A circle is a two-dimensional geometric figure characterized by a straight line whose ends come together to form a ring. Each point on the line is equidistant from the center of the circle. The circumference (C) of a circle represents its perimeter. The area (A) of a circle represents the space enclosed within it. Both the area and the perimeter can be calculated using simple mathematical formulas that involve knowing the radius or diameter and the value of the constant π.
Steps
Part 1 of 3: Calculate the circumference
Step 1. Learn the formula for calculating the circumference
For this purpose, two formulas can be used: C = 2πr or C = πd, where π is a mathematical constant, which, once rounded, takes the value 3, 14, r is the radius of the circle in question and instead represents the diameter.
- Since the radius of a circle is exactly half the diameter, the two formulas shown are essentially identical.
- To express the value relative to the circumference of a circle, you can use any of the units of measurement used in relation to a length: meters, centimeters, feet, miles, etc.
Step 2. Understand the different parts of the formula
To find the circumference of a circle, three components are used: the radius, the diameter and the π. The radius and the diameter are related to each other, since the radius is exactly half the diameter and, consequently, the latter is exactly twice the radius.
- The radius (r) of a circle is the distance between any point on the circumference and the center.
- The diameter (d) of a circle is the line that joins two opposite points of the circumference passing through the center.
- The Greek letter π represents the relationship between the circumference of a circle and its diameter and is represented by the number 3, 14159265…. It is an irrational number that has an infinite number of decimal places that repeat without a fixed pattern. Normally the value of the constant π is rounded to the number 3, 14.
Step 3. Measure the radius or diameter of the given circle
To do this, use a common ruler by placing it on the circle so that one end is aligned with a point on the circumference and the side with the center. The distance between the circumference and the center is the radius, while the distance between the two points of the circumference that touch the ruler is the diameter (in this case remember that the side of the ruler must be aligned with the center of the circle).
In most geometry problems found in textbooks, the radius or diameter of the circle to be studied are known values
Step 4. Replace the variables with their respective values and perform the calculations
Once you have determined the value of the radius or diameter of the circle you are studying, you can insert them into the relative equation. If you know the radius value, use the formula C = 2πr. While if you know the value of the diameter, use the formula C = πd.
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For example: what is the circumference of a circle with a radius of 3 cm?
- Write the formula: C = 2πr.
- Replace the variables with known values: C = 2π3.
- Perform the calculations: C = (2 * 3 * π) = 6 * 3, 14 = 18.84 cm.
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For example: what is the circumference of a circle with a diameter of 9 m?
- Write the formula: C = πd.
- Replace the variables with the known values: C = 9π.
- Perform the calculations: C = (9 * 3, 14) = 28, 26 m.
Find the Circumference and Area of a Circle Step 5 Step 5. Practice with other examples
Now that you have learned the formula for calculating the circumference of a circle, it is time to practice some example problems. The more problems you solve, the easier it will be to tackle future ones.
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Calculate the circumference of a circle with a diameter of 5 km.
C = πd = 5 * 3.14 = 15.7 km
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Calculate the circumference of a circle with a radius of 10 mm.
C = 2πr = C = 2π10 = 2 * 10 * 3, 14 = 62.8 mm
Part 2 of 3: Calculate the Area
Find the Circumference and Area of a Circle Step 6 Step 1. Learn the formula for calculating the area of a circle
As in the case of the circumference, the area of a circle can also be calculated from the diameter or radius using the following formulas: A = πr2 or A = π (d / 2)2, where π is a mathematical constant, which, once rounded, takes the value 3, 14, r is the radius of the circle in question and d represents the diameter instead.
- Since the radius of a circle is exactly half the diameter, the two formulas shown are essentially identical.
- The area of an area is expressed using any square unit of measurement for length: square feet (ft2), square meters (m2), square centimeters (cm2), etc.
Find the Circumference and Area of a Circle Step 7 Step 2. Understand the different parts of the formula
Three components are used to identify the area of a circle: the radius, the diameter and the π. The radius and the diameter are related to each other, since the radius is exactly half the diameter and, consequently, the latter is exactly twice the radius.
- The radius (r) of a circle is the distance between any point on the circumference and the center.
