ACT Math : Plane Geometry
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Example Questions
Example Question #3 : Circles
There are two identical circles on a plane that overlap. The radius of both circles is 1. The region in which they overlap has an area of π.
What is the total area of the 2 overlapping circles?
1
2
π
2π
0
π
The total area of both circles is π + π = 2π
Since the region overlaps, we cannot count it twice, so we must subtract it.
we get 2π – π = π
Example Question #1 : Radius
A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?
4π-4
2π-4
8π - 16
8π-4
8π-8
8π - 16
Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2. Thus, the radius of the circle is half of the diameter, or 2√2. The area of the circle is then π(2√2)2, which equals 8π. Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.
Example Question #1 : How To Find The Area Of A Circle
If a circle has a circumference of 16π, what would its area be if its radius were halved?
64π
8π
16π
4π
16π
The circumference of a circle = πd where d = diameter. Therefore, this circle’s diameter must equal 16. Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8. Halving the radius would give us a new radius of 4. To find the area of this new circle, use the formula A=πr² where r = radius. Plug in 4 for r. Area will equal 16π.
Example Question #4 : How To Find The Area Of A Circle
A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.
351.5
346.5
356.5
361.5
341.5
361.5
The area of the circle is (30/2)2*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5
Example Question #1 : Radius
The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?
64π
50π
160π
54π
49π
64π
The original radius would be 4, making the new radius 8 and by the area of a circle (A=π(r)2) the new area would be 64π.
Example Question #11 : How To Find The Area Of A Circle
A circle with a diameter of 6” sits inside a circle with a radius of 8”. What is the area of the interstitial space between the two circles?
55π in2
25π in2
7π in2
28π in2
72π in2
55π in2
The area of a circle is πr2.
The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 32 = 9π in2
The radius of the second circle = 8” so the area = π * 82 = 64π in2
The area of the interstitial space = area of the first circle – area of the second circle.
Area = 64π in2 - 9π in2 = 55π in2
Example Question #2 : How To Find The Area Of A Circle
If the radius of a circle is tripled, and the new area is 144π what was the diameter of the original circle?
7
12
6
8
4
8
The area of a circle is A=πr2. Since the radius was tripled 144π =π(3r)2. Divide by π and then take the square root of both sides of the equal sign to get 12=3r, and then r=4. The diameter (d) is equal to twice the radius so d= 2(4) = 8.
Example Question #541 : Geometry
If the radius of Circle A is three times the radius of Circle B, what is the ratio of the area of Circle A to the area of Circle B?
3
15
9
12
6
9
We know that the equation for the area of a circle is π r2. To solve this problem, we pick radii for Circles A and B, making sure that Circle A’s radius is three times Circle B’s radius, as the problem specifies. Then we will divide the resulting areas of the two circles. For example, if we say that Circle A has radius 6 and Circle B has radius 2, then the ratio of the area of Circle A to B is: (π 62)/(π 22) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.
Example Question #11 : How To Find The Area Of A Circle
- A circle is inscribed inside a 10 by 10 square. What is the area of the circle?
10π
25π
100π
50π
40π
25π
Area of a circle = A = πr2
R = 1/2d = ½(10) = 5
A = 52π = 25π
Example Question #541 : Plane Geometry
A square has an area of 1089 in2. If a circle is inscribed within the square, what is its area?
1089π in2
16.5 in2
272.25π in2
33π in2
33 in2
272.25π in2
The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:
A = s2 or 1089 = s2. Taking the square root of both sides, we get: s = 33.
Now, based on this, we know that 2r = 33 or r = 16.5. The area of the circle is πr2 or π16.52 = 272.25π.
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