How to Solve Circles Word Problems Step-by-Step Guide
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In order to solve circle word problems, it's important to understand the basic formulas associated with circles:
- The circumference of a circle (C) is given by \(C = 2\pi r\), where \(r\) is the radius of the circle.
- The area of a circle (A) is given by \(A = \pi r^2\).
Step 1: Identify the given information
Read the problem carefully and identify what is given. This could be the radius, diameter, circumference, or area of the circle.
Step 2: Determine what needs to be found
Identify what the problem is asking you to find. It could be any of the aforementioned properties of a circle.
Step 3: Use the appropriate formula
Depending on what is given and what needs to be found, use the appropriate formula. You may need to rearrange the formula to solve for the unknown.
Step 4: Plug in the known values
Once you have the appropriate formula, plug in the known values and solve for the unknown.
Step 5: Check your answer
Always check your answer to make sure it makes sense in the context of the problem.
Remember, practice is key to getting better at solving circle word problems!
Example
Let's consider an example where the diameter of a circle is given, and we need to find the circumference and the area.
Problem:
A circular park has a diameter of 10 meters. What is the circumference and the area of the park?
Solution:
Given, diameter \(d = 10 \, m\). Hence, the radius \(r = \frac{d}{2} = \frac{10}{2} = 5 \, m\).
To find the circumference:
The formula for the circumference \(C\) is \(C = 2\pi r\).
Substituting the value of \(r\), we get \(C = 2\pi \times 5 = 10\pi \, m\).
To find the area:
The formula for the area \(A\) is \(A = \pi r^2\).
Substituting the value of \(r\), we get \(A = \pi \times 5^2 = 25\pi \, m^2\).
Answer:
So, the circumference of the park is \(10\pi \, m\) and the area of the park is \(25\pi \, m^2\).
Exercises
1) What is the circumference of a circle with a radius of \(10 \, m\)?
2) A circular pond has a diameter of \(7 \, m\). What is the area of the pond?
3) If a bicycle wheel has a radius of \(14 \, inches\), how far does the bicycle travel in one revolution of the wheel?
4) What is the diameter of a circle if its circumference is \(31.4 \, m\)?
5) A circular field has a circumference of \(66 \, m\). What is the radius of the field?
6) How much fencing is needed to surround a circular garden with a diameter of \(12 \, m\)?
7) If a circle has an area of \(78.5 \, sq. m\), what is the radius of the circle?
8) The radius of a circular track is \(100 \, m\). How far does a person travel if they run around the track 4 times?
9) What is the area of a circle if its diameter is \(20 \, cm\)?
10) A circular swimming pool has a radius of \(3.5 \, m\). What is the pool's circumference?
1) The circumference of the circle is \(2\pi r = 2 \times 3.14 \times 10 \approx 62.8 \, m\).
2) The area of the pond is \(\pi r^2 = 3.14 \times \left(\frac{7}{2}\right)^2 \approx 38.465 \, m^2\).
3) The bicycle travels a distance equal to the circumference of the wheel, \(2\pi r = 2 \times 3.14 \times 14 \approx 87.92 \, inches\).
4) The diameter of the circle is \(d = \frac{C}{\pi} = \frac{31.4}{3.14} = 10 \, m\).
5) The radius of the field is \(r = \frac{C}{2\pi} = \frac{66}{2 \times 3.14} \approx 10.5 \, m\).
6) The fencing needed is equal to the circumference of the garden, \(2\pi r = 2 \times 3.14 \times 6 \approx 37.68 \, m\).
7) The radius of the circle is \(r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{78.5}{3.14}} \approx 5 \, m\).
8) The person travels a distance equal to the circumference of the track times the number of laps, \(4 \times 2\pi r = 4 \times 2 \times 3.14 \times 100 = 2512 \, m\).
9) The area of the circle is \(\pi r^2 = 3.14 \times \left(\frac{20}{2}\right)^2 = 314 \, cm^2\).
10) The pool's circumference is \(2\pi r = 2 \times 3.14 \times 3.5 \approx 21.98 \, m\).
These answers use the formulas for the circumference and area of a circle: \(C = 2\pi r\) and \(A = \pi r^2\), respectively, where \( \pi \) is approximated as \(3.14\)
