How Many Times a Wheel Turns Over a Kilometre

As many times as its circumference fits into that distance. A wheel of radius 35 cm has a circumference of 2.2 m, so over 1.1 km it makes

$$\frac{1100}{2.2}=500\text{ revolutions}$$

One turn of the wheel moves the cycle forward by exactly one circumference. This page covers everything in the ICSE Class 7 Mathematics chapter's first part: perimeter of rectilinear figures, circumference of a circle, dimensions from a ratio, and practical problems.

Perimeter is the total distance around a closed figure — the length of its boundary, measured in units of length such as cm or m.

$$\text{rectangle}=2(l+b),\text{square}=4s,\text{triangle}=a+b+c$$

Worked examples. A rectangle 8 cm by 5 cm has perimeter

$$2(8+5)=26\text{ cm}$$

A square of side 7 cm has perimeter $4\times 7=28$ cm. A triangle with sides 6 cm, 8 cm and 10 cm has perimeter 24 cm.

Finding a missing side. A rectangle has perimeter 26 cm and length 8 cm. Since $2(l+b)=26$, we get $l+b=13$, so

$$b=13-8=5\text{ cm}$$

Another. A square has perimeter 36 cm, so its side is $\frac{36}{4}=9$ cm.

A cricket boundary rope and the wooden beading round a photo frame are both perimeters in ordinary use.

The first step in any missing-side question is halving. From $2(l+b)=26$ you must reach $l+b=13$ before subtracting — subtracting 8 from 26 directly gives 18, which is not a side of anything.

Write the sides as multiples of a single letter, form the perimeter equation, and solve for that letter.

Worked example. A rectangle has perimeter 48 cm and its sides are in the ratio $3:5$.

Let the sides be $3x$ and $5x$. Then

$$2(3x+5x)=48\Rightarrow 16x=48\Rightarrow x=3$$

So the sides are $3\times 3=9$ cm and $5\times 3=15$ cm.

Check: $2(9+15)=48$ cm, and $9:15$ reduces to $3:5$. Both conditions hold.

Another. A triangle has perimeter 60 cm with sides in the ratio $3:4:5$. The total parts are 12, so

$$12x=60\Rightarrow x=5$$

giving sides of 15 cm, 20 cm and 25 cm. And since $15^2+20^2=225+400=625=25^2$, that triangle is right-angled.

The checking step is worth the line it takes, because it tests both conditions. A pair of sides can add correctly and still be in the wrong ratio if the multiplication slipped, so confirm the perimeter and reduce your two sides back to the given ratio.

For cost, find the perimeter and multiply by the rate. For revolutions, divide the distance by the circumference.

Cost of fencing. A rectangular field is 40 m by 30 m, and fencing costs ₹25 per metre.

$$\text{perimeter}=2(40+30)=140\text{ m}$$

$$\text{cost}=140\times 25=₹3500$$

Cost of fencing a circular plot. A circular garden of radius 14 m is to be fenced at ₹30 per metre:

$$C=2\times \frac{22}{7}\times 14=88\text{ m},\text{cost}=88\times 30=₹2640$$

Number of revolutions. A cycle wheel has radius 35 cm.

$$C=2\times \frac{22}{7}\times 35=220\text{ cm}=2.2\text{ m}$$

Over a distance of 1.1 km, which is 1100 m:

$$\text{revolutions}=\frac{1100}{2.2}=500$$

Distance from revolutions. The same wheel making 250 turns covers $250\times 2.2=550$ m.

The unit conversion is where these questions are won or lost. The circumference came out in centimetres and the distance was given in kilometres, so both had to be brought to metres first. Dividing 1100 by 220 without converting gives 5 revolutions — an answer a hundred times too small, and one a moment's thought would reject.