All Area of Circle Formulas | Definition and Examples

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What is Circle?

What is Circle and Formulas?

A circle is a fundamental shape in geometry, defined as the set of all points in a plane that are at a constant distance from a central point. This fixed distance is known as the radius. The circle is distinctively round and is defined by several essential components:

• Center: The central point from which all points on the circle are equidistant.
• Radius: The distance from the center to any point on the circle.
• Diameter: The distance across the circle, passing through the center, which is twice the radius.
• Circumference: The perimeter or the total distance around the circle.
• Area: The amount of space enclosed within the circle's boundary.

Circle Formulas

Before diving into all the circle formulas, let’s quickly review what a circle is. A circle is a shape where all points are the same distance from a central point, called the center. The radius is the distance from the center to the edge of the circle. Now, let's explore all the circle formulas with some examples to help you understand them better

List of All Circle Formulas

Parameters	Circle Formulas
Circumference of a Circle	C=2πr
Area of a Circle	A=πr²
Diameter of a Circle	D=2r
Radius of a Circle (from Diameter)	r=2D
Radius of a Circle (from Circumference)	r=2πC​
Arc Length	Arc Length=360∘θ​×2πr
Sector Area	Sector Area=360∘θ​×πr2

A circle is a fundamental geometric shape that is defined as the set of all points in a plane that are equidistant from a given point, called the center. Here are some key characteristics and terms related to circles:

1. Center: The fixed point from which all points on the circle are equidistant.
2. Radius: The distance from the center of the circle to any point on the circle. All radii of a circle are equal.
3. Diameter: A line segment that passes through the center of the circle and has its endpoints on the circle. The diameter is twice the length of the radius.
4. Circumference: The distance around the circle, calculated using the formula C=2πr, where r is the radius.
5. Area: The space enclosed by the circle, calculated using the formula 𝐴=πr²
6. Chord: A line segment with both endpoints on the circle. The diameter is the longest chord.
7. Arc: A part of the circumference of a circle.
8. Sector: A region enclosed by two radii and an arc.
9. Tangent: A line that touches the circle at exactly one point. This point is called the point of tangency.
10. Secant: A line that intersects the circle at two points.

What are all Areas of Circle Formulas?

Circle Formulas

Parameters like the area, circumference, and radius of a circle can be calculated using specific formulas. Here are the different formulas used to calculate these parameters:

• Circumference: $𝐶=2𝜋𝑟$
• Area: $𝐴=𝜋{𝑟}^2$
• Diameter: $𝐷=2𝑟$

where $𝑟$ is the radius of the circle.

Here are the key formulas related to a circle and how they are used:

1. Circumference: The total distance around the circle.

$$\mathrm{<span\; style="font-size:\; small;">𝐶</span>}<span\; style="font-size:\; small;">=</span>\mathrm{<span\; style="font-size:\; small;">2</span>}\mathrm{<span\; style="font-size:\; small;">𝜋</span>}\mathrm{<span\; style="font-size:\; small;">𝑟</span>}$$

• Where $𝐶$ is the circumference and $𝑟$ is the radius.
• Use: To find the length around the circle.

2. Area: The space enclosed within the circle.

$$\mathrm{<span\; style="font-size:\; small;">𝐴</span>}<span\; style="font-size:\; small;">=</span>\mathrm{<span\; style="font-size:\; small;">𝜋</span>}{\mathrm{<span\; style="font-size:\; small;">𝑟</span>}}^{\mathrm{<span\; style="font-size:\; small;">2</span>}}$$

• Where $𝐴$ is the area and $𝑟$ is the radius.
• Use: To determine the amount of space inside the circle.

3. Diameter: The distance across the circle, passing through the center.

$$\mathrm{<span\; style="font-size:\; small;">𝑑</span>}<span\; style="font-size:\; small;">=</span>\mathrm{<span\; style="font-size:\; small;">2</span>}\mathrm{<span\; style="font-size:\; small;">𝑟</span>}$$

• Where $𝑑$ is the diameter and $𝑟$ is the radius.
• Use: To find the longest distance across the circle.

4. Radius: The distance from the center of the circle to any point on its edge.

$$\mathrm{<span\; style="font-size:\; small;">𝑟</span>}<span\; style="font-size:\; small;">=</span>\frac{\mathrm{<span\; style="font-size:\; small;">𝑑</span>}}{\mathrm{<span\; style="font-size:\; small;">2</span>}}$$

• Where $𝑟$ is the radius and $𝑑$ is the diameter.
• Use: To find the distance from the center to the edge.

5. Arc Length: The distance along a section of the circle's circumference.

$$\mathrm{<span\; style="font-size:\; small;">𝐿</span>}<span\; style="font-size:\; small;">=</span>\mathrm{<span\; style="font-size:\; small;">𝜃</span>}\mathrm{<span\; style="font-size:\; small;">𝑟</span>}$$

• Where $𝐿$ is the arc length, $𝜃$ is the central angle in radians, and $𝑟$ is the radius.
• Use: To find the length of a curved segment of the circle.

