How to Calculate Circle Area from Diameter (7th Grade Geometry)

📚 Understanding Circle Area from Diameter

In geometry, a circle's area is the amount of space it occupies in a two-dimensional plane. When given the diameter, we can easily calculate the area using a simple formula derived from the relationship between the diameter and the radius.

📜 Historical Context

The study of circles dates back to ancient civilizations. Mathematicians like Archimedes developed methods to approximate the value of pi ($\pi$), which is crucial for calculating circle areas. Over centuries, the understanding of circles has evolved, leading to precise formulas we use today.

📐 Key Principles

The key principle is understanding the relationship between diameter, radius, and $\pi$:

• 📏 The diameter ($d$) is the distance across the circle through the center.
• 📍 The radius ($r$) is the distance from the center to any point on the circle's edge. It's half the diameter ($r = \frac{d}{2}$).
• 🧮 Pi ($\pi$) is a constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.

➗ The Formula

To calculate the area ($A$) of a circle when you know the diameter ($d$):

1. First, find the radius: $r = \frac{d}{2}$.
2. Then, use the formula for the area of a circle: $A = \pi r^2$.
3. Substitute $r$ in the area formula: $A = \pi (\frac{d}{2})^2$.
4. Simplify: $A = \pi \frac{d^2}{4}$.

📝 Step-by-Step Calculation

Let's go through an example:

1. Example: Calculate the area of a circle with a diameter of 10 cm.
2. Find the radius: $r = \frac{d}{2} = \frac{10}{2} = 5$ cm.
3. Use the area formula: $A = \pi r^2 = \pi (5)^2 = 25\pi$ cm$^2$.
4. Approximate the value: $A ≈ 25 * 3.14159 ≈ 78.54$ cm$^2$.

🌍 Real-World Examples

• 🍕 Pizza: You have a pizza with a diameter of 12 inches. To find the area, first calculate the radius ($r = \frac{12}{2} = 6$ inches). Then, $A = \pi (6)^2 = 36\pi$ square inches.
• 🏊 Pool: A circular pool has a diameter of 20 feet. The radius is 10 feet, so the area is $A = \pi (10)^2 = 100\pi$ square feet.
• 🪙 Coin: A coin has a diameter of 2 cm. The radius is 1 cm, so the area is $A = \pi (1)^2 = \pi$ square centimeters.

💡 Tips and Tricks

• ✍️ Always remember to square the radius, not the diameter. A common mistake is forgetting to divide the diameter by 2 before squaring.
• 🔢 Use a calculator to get a more precise value when multiplying by $\pi$.
• 📐 Keep the units consistent. If the diameter is in centimeters, the area will be in square centimeters.

🎯 Practice Quiz

Calculate the area of a circle given the following diameters:

✍️ Conclusion

Calculating the area of a circle from its diameter is straightforward once you understand the relationship between diameter, radius, and $\pi$. With practice, you'll be able to solve these problems quickly and accurately. Keep exploring the fascinating world of geometry!

📚 Understanding Circle Area from Diameter

The area of a circle represents the total space enclosed within its boundary. Knowing how to calculate it from the diameter is a fundamental concept in geometry. Let's explore this!

📜 History and Background

The study of circles dates back to ancient civilizations like the Egyptians and Greeks. Mathematicians like Archimedes made significant contributions to understanding the properties of circles, including the relationship between its diameter, radius, and area. The formula we use today is a result of centuries of mathematical exploration.

🔑 Key Principles

To calculate the area of a circle from its diameter, you need to understand two key concepts:

• 📏 Diameter: The distance across the circle passing through the center.
• 半径 Radius: The distance from the center of the circle to any point on its edge. The radius is always half the diameter.

The formula to calculate the area of a circle is:

$A = \pi r^2$

Where:

• 🅰️ $A$ represents the area of the circle.
• 🥧 $\pi$ (pi) is a mathematical constant approximately equal to 3.14159.
• Radius $r$ is the radius of the circle.