- The diameter (d) of a circle is the line that joins two opposite points of the circumference passing through the center.
- The Greek letter π represents the relationship between the circumference of a circle and its diameter, represented by the number 3, 14159265…. It is an irrational number, which has an infinite number of decimal places that repeat without a fixed pattern. Normally the value of the constant π is rounded to the number 3, 14.
Find the Circumference and Area of a Circle Step 8 Step 3. Measure the radius or diameter of the given circle
To do this, use a common ruler by placing it on the circle so that one end is aligned with a point on the circumference and the side with the center. The distance between the circumference and the center is the radius, while the distance between the two points of the circumference that touch the ruler is the diameter (in this case remember that the side of the ruler must be aligned with the center of the circle).
In most textbook geometry problems, the radius or diameter of the circle to be studied are known values
Find the Circumference and Area of a Circle Step 9 Step 4. Replace the variables with their respective values and perform the calculations
Once you have determined the value of the radius or diameter of the circle you are studying, you can insert them into the relevant equation. If you know the radius value, use the formula A = πr2. While if you know the value of the diameter, use the formula A = π (d / 2)2.
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For example: what is the area of a circle having a radius of 3 m?
- Write the formula: A = πr2.
- Replace the variables with the known values: A = π32.
- Calculate the square of the radius: r2 = 32 = 9.
- Multiply the result by π: A = 9π = 28.26 m2.
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For example: what is the area of a circle having a diameter of 4 m?
- Write the formula: A = π (d / 2)2.
- Replace variables with known values: A = π (4/2)2
- Divide the diameter in half: d / 2 = 4/2 = 2.
- Calculate the square of the result obtained: 22 = 4.
- Multiply it by π: A = 4π = 12.56m2
Find the Circumference and Area of a Circle Step 10 Step 5. Practice with other examples
Now that you've learned the formula for calculating the circumference of a circle, it's time to practice some example problems. The more problems you solve, the easier it will be to tackle future ones.
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Calculate the area of a circle having a diameter of 7 cm.
A = π (d / 2)2 = π (7/2)2 = π (3, 5)2 = 12.25 * 3.14 = 38.47cm2.
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Calculate the area of a circle with a radius of 3 cm.
A = πr2 = π32 = 9 * 3.14 = 28.26cm2.
Part 3 of 3: Calculating Area and Circumference with Variables
Find the Circumference and Area of a Circle Step 11 Step 1. Determine the radius and diameter of a circle
Some geometry problems may give you the radius or diameter of a circle as a variable: r = (x + 7) or d = (x + 3). In this case you can still proceed with the calculation of the area or circumference, but your final solution will also have the same variable inside it. Note the radius or diameter value provided by the problem text.
For example: calculate the circumference of a circle having a radius equal to (x = 1)
Find the Circumference and Area of a Circle Step 12 Step 2. Write the formula using the information you have
Whether you are calculating the area or the circumference, you still need to replace the variables of the formula used with the known values. Write the formula you need (for calculating the area or circumference), then replace the variables present with their known values.
- For example: calculate the circumference of a circle having the even radius (x + 1).
- Write the formula: C = 2πr.
- Replace the variables with the known values: C = 2π (x + 1).
Find the Circumference and Area of a Circle Step 13 Step 3. Solve the equation as if the variable were any number
At this point you can proceed to solve the resulting equation, as you normally would. Handle the variable as if it were any other number. To simplify your solution, you may need to use the distributive property:
- For example: calculate the circumference of a circle having a radius equal to (x + 1).
- C = 2πr = 2π (x + 1) = 2πx + 2π1 = 2πx + 2π = 6.28x + 6.28.
- If the problem text gives the value of "x", you can use it to calculate your final solution as an integer.
Find the Circumference and Area of a Circle Step 14 Step 4. Practice with other examples
Now that you've learned the formula, it's time to practice some example problems. The more problems you solve, the easier it will be to tackle future ones.
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Calculate the area of a circle with a radius equal to 2x.
A = πr2 = π (2x)2 = π4x2 = 12.56x2.
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Calculate the area of a circle with a diameter equal to (x + 2).
A = π (d / 2)2 = π ((x +2) / 2)2 = ((x +2)2/ 4) π.
