SSAT Upper Level Quantitative › How to find the circumference of a circle
A central angle of a circle has a chord with length
. Give the circumference of the circle.
The figure below shows , which matches this description, along with its chord
:
By way of the Isosceles Triangle Theorem, can be proved a 45-45-90 triangle with hypotenuse 40. By the 45-45-90 Theorem, its legs, both radii, have length that can be determined by dividing this by
, so
This is the radius, so the circumference is
The track at Madison High School is a perfect circle of radius 600 feet, and is shown in the above figure. Boris wants to run around the track for one mile. If Boris starts at point A and runs clockwise, which of the following is closest to the point at which he will stop running?
(Assume the five points are evenly spaced)
A circle of radius 600 feet will have a circumference of
feet.
Boris will run one mile, or 5,280 feet, which will be about
tiimes the circumference of the track, or, equivalently, once around the track, plus another two-fifths of the circle. This means he will end up closest to point C.
Find the circumference of a circle with a radius of 12.
There are two possible formulas for finding the circumference of a circle. They are as follows:
And:
Where C is circumference, r is radius, and d is diameter.
For this problem, we are given the radius is 12, meaning we can plug our numbers into the first equation. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.
Give the circumference of a circle that circumscribes an equilateral triangle with perimeter .
An equilateral triangle of perimeter 84 has sidelength one-third of this, or 28.
Construct this triangle and its circumscribed circle, as well as a perpendicular bisector to one side and a radius to one of that side's endpoints:
Each side of the triangle has measure 28, so . Also, the triangle formed by the segments, by symmetry, is a 30-60-90 triangle. Therefore, by the 30-60-90 Theorem,
and .
This is the radius, so the circumference is times this, or
The track at Douglas MacArthur High School is shown above; it is the composite of a square and a semicircle.
Jennifer wants to run one mile. If she begins at Point A and begins running clockwise, where will she be when she is finished?
Between Point E and Point A
Between Point A and Point B
Between Point B and Point C
Between Point C and Point D
Between Point D and Point E
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 400 feet; this length is
feet.
The distance around the track is about
feet.
Divide this into 5,280 feet (one mile):
This means that Jennifer will run two times around the track to return to Point A. She will have 1,624 feet to go, so she will do the following:
She will run 400 feet from A to B, leaving feet;
She will run 628 feet from B to D, leaving feet;
She will run 400 feet from D to E, leaving feet;
She will then run the remaining 196 feet from Point E toward Point A.
The correct response is that she will be between Point E and Point A.
The track at Eisenhower High School is shown above; it is comprised of a square and a semicircle.
Mike begins at Point A, runs five times around the track clockwise, and continues further until he reaches Point C. Which of the following comes closest to the distance Mike runs?
First, it is necessary to know the length of the semicircle connecting Points B and D, which has diameter 500 feet; this length is
feet.
The distance around the track is about
feet.
Mike runs around the track five times, which is a distance of about
feet.
He then proceeds to Point B, which is an additional 500 feet, and to Point C, which is half the semicircle, or
feet.
Therefore, Mike runs about
feet.
Divide by 5,280 to convert to miles:
miles.
Of the choices, the closest is miles.
Give the circumference of a circle that is inscribed in an equilateral triangle with perimeter 60.
An equilateral triangle of perimeter 60 has sidelength one-third of this, or 20.
Construct this triangle and its inscribed circle, as well as a radius to one side - which, by symmetry, is a perpendicular bisector - and a segment to one of that side's endpoints:
Each side of the triangle has measure 20, so . Also, the triangle formed by the segments, by symmetry, is a 30-60-90 triangle. Therefore,
which is the radius of the circle. The cricumference of the circle is times this, or
Find the circumference of a circle with a diameter of 12.
There are two possible formulas for finding the circumference of a circle. They are as follows:
And:
Where C is circumference, r is radius, and d is diameter.
For this problem, we are given the diameter is 12, meaning we can plug our numbers into the second equation. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.
Find the circumference of a circle with a diameter of 6.
There are two possible formulas for finding the circumference of a circle. They are as follows:
And:
Where C is circumference, r is radius, and d is diameter.
For this problem, we are given the diameter is 6, meaning we can plug our numbers into the second equation. Since pi is an irrational constant, it is okay to leave the answer in terms of pi.
Find the circumference of the circle with an area of .
Write the formula for the area of the circle. The radius is needed to find the circumference of the circle.
Substitute the area.
The radius of the circle is 3. Substitute this in the circumference formula.
The correct answer is: