on the circle.

Problem Set 31

1Points S, L, M and N are on the circle. (1) Write the names of the arcs with end-points S and M. (2) Write the names of the arcs with the end-points L and N.

End-points S and M: arc SLM and arc SNM
End-points L and N: arc LMN and arc LSN

Two points always divide a circle into two arcs. The middle letter is a point on that arc, which tells us which of the two arcs is meant.

2Write the names of arcs that points A, B, C and D in the given circle give rise to.

Going round the circle, the points are in the order A, B, C, D.

Arcs between neighbouring points: arc AB, arc BC, arc CD, arc DA
Arcs passing through one point: arc ABC, arc BCD, arc CDA, arc DAB

The four short arcs AB, BC, CD and DA together make up the whole circle. Longer arcs can be named in the same way; for example, the arc from A to D that passes through B and C can be named arc ACD (or arc ABD).

3Give the names of the arcs that are made by points P, Q, R, S and T in the figure.

Going round the circle, the points are in the order P, Q, R, S, T.

Arcs between neighbouring points: arc PQ, arc QR, arc RS, arc ST, arc TP
Arcs passing through one point: arc PQR, arc QRS, arc RST, arc STP, arc TPQ

Longer arcs can also be named in the same way, for example arc PRS (from P to S through Q and R) or arc QST.

4Measure and note down the circumference of different circular objects.

Wind a measuring tape (or a string) once round the edge of the object and read the length. Sample readings:

A bangle: about 20 cm
A steel glass (rim): about 22 cm
A dinner plate: about 80 cm
A bucket (rim): about 95 cm

This is a practical activity. Your readings will depend on the objects you measure; write the name of each object and its circumference in your notebook.

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