✏️ How to Calculate Circle Area from Diameter

1. 📐 Find the Radius: Divide the diameter by 2 to find the radius ($r = \frac{d}{2}$).
2. ➕ Square the Radius: Calculate $r^2$ (radius multiplied by itself).
3. ➗ Multiply by Pi: Multiply the result by $\pi$ to find the area ($A = \pi r^2$).

🍩 Real-World Examples

Pizza Time

Imagine you have a pizza with a diameter of 12 inches. What's the area of the pizza?

1. 📐 Find the Radius: $r = \frac{12}{2} = 6$ inches
2. ➕ Square the Radius: $6^2 = 36$ square inches
3. ➗ Multiply by Pi: $A = \pi * 36 \approx 3.14159 * 36 \approx 113.1$ square inches

So, the area of the pizza is approximately 113.1 square inches.

Coin Calculation

Let's say you have a coin with a diameter of 2 centimeters. What's the area of the coin?

1. 📐 Find the Radius: $r = \frac{2}{2} = 1$ centimeter
2. ➕ Square the Radius: $1^2 = 1$ square centimeter
3. ➗ Multiply by Pi: $A = \pi * 1 \approx 3.14159 * 1 \approx 3.14$ square centimeters

Therefore, the area of the coin is approximately 3.14 square centimeters.

💡 Conclusion

Calculating the area of a circle from its diameter is a straightforward process once you understand the relationship between diameter, radius, and the formula $A = \pi r^2$. With this knowledge, you can easily solve various real-world problems involving circles. Keep practicing, and you'll master it in no time!

📚 Understanding Circle Area from Diameter

The area of a circle is the amount of space it occupies within its boundary. Knowing the diameter, which is the distance across the circle through its center, allows us to easily calculate this area. Let's explore how!

📜 History and Background

The study of circles dates back to ancient civilizations. Mathematicians like Archimedes developed methods to approximate the value of pi ($\pi$), which is fundamental to circle calculations. The formula for the area of a circle has been refined over centuries, becoming a cornerstone of geometry.

🔑 Key Principles

The key to calculating the area from the diameter lies in understanding the relationship between diameter, radius, and $\pi$.

• 📏 Diameter (d): The distance across the circle passing through the center.
• 📍 Radius (r): The distance from the center of the circle to any point on its edge. The radius is half the diameter: $r = \frac{d}{2}$.
• ♾️ Pi ($\pi$): A mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.

🧮 The Formula

The formula to calculate the area (A) of a circle using the radius (r) is:

$A = \pi r^2$

Since $r = \frac{d}{2}$, we can rewrite the formula in terms of the diameter (d) as:

$A = \pi (\frac{d}{2})^2 = \pi \frac{d^2}{4}$

✏️ Step-by-Step Calculation

1. 1️⃣ Measure the Diameter: Determine the length of the diameter of the circle.
2. ➗ Find the Radius: Divide the diameter by 2 to find the radius: $r = \frac{d}{2}$.
3. 🔢 Square the Radius: Calculate $r^2$ (radius multiplied by itself).
4. ✖️ Multiply by Pi: Multiply the result by $\pi$ (approximately 3.14159): $A = \pi r^2$.

🌍 Real-world Examples

• 🍕 Pizza: Imagine a pizza with a diameter of 12 inches. The radius is 6 inches. The area would be $A = \pi (6^2) = 36\pi \approx 113.1$ square inches.
• 🕳️ Pond: A circular pond has a diameter of 20 meters. The radius is 10 meters. The area is $A = \pi (10^2) = 100\pi \approx 314.16$ square meters.
• 🪙 Coin: A coin has a diameter of 25 mm. The radius is 12.5 mm. The area is $A = \pi (12.5^2) = 156.25\pi \approx 490.87$ square mm.

💡 Conclusion

Calculating the area of a circle from its diameter is a straightforward process using the formula $A = \pi \frac{d^2}{4}$. By understanding the relationship between diameter, radius, and $\pi$, you can easily determine the area of any circle. This skill is useful in various real-world applications, from cooking to construction!