6. Sector Area: The area of a "pie slice" part of the circle.

$${\mathrm{<span\; style="font-size:\; small;">𝐴</span>}}_{\text{<span style="font-size: small;">sector</span>}}<span\; style="font-size:\; small;">=</span>\frac{\mathrm{<span\; style="font-size:\; small;">𝜃</span>}{\mathrm{<span\; style="font-size:\; small;">𝑟</span>}}^{\mathrm{<span\; style="font-size:\; small;">2</span>}}}{\mathrm{<span\; style="font-size:\; small;">2</span>}}$$

• Where ${𝐴}_{\text{sector}}$ is the sector area, $𝜃$ is the central angle in radians, and $𝑟$ is the radius.
• Use: To find the area of a section of the circle.

These formulas are fundamental in geometry and are used in various applications, including engineering, architecture, and everyday calculations involving circular shapes.

Examples on Circle Formulas

Let us solve some interesting problems using the perimeter of the circle formula.

Example 1. If the radius of a circle is 5 cm, what is its circumference?

Solution: To find the circumference of a circle, we use the formula:

$𝐶=2𝜋𝑟$

where $𝑟$ is the radius of the circle and $𝜋$ is a mathematical constant approximately equal to $3.14159$.

Given that the radius $𝑟$ is $5$ cm, we can substitute this value into the formula:

$𝐶=2𝜋\times 5$

$𝐶=10𝜋$

Now, let's approximate $𝜋$ to two decimal places, which is $3.14$, and multiply:

$𝐶\approx 10\times 3.14$

$𝐶\approx 31.4$

So, the circumference of the circle with a radius of $5$ cm is approximately $31.4$ cm.

Example 2. Find the area of a circle with a diameter of 12 meters.

Solution: To find the area of a circle, we use the formula:

$𝐴=𝜋{𝑟}^2$

where $𝑟$ is the radius of the circle.

Given that the diameter $𝑑$ is $12$ meters, we know that the diameter is twice the radius ($𝑑=2𝑟$). So, we can find the radius by dividing the diameter by $2$:

$𝑟=\frac{𝑑}{2}=\frac{12}{2}=6\text{ meters}$

Now, we can substitute the radius $𝑟=6$ meters into the formula for the area:

$𝐴=𝜋\times (6{)}^2$

$𝐴=𝜋\times 36$

Now, let's approximate $𝜋$ to two decimal places, which is $3.14$, and multiply:

$𝐴\approx 3.14\times 36$

$𝐴\approx 113.04$

So, the area of the circle with a diameter of $12$ meters is approximately $113.04$ square meters.

Example 3. Given the circumference of a circle is 31.4 cm, what is its radius?

Solution: To find the radius of a circle when given the circumference, we can use the formula for circumference:

$𝐶=2𝜋𝑟$

Given that the circumference $𝐶$ is $31.4$ cm, we can plug this value into the formula and solve for the radius $𝑟$:

$31.4=2\times 𝜋\times 𝑟$

Now, we can solve for $𝑟$ by dividing both sides by $2𝜋$:

$𝑟=\frac{31.4}{2𝜋}$

Approximating $𝜋$ to two decimal places, which is $3.14$, we have:

$𝑟=\frac{31.4}{2\times 3.14}$

$𝑟\approx \frac{31.4}{6.28}$

$𝑟\approx 5\text{ cm}$

So, the radius of the circle with a circumference of $31.4$ cm is $5$ cm.

Example 4. If the area of a circle is 154 square centimeters, what is its diameter?

Solution: To find the diameter of a circle when given the area, we need to use the formula for the area of a circle:

$𝐴=𝜋{𝑟}^2$

Given that the area $𝐴$ is $154$ square centimeters, we can rearrange the formula to solve for the radius $𝑟$:

${𝑟}^2=\frac{𝐴}{𝜋}$

${𝑟}^2=\frac{154}{𝜋}$

Now, to find $𝑟$, we take the square root of both sides:

$𝑟=\sqrt{\frac{154}{𝜋}}$

Approximating $𝜋$ to two decimal places, which is $3.14$, we have:

$𝑟\approx \sqrt{\frac{154}{3.14}}$

$𝑟\approx \sqrt{49.04458}$

$𝑟\approx 7\text{ cm}$

Now that we have the radius, we can find the diameter by doubling the radius:

$\text{Diameter}=2\times 𝑟$

$\text{Diameter}=2\times 7$

$\text{Diameter}=14\text{ cm}$

So, the diameter of the circle with an area of $154$ square centimeters is $14$ centimeters.

Example 5. What is the length of an arc on a circle with radius 8 meters and a central angle 45°?

Solution: To find the length of an arc on a circle, we use the formula:

$\text{Arc Length}=\frac{𝜃}{360\mathrm{°}}\times 2𝜋𝑟$

where:

• $𝜃$ is the central angle (in degrees),
• $𝑟$ is the radius of the circle.

Given that the radius $𝑟$ is $8$ meters and the central angle $𝜃$ is $45\mathrm{°}$, we can substitute these values into the formula:

$\text{Arc Length}=\frac{45\mathrm{°}}{360\mathrm{°}}\times 2𝜋\times 8$

$\text{Arc Length}=\frac{1}{8}\times 2𝜋\times 8$

$\text{Arc Length}=\frac{1}{8}\times 16𝜋$

$\text{Arc Length}=2𝜋$

So, the length of the arc on the circle with a radius of $8$ meters and a central angle of $45\mathrm{°}$ is $2𝜋$ meters.