The radius of a wheel is 21 cm. What is the distance (in cm) travelled by the wheel in 20 revolutions?
2640 cm
Calculating Wheel Distance in Revolutions
This problem requires us to find the total distance covered by a wheel as it completes a certain number of revolutions. The distance covered in one revolution of a wheel is equal to its circumference.
Understanding the Problem
We are given:
- The radius of the wheel, $r = 21\text{ cm}$.
- The number of revolutions = $20$.
We need to find the total distance travelled by the wheel in $20$ revolutions.
Step-by-Step Solution
Step 1: Find the Circumference of the Wheel
The circumference of a circle (which is the shape of the wheel) is given by the formula:
$\text{Circumference } (C) = 2\pi r$
We will use the value of $\pi = \frac{22}{7}$ for this calculation, as the radius ($21\text{ cm}$) is a multiple of $7$.
Substitute the given radius into the formula:
$C = 2 \times \frac{22}{7} \times 21\text{ cm}$
$C = 2 \times 22 \times \frac{21}{7}\text{ cm}$
$C = 2 \times 22 \times 3\text{ cm}$
$C = 44 \times 3\text{ cm}$
$C = 132\text{ cm}$
So, the distance covered by the wheel in one revolution is $132\text{ cm}$.
Step 2: Calculate the Total Distance Travelled
The total distance travelled is the distance covered in one revolution multiplied by the total number of revolutions.
Total distance = Circumference $\times$ Number of revolutions
Total distance = $132\text{ cm} \times 20$
Total distance = $2640\text{ cm}$
Final Answer Calculation Summary
| Item | Value | Unit |
|---|---|---|
| Radius ($r$) | 21 | cm |
| Number of Revolutions | 20 | - |
| Circumference ($C = 2\pi r$) | 132 | cm |
| Total Distance (Circumference $\times$ Revolutions) | 2640 | cm |
The total distance travelled by the wheel in 20 revolutions is $2640\text{ cm}$.
Revision Table: Wheel Distance & Revolutions
| Concept | Description | Formula |
|---|---|---|
| Radius | Distance from the center to the edge of a circle/wheel. | $r$ |
| Circumference | The distance around the circle/wheel. Distance covered in one revolution. | $C = 2\pi r$ |
| Revolution | One complete turn of the wheel. | - |
| Total Distance | Circumference multiplied by the number of revolutions. | Total Distance = $C \times \text{Number of revolutions}$ |
Additional Information: Wheel Motion
When a wheel rolls without slipping, the distance it travels on the ground in one complete revolution is exactly equal to its circumference. This principle is fundamental in understanding how wheels move and how to calculate distances based on rotations.
Using $\pi = \frac{22}{7}$ is common in geometry problems when the radius or diameter is a multiple of 7. Otherwise, $\pi \approx 3.14$ is often used.
Problems involving wheel revolutions connect geometry (circumference calculation) with linear distance measurement.
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Take \(\left(\pi=\frac{22}{7}\right)\)