📚 Understanding Circle Area from Diameter

Let's explore how to calculate the area of a circle when you know its diameter. It's a fundamental concept in geometry, and understanding it opens doors to solving many real-world problems.

📜 A Brief History of Circles

The study of circles dates back to ancient civilizations. Egyptians and Babylonians made significant strides in understanding circles, using approximations of $\pi$ (pi) in their calculations. The Greek mathematician Archimedes further refined the calculation of $\pi$ and developed methods to determine the area of a circle with greater accuracy.

🔑 Key Principles: Radius and $\pi$

Before calculating the area, remember these key principles:

• 📏 Radius: The radius ($r$) is the distance from the center of the circle to any point on its edge. It's half the diameter.
• ♾️ Diameter: The diameter ($d$) is the distance across the circle, passing through the center.
• 🥧 Pi ($\pi$): Pi is a mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.

📐 The Formula: Diameter to Area

Here's how to calculate the area ($A$) of a circle using the diameter ($d$):

1. First, find the radius: $r = \frac{d}{2}$
2. Then, use the area formula: $A = \pi r^2$
3. Substitute $r = \frac{d}{2}$ into the area formula: $A = \pi (\frac{d}{2})^2 = \pi \frac{d^2}{4}$

➕ Example 1: Simple Calculation

Let's say a circle has a diameter of 10 cm. Find its area.

1. Find the radius: $r = \frac{10}{2} = 5$ cm
2. Calculate the area: $A = \pi (5)^2 = 25\pi \approx 78.54$ cm$^2$

🏢 Example 2: Real-World Application

Imagine you're designing a circular garden with a diameter of 8 meters. How much space will the garden occupy?

1. Find the radius: $r = \frac{8}{2} = 4$ meters
2. Calculate the area: $A = \pi (4)^2 = 16\pi \approx 50.27$ m$^2$

💡 Tips and Tricks

• 🧮 Approximation: Use 3.14 as an approximation for $\pi$ when a calculator isn't available.
• ✍️ Units: Always include the correct units (e.g., cm$^2$, m$^2$) in your final answer.
• 🧐 Double-Check: Ensure you're using the radius and not the diameter directly in the $A = \pi r^2$ formula.

📝 Practice Quiz

Calculate the area of the following circles, given their diameters:

1. Diameter = 6 cm
2. Diameter = 14 m
3. Diameter = 20 mm
4. Diameter = 3 m
5. Diameter = 18 cm
6. Diameter = 100 m
7. Diameter = 1 mm

✅ Conclusion

Calculating the area of a circle from its diameter involves understanding the relationship between diameter, radius, and $\pi$. By mastering this concept, you can solve a variety of geometrical problems and appreciate the beauty of mathematics in everyday life.

📚 Understanding Circle Area from Diameter

In geometry, a circle's area represents the total space enclosed within its boundary. Knowing how to calculate this area from the diameter is a fundamental skill. The diameter is the distance across the circle through its center.

📜 Historical Context

The study of circles dates back to ancient civilizations. Mathematicians like Archimedes developed methods to approximate $\pi$ (pi), which is crucial for area calculations. Understanding circles was vital for early engineering, astronomy, and architecture.

📐 Key Principles

The area of a circle is calculated using the formula $A = \pi r^2$, where $A$ is the area and $r$ is the radius. Since the diameter ($d$) is twice the radius ($r$), we have $r = \frac{d}{2}$. Substituting this into the area formula gives us $A = \pi (\frac{d}{2})^2 = \frac{\pi d^2}{4}$.

• 🔍 Definition of Diameter: The diameter is a straight line passing through the center of the circle, connecting two points on the circumference.
• 📏 Relationship between Radius and Diameter: The radius is half the length of the diameter ($r = \frac{d}{2}$).
• ➗ Formula Derivation: Substituting $r = \frac{d}{2}$ into $A = \pi r^2$ gives $A = \frac{\pi d^2}{4}$.

