Circle Formulas Reminder
The formula for the circumference ($C$) of a circle is $C = 2 \pi r$. The formula for the area ($A$) of a circle is $A = \pi r^2$. Here, '$r$' represents the radius of the circle.
Equating Circumference and Area
The problem states that the circumference and area of the circle are numerically equal:
$ C = A $Substitute the formulas:
$ 2 \pi r = \pi r^2 $Solving for the Radius
To find the radius '$r$', we solve the equation:
- Divide both sides by $\pi$ (since $\pi \neq 0$): $ 2r = r^2 $
- Rearrange the equation to form a quadratic equation: $ r^2 - 2r = 0 $
- Factor out '$r$': $ r(r - 2) = 0 $
- This equation gives two possible solutions for '$r$':
- $r = 0$
- $r - 2 = 0 \implies r = 2$
A circle cannot have a radius of 0. Therefore, the only valid radius is 2 units.
Conclusion
When the circumference and area of a circle are numerically equal, the radius must be 2 units.
Similar Questions
- PQ is a diameter of a circle whose centre is O. If a point R lies on the circle and $\angle RPO$ is $39^\circ$, then find the measure of $\angle RQP$.
- The points A, B and C lie on a circle with centre O. The $\angle \text{ACB} = 47.5^{\circ}$. What is the $\angle \text{AOB}$ on the minor $\widehat{\text{AB}}$?
- $1^{\circ}$ of latitude is equal to ___ km approximately.
- A wheel makes 1000 revolutions in covering a distance of 154 km. Find the radius (in metre) of the wheel.
- Two intersecting circles are said to be orthogonal if the angle between them is a/an _____.
A circle having centre $c_1$, radius $r_1$ = 5 cm is placed against a right angle. Another smaller circle having centre $c_2$, radius $r_2$ is also placed touching the sides of angle and the bigger circle as shown in the figure. Find the radius, $r_2$ in cm, of the smaller circle.
In the given circle with centre O, the obtuse angle at the centre measures $104^\circ$. In the quadrilateral drawn inside the circle, what is the measure of the angle opposite to $\angle\text{O}$? [Figure is not drawn to scale.]
- The chord of contact of tangents drawn from a point on the circle $x^2 + y^2 = a^2$ to the circle $x^2 + y^2 = b^2$ touches the circle $x^2 + y^2 = c^2$ such that $b^m = a^n c^p$, where $m,n,p \in \mathbb{N}$. Find the value of $m + n + p + 10$.
- Find the length of the tangent from any point on the circle $x^2 + y^2 + 2011x + 2012y + 2013 = 0$ to the circle $x^2 + y^2 + 2011x + 2012y + 2014 = 0$.
- If a point (1, 2) is translated 2 units through the positive direction of x-axis and then the tangents drawn from that point to the circle $x^2 + y^2 = 9$, find the angle between the tangents.
Important Questions from Circles
If 3x + y - 5 = 0 is the equation of a chord of the circle x2 + y2 - 25 = 0, then what are the coordinates of the mid-point of the chord ?
What is the area of minor segment ?
What is the area of major segment ?
A straight line x = y + 2 touches the circle 4(x 2+ y 2) = r 2. The value of r is
If the centre of the circle passing through the origin is (3, 4), then the intercepts cut off by the circle on x-axis and y-axis respectively are
