Diameter and Circumference വ്യാസവും ചുറ്റളവും
The perimeter of a polygon is easy: add up its sides. A circle has no sides. So how long a wire do we need to bend into a circle 5 cm across? This section finds the key fact – circumference changes in the same proportion as diameter – by trapping circles between polygons.
- Perimeter (ചുറ്റളവ്) of a polygon = sum of its sides; a regular polygon (സമബഹുഭുജം) with n sides of length s has perimeter ns.
- A regular polygon can be drawn inside a circle; joining the centre to its corners gives n equal triangles.
- If a figure is scaled, every length in it is scaled by the same factor (chapter സദൃശത്രികോണങ്ങൾ).
1. Polygons are easy, circles are not
When a wire is bent into a circle, its length becomes the circumference. To know the wire's length without measuring, we need to know how the circumference is related to the diameter (വ്യാസം). First question: if the diameter is made 1½ times, 2 times or ½ times, does the circumference change in the same proportion?
2. Polygons in a circle and in a bigger circle
Draw the same regular polygon inside a circle, and again inside a circle whose radius is 1½ times as long.
Whatever the polygon, the big perimeter ÷ the small perimeter equals the big diameter ÷ the small diameter.
3. More sides, closer to the circle
As the number of sides grows, the polygon hugs the circle more and more closely, and its perimeter gets closer and closer to the circumference.
4. Putting it together with algebra (page 154)
വൃത്തങ്ങളുടെ ചുറ്റളവുകൾ മാറുന്നത്, വ്യാസങ്ങൾ മാറുന്ന അതേ തോതിലാണ്
The circumferences of circles change in the same proportion as their diameters. So if we know the circumference of the circle of diameter 1, every other circumference is that number times the diameter. The next section names that number: π.
A bicycle wheel twice as wide goes twice as far in one turn. A round table 1½ times as wide needs 1½ times as much edging tape. A running track's outer lanes are longer because their diameters are bigger – which is why races in lanes have staggered starts.
5. Check yourself
- A wire bent into a circle has length equal to the circumference; to compute it we need the link between circumference and diameter.
- Regular polygons in a circle scale with the radius: 1½ times the radius ⇒ 1½ times each side ⇒ 1½ times the perimeter.
- With more sides the polygon perimeters approach the circumference; so circumferences also scale like diameters.
- Circumference changes in the same proportion as diameter.