✍️ Step-by-Step Calculation

1. Identify the Diameter: Determine the length of the diameter of the circle.
2. Calculate the Radius: Divide the diameter by 2 to find the radius ($r = \frac{d}{2}$).
3. Apply the Formula: Use the formula $A = \pi r^2$ or $A = \frac{\pi d^2}{4}$ to calculate the area.

🌍 Real-World Examples

Example 1: Pizza Size

A pizza has a diameter of 12 inches. What is its area?

1. Diameter, $d = 12$ inches.

2. Radius, $r = \frac{12}{2} = 6$ inches.

3. Area, $A = \pi (6)^2 = 36\pi \approx 113.1$ square inches.

Example 2: Garden Design

You're designing a circular garden with a diameter of 8 meters. What is the area you'll be planting?

1. Diameter, $d = 8$ meters.

2. Radius, $r = \frac{8}{2} = 4$ meters.

3. Area, $A = \pi (4)^2 = 16\pi \approx 50.27$ square meters.

💡 Tips and Tricks

• 🔢 Using $\pi$: For practical purposes, you can use 3.14 as an approximation for $\pi$.
• 📝 Units: Always include the correct units (e.g., square inches, square meters) when stating the area.
• calculator icon Calculator Use: Use a calculator to get a more precise value, especially for complex problems.

✅ Conclusion

Calculating the area of a circle from its diameter is straightforward once you understand the relationship between diameter, radius, and the area formula. By following the steps and examples provided, you can confidently solve related problems in geometry and real-world applications.

📚 Understanding Circle Area from Diameter

The area of a circle is the amount of space inside the circle. Knowing how to calculate it from the diameter is super useful! The diameter is simply the distance across the circle through its center. We'll explore how these two relate and how to easily calculate the area.

📜 History and Background

The study of circles dates back to ancient civilizations. Mathematicians like Archimedes developed methods to approximate the value of pi ($\pi$), which is fundamental to circle calculations. Over centuries, understanding circle properties has been crucial in fields ranging from astronomy to engineering.

🔑 Key Principles

The core principle involves understanding the relationship between the diameter, radius, and area of a circle.

• 📏 The diameter ($d$) is twice the radius ($r$): $d = 2r$
• 🧮 The radius ($r$) is half the diameter ($d$): $r = \frac{d}{2}$
• 📐 The area ($A$) of a circle is calculated using the formula: $A = \pi r^2$

Therefore, if you know the diameter, you can find the radius and then calculate the area.

➗ Step-by-Step Calculation

1. Find the Radius: Divide the diameter by 2 to get the radius ($r = \frac{d}{2}$).
2. Square the Radius: Calculate $r^2$.
3. Multiply by Pi: Multiply $r^2$ by $\pi$ (approximately 3.14159) to get the area ($A = \pi r^2$).

🍩 Real-World Examples

Example 1: Pizza Time!

A pizza has a diameter of 12 inches. What's its area?

1. Radius: $r = \frac{12}{2} = 6$ inches
2. Area: $A = \pi (6^2) = \pi (36) \approx 36 \times 3.14159 \approx 113.1$ square inches

Example 2: Circular Garden

A circular garden has a diameter of 8 meters. What's the area you can plant in?

1. Radius: $r = \frac{8}{2} = 4$ meters
2. Area: $A = \pi (4^2) = \pi (16) \approx 16 \times 3.14159 \approx 50.3$ square meters

Example 3: Coin Area

A coin has a diameter of 20 millimeters. What's its area?

1. Radius: $r = \frac{20}{2} = 10$ millimeters
2. Area: $A = \pi (10^2) = \pi (100) \approx 100 \times 3.14159 \approx 314.2$ square millimeters

📝 Practice Quiz

Calculate the area of the circles given their diameters:

💡 Conclusion

Calculating the area of a circle from its diameter is a fundamental skill in geometry. By understanding the relationship between diameter, radius, and the area formula, you can easily solve real-world problems. Keep practicing, and you'll master it in no time!

📚 Understanding Circle Area

The area of a circle is the amount of space inside the circle. Knowing how to calculate it is super useful in many real-world situations!

