ISEE Upper Level Quantitative Reasoning › How to find the length of a radius
You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be .
What is the radius of the crater?
Cannot be determined from the information provided
You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be .
What is the radius of the crater?
To solve this, we need to recall the formula for the area of a circle.
Now, we know A, so we just need to plug in and solve for r!
Begin by dividing out the pi
Then, square root both sides.
So our answer is 13m.
The area of Circle B is four times that of Circle A. The area of Circle C is four times that of Circle B. Which is the greater quantity?
(a) Twice the radius of Circle B
(b) The sum of the radius of Circle A and the radius of Circle C
(b) is greater.
(a) is greater.
(a) and (b) are equal.
It cannot be determined from the information given.
Let be the radius of Circle A. Then its area is
.
The area of Circle B is , so the radius of Circle B is twice that of Circle A; by a similar argument, the radius of Circle C is twice that of Circle B, or
.
(a) Twice the radius of circle B is .
(b) The sum of the radii of Circles A and B is .
This makes (b) greater.
Refer to the above diagram. has length
. Give the radius of the circle.
Inscribed , which measures
, intercepts a minor arc with twice its measure. That arc is
, which consequently has measure
.
The corresponding major arc, , has as its measure
, and is
of the circle.
If we let be the circumference and
be the radius, then
has length
.
This is equal to , so we can solve for
in the equation
The radius of the circle is 50.
A circle has a circumference of . What is the radius of the circle?
Not enough information to determine.
A circle has a circumference of . What is the radius of the circle?
Begin with the formula for circumference of a circle:
Now, plug in our known and work backwards:
Divide both sides by two pi to get:
The area of a circle is . Give its radius in terms of
.
(Assume is positive.)
The relation between the area of a circle and its radius
is given by the formula
Since
:
We solve for :
Since is positive, as is
:
If the diameter of a circle is equal to , then what is the value of the radius?
Given that the radius is equal to half the diameter, the value of the radius would be equal to divided by 2. This gives us:
What is the radius of a circle with circumference equal to ?
The circumference of a circle can be found using the following equation:
What is the value of the radius of a circle if the area is equal to ?
The equation for finding the area of a circle is .
Therefore, the equation for finding the value of the radius in the circle with an area of is:
What is the radius of a circle with a circumference of ?
The circumference of a circle can be found using the following equation:
We plug in the circumference given, into
and use algebraic operations to solve for
.
Compare the two quantities:
Quantity A: The radius of a circle with area of
Quantity B: The radius of a circle with circumference of
The two quantities are equal.
The quantity in Column A is greater.
The quantity in Column B is greater.
The relationship cannot be determined from the information given.
Recall for this question that the formulae for the area and circumference of a circle are, respectively:
For our two quantities, we have:
Quantity A
Therefore,
Taking the square root of both sides, we get:
Quantity B
Therefore,
Therefore, the two quantities are equal.