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ACT Math : Plane Geometry

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Example Questions

Example Question #3 : How To Find The Length Of An Arc

What is the area of the sector of a circle with a central angle of 120 degrees and a radius of 5cm ? Simplify any fractions and leave your answer in terms of $\pi$ .

Possible Answers:

$\frac{3\pi}{25}cm^2$

$24\pi cm^2$

$3000\pi cm^2$

$25\pi \cm^2$

$\frac{25\pi}{3} cm^2$

Correct answer:

$\frac{25\pi}{3} cm^2$

Explanation:

The formula for the area of a sector of a circle is:

$\frac{central\: angle}{360}*\pi r^2$

The central angle given is 120 thus:

$\frac{120}{360}*\pi (5cm)^2= \frac{1}{3}\pi 25cm^2 = \frac{25\pi}{3}cm^2$

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Example Question #51 : Circles

A water wheel turns a $120^{\circ}$ arc every minute. If the radius of the wheel is , how far in meters does the wheel turn along its edge each minute?

Possible Answers:

$\frac{5\pi}{3} m$

$8\pi m$

$10\pi m$

$\frac{\pi}{3} m$

$4\pi m$

Correct answer:

$4\pi m$

Explanation:

If the radius is , then the circumference of the wheel is:

$C = 2r\pi = 12\pi$

If the wheel turns $120^{\circ}$ each minute, then it turns $\frac{120^{\circ}}{360^{\circ}} = \frac{1}{3}$ of the circumference each minute.

$d = 12\pi \cdot \frac{1}{3} = 4\pi$

Thus, the wheel turns $4\pi m$ each minute.

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Example Question #4 : How To Find The Length Of An Arc

What is the length of the arc ABC ?

arc2

The total area of the circle is $64\pi \, in^{2}$ and the area of the shaded region is $12\pi \,in^{2}$ .

Possible Answers:

$8\:in$

$\frac{12\pi}{5}\:in$

$\frac{3\pi}{16}\:in$

$16\pi\:in$

$3\pi\:in$

Correct answer:

$3\pi\:in$

Explanation:

If the area of the circle is $64\pi\:in^2$ , the radius can be found using the formula for the area of a circle:
$A=\pi r^2$

For our data, this is:

$64\pi=\pi r^2$

r^2=64

Therefore, $r=8\:in$

Now, the circumference of the circle is defined as:

$C=2\pi r$

For our data, this is:

$C=2*8*\pi=16\pi\:in$

Now, we know that a sector is a percentage of the total area. This percentage is easily calculated:

$\frac{12\pi\:in^2}{64\pi\:in^2}=\frac{3}{16}$

So, the length of the arc will merely be the same percentage, but now applied to the circumference:

$\frac{3}{16} * 16\pi\:in = 3\pi\:in$

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Example Question #21 : Circles

Circle

In the circle above, the length of arc BC is 100 degrees, and the segment AC is a diameter. What is the measure of angle ADB in degrees?

Possible Answers:

100

cannot be determined

Correct answer:

Explanation:

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.

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Example Question #2 : How To Find The Angle Of A Sector

What is the angle of a sector of area in^2 on a circle having a radius of $15\:in$ ?

Possible Answers:

$3.00^{\circ}$

$22.92^{\circ}$

$0.06^{\circ}$

$15.22^{\circ}$

$72.00^{\circ}$

Correct answer:

$22.92^{\circ}$

Explanation:

To begin, you should compute the complete area of the circle:

$A=\pi r^2$

For your data, this is:

$A=15^2\pi=225\pi$

Now, to find the angle measure of a sector, you find what portion of the circle the sector is. Here, it is:

$\frac{45}{225\pi}$

Now, multiply this by the total 360 degrees in a circle:

$\frac{45}{225\pi}*360=22.918311805212$

Rounded, this is $22.92^{\circ}$ .

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Example Question #1 : How To Find The Angle Of A Sector

What is the angle of a sector that has an arc length of 13.5 on a circle of diameter ?

Possible Answers:

$10.74^{\circ}$

$128.92^{\circ}$

$14.24^{\circ}$

$35.81^{\circ}$

$194.14^{\circ}$

Correct answer:

$128.92^{\circ}$

Explanation:

The first thing to do for this problem is to compute the total circumference of the circle. Notice that you were given the diameter. The proper equation is therefore:

$C=\pi d$

For your data, this means,

$C=12\pi$

Now, to compute the angle, note that you have a percentage of the total circumference, based upon your arc length:

$\frac{13.5}{12\pi}*360=128.9155039044336$

Rounded to the nearest hundredth, this is $128.92^{\circ}$ .

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Example Question #2 : Diameter

The perimeter of a circle is 36 π. What is the diameter of the circle?

Possible Answers:

Correct answer:

Explanation:

The perimeter of a circle = 2 πr = πd

Therefore d = 36

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Example Question #2 : How To Find The Length Of The Diameter

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?

Possible Answers:

$6.25\pi$

$5\pi$

$6\pi$

$12.5\pi$

$25\pi$

Correct answer:

$6.25\pi$

Explanation:

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr².

$A=\pi (5/2)^2=6.25\pi$

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Example Question #1 : Diameter

If a circle has an area of $49\pi$ , what is the diameter of the circle?

Possible Answers:

3.5

Correct answer:

Explanation:

1. Use the area to find the radius:

$Area=\pi r^{2}$

$49\pi=\pi r^{2}$

$49=r^{2}$

r=7

2. Use the radius to find the diameter:

$d=2r=2\cdot 7=14$

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Example Question #2 : How To Find The Length Of The Diameter

What is the diameter of a semi-circle that has an area of $25\pi$ ?

Possible Answers:

$5\sqrt{2}$

$10\sqrt{2}$

$25\sqrt{3}$

Correct answer:

$10\sqrt{2}$

Explanation:

To begin, be very careful to note that the question asks about a semi-circle—not a complete circle! This means that a complete circle composed out of two of these semi-circles would have an area of $50\pi$ . Now, from this, we can use our area formula, which is:

$A=\pi r^2$

For our data, this is:

$50\pi=\pi r^2$

Solving for , we get:

$r=\sqrt{50}$

This can be simplified to:

$r = 5\sqrt{2}$

The diameter is , which is $2 * 5\sqrt{2}$ or $10\sqrt{2}$ .

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #3 : How To Find The Length Of An Arc

Example Question #51 : Circles

Example Question #4 : How To Find The Length Of An Arc

Example Question #21 : Circles

Example Question #2 : How To Find The Angle Of A Sector

Example Question #1 : How To Find The Angle Of A Sector

Example Question #2 : Diameter

Example Question #2 : How To Find The Length Of The Diameter

Example Question #1 : Diameter

Example Question #2 : How To Find The Length Of The Diameter

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