Complete Guide to Circles on the ACT® Math
Read time: 3 minutes 30 seconds Last updated: September 23rd, 2024
There are several different ways that the ACT® Math may ask about circles. Make sure you master the basics of circumference and area before moving on. The concepts beyond that will require you to have a strong grasp on the component parts of a circle. The ACT® Math will give you questions about angles, arcs and graphs of circles. Keep reading to learn how to answer these questions effectively.
Area of a Circle
The area of a circle is given by the formula:
Circumference of a Circle
The circumference of a circle is given by the formula:
Example 1
Answer Choices:
A) 44
B) 154
C) 49
D) 21
E) 14
Click for the Answer
Correct Answer Explanation:Answer Choices:
- A) 44: Incorrect because it might be confusing the formula with the diameter or circumference.
- B) 154: Correct. The area of a circle is given by . Plugging in the values: .
- C) 49: Incorrect because it is simply the radius squared without multiplying by .
- D) 21: Incorrect because it is a miscalculation, likely confusing area with diameter.
- E) 14: Incorrect because it is the diameter, not the area.
Correct Answer:
B) 154
Example 2
Answer Choices:
A) 10
B) 20
C) 31.4
D) 15.7
E) 25
Click for the Answer
Correct Answer Explanation:Answer Choices:
- A) 10: Incorrect because it is the radius, not the circumference.
- B) 20: Incorrect because it is double the radius, not accounting for .
- C) 31.4: Correct. The circumference of a circle is given by . Plugging in the values: .
- D) 15.7: Incorrect because it is half the correct circumference.
- E) 25: Incorrect because it does not relate to the circumference formula.
Correct Answer:
C) 31.4
Example 3
Answer Choices:
A) 78.5
B) 25
C) 50
D) 31.4
E) 20
Click for the Answer
Correct Answer Explanation:Answer Choices:
- A) 78.5: Correct. The area of a circle is given by . Plugging in the values:
- B) 25: Incorrect because it is just the radius squared without multiplying by .
- C) 50: Incorrect because it is a miscalculation.
- D) 31.4: Incorrect because it is the circumference, not the area.
- E) 20: Incorrect because it does not relate to the area formula.
Correct Answer:
A) 78.5
Example 4
Answer Choices:
A) 50.24
B) 25.12
C) 16
D) 32
E) 40
Click for the Answer
Correct Answer Explanation:Answer Choices:
- A) 50.24: Correct. The circumference of a circle is given by . Plugging in the values: .
- B) 25.12: Incorrect because it is half the correct circumference.
- C) 16: Incorrect because it is the diameter, not the circumference.
- D) 32: Incorrect because it is the diameter without accounting for .
- E) 40: Incorrect because it does not relate to the circumference formula.
Correct Answer:
A) 50.24
Degrees in a Circle
Consider this: you keep track of how you spend your time in a regular day. This is what you come up with:
- School: 8 hours
- Homework: 3 hours
- Video games: 2 hours
- Hanging with friends/family: 2 hours
- Commuting: 1 hour
- Sleeping: 8 hours
- Chores: 1 hour
You make a pie chart of this but then realize you combined chores and homework time on.
If you remade the pie chart but added chores as its own slice, what percent of the pie chart would it represent?
The biggest issue for this question is that most students don’t RTQ. That’s understandable, because you think of the math test as just numbers. While that math test is a test of math, it’s also a test of getting the numbers and context from the problem. Don’t forget to read the question.
A lot of students would answer 16%, then move on. That is not the correct answer. To solve this question, we just need to find out what percent of the total day chores represents. So we take the number of hours for chores, 1, and divide it by the number of hours in a day, per the question, 24.
The math becomes 1/24 = 4.17%
Arcs in a Circle
The ACT Math® sometimes asks for an arc. Not every student learns these in high school. They’re relatively straightforward.
When we were looking at degrees in a line, we saw that a straight line had 180 degrees. Then we showed that the semicircle that would connect two points on a line proved the line had 180 degrees. Arcs are an extension of that reasoning that take it a step further.Instead of knowing the degrees for the part of a circle, we’re going to know the degree of an angle. Except, the angle likely won’t be 180 degrees. It’ll be smaller. Let’s say the angle is 30 degrees. How big is the respective arc? 30 degrees.
Circle Graph
Graphs of circles come up all the time on the ACT® Math Test. I’d highly recommend most students to learn how to work with the explicit formula of a circle.
A circle can be graphed using the equation above. The ACT® Math will ask you to understand the different components.
- h k = center
- y x = point on the radius
So at (0,0) you will see .
But If we use some real numbers for h k x and y, we could see a graph that looks like this:
- h = 1
- k = -2
- x = 2
- y = 1
In the graph above:
- h moved the center 1 unit to the right.
- k moved the center 2 units down.
- x and y determined a specific point on the circle's boundary.
The way this sort of question typically shows up is one of two ways. Either, the ACT® Math asks you to choose the appropriate graph of a circle given a certain equation. You’ll have to map out the different points based on the h, k, x and y.
The other way the ACT® Math may ask this question is the reverse of the above: the Test may give you a graph with certain points filled in. Then it’s up to you to know the difference between the h k x and y; the minus signs with the parentheses and the addition sign between them; and the fact that the radius is squared.
How Will Circles Show Up on the Test?
The above examples are indicative of the questions on the ACT® Math Test. In the beginning of the section, you’ll find more straightforward questions that require you to know the basic tenets of circle geometry: area, circumference, degrees in a circle, arcs and the graphs of a circle.
As the questions become more difficult, you’ll be required to think across disciplines to include algebra, for example, in your work. You may also face a greater number of steps to reach the correct answer. Knowing the above information about circles can help greatly to increase your score.