What properties are common to all circles, big and small?
Step-by-step solution
Idea: Look past the size. A small circle and a big circle differ only in how far their points are from the centre; what never changes is the way each is built around one point: the centre, with the whole circle at one fixed distance from it. This shared property is used as the definition of a circle.
- Every circle has a centre. Whatever its size, a circle is built around one special point in the middle, called the centre.½ mark
- All points of a circle are equally far from its centre. Measure from the centre to any point on the circle and you always get the same length, the radius. In the picture, all three radii of the small circle are equal (r), and all three radii of the big circle are equal (R). Only the radius differs from circle to circle; the property itself is the same for both.½ mark
- This observation is turned around and made the definition: a circle is the set of all points in a plane that are at a fixed distance (the radius) from a fixed point (the centre).½ mark
- Other properties that every circle has follow from this one: all its radii are equal; every diameter is two radii, so it is 2 × radius and all diameters are equal; and the circumference divided by the diameter is the same number, π (about 3.14), for every circle.½ mark
Check: Test it on any round object: put the point of a compass at the centre and the pencil on the edge. Turning the compass traces the edge all the way round, which can only happen because every point of the edge is the same distance from the centre.
Answer to write in the exam
Every circle, big or small, has a centre.
All points on a circle are at the same distance (the radius) from the centre.
Hence all radii of a circle are equal, diameter = 2 × radius, and circumference ÷ diameter = π for every circle.
∴ Common property: a circle is the set of all points in a plane at a fixed distance (radius) from a fixed point (centre).
Common mistakes that cost marks
- Answering only “all circles are round”. That is true but not a property you can measure or use. Name the exact property: a centre, and every point at the same distance from it.
- Thinking all circles share the same radius or the same circumference. These change with size; what is common is that each circle’s own points are all equally far from its own centre (and the ratio circumference ÷ diameter, π).
- Counting the inside region as part of the circle. The circle is only the boundary; points inside are closer to the centre than the radius.
How this can come in the exam
Assertion (A): All diameters of a circle have the same length.
Reason (R): Every diameter is made of two radii, and all radii of a circle are equal.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true, but R is false.
- A is false, but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
A diameter passes through the centre, so it is two radii joined end to end: its length is 2 × radius. All radii are equal, so all diameters are equal. Both are true and R explains A.
Two circles have radii 4 cm and 9 cm. Which of these is the same for both circles?
- The length of a diameter
- The circumference
- Circumference ÷ diameter
- The area
Show answer
(C) Circumference ÷ diameter
Diameters are 8 cm and 18 cm, circumferences 8π cm and 18π cm, areas 16π cm2 and 81π cm2: all different. But circumference ÷ diameter = 8π ÷ 8 = 18π ÷ 18 = π for both.
Try one yourself
A circle has centre O and radius 5.5 cm. In the same plane, OP = 5.5 cm, OQ = 3.2 cm and OS = 7.1 cm. Which of the points P, Q, S lie on the circle, which inside and which outside?
Show answer
A point is on the circle only if its distance from the centre equals the radius. P is on the circle (5.5 cm = radius), Q is inside (3.2 cm < 5.5 cm) and S is outside (7.1 cm > 5.5 cm).
More questions like this
- List some objects from nature that resemble a circle.
- Jamuna has a circular piece of paper. She is trying to locate its centre. Amina gives her a suggestion. She follows the instructions and is thrilled to find that it works. Can you guess what Amina told her?
- Say you are looking at a wheel of a vehicle. You see a point of the wheel touching the ground. When you look at the wheel again after some time, you again see a point of the wheel touching the ground. Can you tell if the two points are the same point?
- Draw a circle on the paper and cut along the circle. Fold the circular paper so that the boundaries overlap, then open it. You see a crease; it is a line of reflection symmetry of the circle. Does this line pass through the centre of the circle?
- 1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?
2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
(Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.)