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
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- $1^{\circ}$ of latitude is equal to ___ km approximately.
- The radius of a circle is equal to the diagonal of a square. If the area of the square is 32 sq units, then the area of the circle is:
- What is the area of the largest square that is inscribed in a semicircle of radius 10 cm?
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- Two intersecting circles are said to be orthogonal if the angle between them is a/an _____.
A park is designed in the shape of a right-angled triangle, where the two shorter sides measure 13 m and 84 m. A circular track with a uniform width of 2 m is constructed such that its outer boundary coincides with the circumcircle of the triangular park. Determine the perimeter of the inner boundary of this circular track.
- From a point Q, the length of the tangent to a circle is 21cm and the distance of Q from the centre 'O' of the circle is 29cm. Find the radius of the circle.
- The angle of a sector is 30°. If its radius is 42 cm then the length of the arc of the sector is:
Important Questions from Circles
The sum of the radius and diameter of a circle is 84 cm. What is the circumference of this circle?
The maximum area of a right-angled triangle inscribed in a circle of radius r is
The tangent at a point C of a circle and diameter AB when extended intersect at D, if ∠DCA = 110°, then ∠CBA is equal to
The equation of a circle with diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area of 154 sq. units is
The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
