All SAT Math Resources
Example Questions
Example Question #91 : Circles
A circle is circumscribed about a regular hexagon of perimeter 120, Give its area.
Below is the figure referenced; note that the hexagon is divided by its diameters.
The hexagon is divided into six equilateral triangles; the length of each side is equal to the radius of the hexagon, and, consequently, the radius of the circle. The regular hexagon has perimeter 120, so the length of each congruent side is . The radius of the hexagon, and of the circle, is 20 also.
The area of the circle can be found using the area formula, setting :
Example Question #337 : Geometry
A circle is inscribed in a regular hexagon of perimeter 120, Give its area.
Below is the figure referenced; note that the hexagon is divided by its diameters, and that an apothem - a perpendicular segment from the center to one side - has been drawn.
The hexagon is divided into six equilateral triangles. The regular hexagon has perimeter 120, so the length of each congruent side is .
Specifically, is a 30-60-90 triangle. is the midpoint of , so
By the 30-60-90 Theorem, since and are the short and long legs of a right triangle,
.
This is the apothem of the hexagon; this is also the radius of the circle.
The area of the circle can be found using the area formula, setting :
Example Question #12 : New Sat Math No Calculator
What is in radians?
To convert degrees to radians, we need to remember the following formula.
.
Now lets substitute for degrees.
Example Question #335 : Sat Mathematics
A circle is circumscribed by a square, and that square is the base of a cube. If the volume of the cube is 216, what is the area of the circle?
The equation for the volume of a cube is
.
If V= 216, then s = 6.
When a circle is circumscribed by a square the diameter of the circle is equal to the length of one side of the square. In this case, one side of the square is equal to 6; therefore the diameter of the circle is also 6. We can find the area of a circle with the equation
and since the radius is equal to half the diameter:
Example Question #1 : Circumference Of A Circle
If a circle has an area of , what is the circumference of the circle?
The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.
Example Question #1 : How To Find Circumference
A circle with an area of 13π in2 is centered at point C. What is the circumference of this circle?
√13π
√26π
2√13π
26π
13π
2√13π
The formula for the area of a circle is A = πr2.
We are given the area, and by substitution we know that 13π = πr2.
We divide out the π and are left with 13 = r2.
We take the square root of r to find that r = √13.
We find the circumference of the circle with the formula C = 2πr.
We then plug in our values to find C = 2√13π.
Example Question #2 : How To Find Circumference
A 6 by 8 rectangle is inscribed in a circle. What is the circumference of the circle?
10π
12π
8π
6π
25π
10π
First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10π.
Example Question #2 : How To Find Circumference
A gardener wants to build a fence around their garden shown below. How many feet of fencing will they need, if the length of the rectangular side is 12 and the width is 8?
8π + 24
96 ft
40 ft.
4π + 24
8π + 24
The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.
Example Question #344 : Plane Geometry
The diameter of a circle is defined by the two points (2,5) and (4,6). What is the circumference of this circle?
5π
π√5
None of the other answers
2.5π
π√2.5
π√5
We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)
For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5
If d = √5, the circumference of our circle is πd, or π√5.
Example Question #345 : Plane Geometry
A car tire has a radius of 18 inches. When the tire has made 200 revolutions, how far has the car gone in feet?
500π
300π
600π
3600π
600π
If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.