📜 A Little History

The study of circles dates back to ancient civilizations like the Egyptians and Greeks. Mathematicians have been fascinated by circles for thousands of years because of their unique properties. The formula for the area of a circle, $A = \pi r^2$, was developed over time through the work of many mathematicians.

📐 Key Principles: Diameter, Radius, and Pi

To calculate the area of a circle from its diameter, you need to understand a few key concepts:

• 📏 Diameter (d): The distance across the circle through the center.
• 半径 Radius (r): The distance from the center of the circle to any point on the circle. The radius is half of the diameter: $r = \frac{d}{2}$.
• 🧮 Pi (π): A mathematical constant approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter.

✍️ The Formula

The formula to calculate the area (A) of a circle when you know the radius (r) is:

$A = \pi r^2$

Since we often know the diameter (d) instead of the radius, and we know $r = \frac{d}{2}$, we can rewrite the formula as:

$A = \pi (\frac{d}{2})^2$

Which simplifies to:

$A = \pi \frac{d^2}{4}$

➗ Step-by-Step Calculation

1. 📏 Find the radius: Divide the diameter by 2 to find the radius.
2. 🔢 Square the radius: Multiply the radius by itself.
3. ➕ Multiply by pi: Multiply the squared radius by π (approximately 3.14159).

🌍 Real-World Examples

1. 🍕 Pizza: Imagine you have a pizza with a diameter of 12 inches. The radius is 6 inches. The area of the pizza is approximately $3.14159 * 6^2 = 113.1$ square inches.
2. ⛲ Fountain: A circular fountain has a diameter of 8 feet. The radius is 4 feet. The area of the fountain is approximately $3.14159 * 4^2 = 50.27$ square feet.
3. 🍪 Cookie: A cookie has a diameter of 3 inches. The radius is 1.5 inches. The area of the cookie is approximately $3.14159 * 1.5^2 = 7.07$ square inches.

💡 Practice Quiz

1. ❓ A circle has a diameter of 10 cm. What is its area?
2. ❓ If the diameter of a circle is 14 inches, calculate its area.
3. ❓ The diameter of a circular garden is 20 feet. What is the area of the garden?

✅ Conclusion

Calculating the area of a circle from its diameter is a straightforward process once you understand the relationship between diameter, radius, and pi. With this knowledge, you can solve a variety of real-world problems involving circles! Keep practicing, and you'll master it in no time.

📚 Understanding Circle Area from Diameter

Let's explore how to calculate the area of a circle when you're given the diameter. It's a fundamental concept in geometry, and once you grasp the relationship between diameter, radius, and area, it becomes quite straightforward!

📜 A Bit of History

The study of circles dates back to ancient civilizations. Early mathematicians like the Babylonians and Egyptians approximated the value of pi ($\pi$) to calculate the area of circles. The formula we use today is a result of centuries of mathematical refinement.

📐 Key Principles

• 📏 Diameter and Radius: The diameter ($d$) of a circle is the distance across the circle through its center. The radius ($r$) is the distance from the center to any point on the circle. The relationship is: $r = \frac{d}{2}$.
• 🧮 Area Formula: The area ($A$) of a circle is calculated using the formula: $A = \pi r^2$, where $\pi$ (pi) is approximately 3.14159.
• 🔄 Combining Them: To find the area from the diameter, substitute $r = \frac{d}{2}$ into the area formula: $A = \pi (\frac{d}{2})^2 = \pi \frac{d^2}{4}$.

✍️ Step-by-Step Calculation

1. Identify the Diameter: Determine the length of the diameter ($d$).
2. Calculate the Radius: Divide the diameter by 2 to find the radius ($r = \frac{d}{2}$).
3. Apply the Area Formula: Use the formula $A = \pi r^2$ or $A = \pi \frac{d^2}{4}$ to calculate the area.

🌍 Real-World Examples

• 🍕 Pizza: If a pizza has a diameter of 12 inches, its radius is 6 inches. The area is $A = \pi (6)^2 = 36\pi$ square inches, approximately 113.1 square inches.
• 🪙 Coin: A coin with a diameter of 20 mm has a radius of 10 mm. The area is $A = \pi (10)^2 = 100\pi$ square mm, approximately 314.16 square mm.
• 🕳️ Circular Table: A circular table with a diameter of 3 feet has a radius of 1.5 feet. The area is $A = \pi (1.5)^2 = 2.25\pi$ square feet, approximately 7.07 square feet.

💡 Tips and Tricks

• ✔️ Units: Always remember to include the appropriate units (e.g., square inches, square centimeters) when stating the area.
• 💻 Approximation: Use 3.14 or a calculator's $\pi$ button for more accurate results.
• 🧐 Double-Check: Ensure you've correctly identified the diameter and performed the calculations accurately.

📝 Practice Quiz

1. A circle has a diameter of 10 cm. What is its area?
2. The diameter of a circular garden is 8 meters. Calculate the area.
3. If the diameter of a round table is 5 feet, what is the area?
4. A circular window has a diameter of 30 inches. Find its area.
5. What is the area of a circle with a diameter of 14 mm?
6. The diameter of a circular plate is 25 cm. Calculate its area.
7. A circle's diameter is 42 inches. What is its area?

✅ Conclusion

Calculating the area of a circle from its diameter involves understanding the relationship between diameter and radius, and applying the area formula. With practice, you'll master this essential geometry skill!

📚 Understanding Circle Area and Diameter

In geometry, a circle's area is the amount of space it occupies in a two-dimensional plane. The diameter is the distance across the circle through its center. Knowing the diameter, we can easily find the area using a simple formula.

📜 A Brief History

The study of circles dates back to ancient civilizations. Mathematicians like Archimedes developed methods to approximate $\pi$ (pi), which is crucial for calculating circle areas. The formula we use today is a result of centuries of mathematical exploration.

🔑 Key Principles and Formulas

The key to calculating the area from the diameter lies in understanding the relationship between the diameter ($d$), the radius ($r$), and $\pi$.

• 📏 Radius: The radius is half of the diameter. $r = \frac{d}{2}$
• 🧮 Area Formula: The area ($A$) of a circle is given by $A = \pi r^2$

📝 Step-by-Step Calculation

Here's how to calculate the area of a circle when you know the diameter:

1. Find the Radius: Divide the diameter by 2 to find the radius.
2. Square the Radius: Multiply the radius by itself ($r^2$).
3. Multiply by Pi: Multiply the squared radius by $\pi$ (approximately 3.14159).

➗ Example 1: Diameter = 10 cm

1. Radius: $r = \frac{10}{2} = 5$ cm
2. Square the Radius: $5^2 = 25$ cm$^2$
3. Multiply by Pi: $A = \pi * 25 \approx 78.54$ cm$^2$

➕ Example 2: Diameter = 7 inches

1. Radius: $r = \frac{7}{2} = 3.5$ inches
2. Square the Radius: $3.5^2 = 12.25$ inches$^2$
3. Multiply by Pi: $A = \pi * 12.25 \approx 38.48$ inches$^2$

🌍 Real-World Applications

Calculating the area of circles is useful in many real-world scenarios:

• 🍕 Pizza: Determining the amount of pizza you get based on its diameter.
• ⚙️ Engineering: Designing circular components for machines.
• 🏡 Construction: Calculating the area of circular gardens or pools.

💡 Tips and Tricks

• 🔢 Use a Calculator: For more accurate results, use a calculator with a $\pi$ button.
• 📐 Units: Always include the correct units (e.g., cm$^2$, inches$^2$) in your answer.
• ✍️ Double-Check: Review your calculations to avoid mistakes.

✅ Practice Quiz

Calculate the area of the circles with the following diameters:

1. Diameter = 4 cm
2. Diameter = 12 inches
3. Diameter = 9 m
4. Diameter = 15 cm
5. Diameter = 20 inches
6. Diameter = 6 m
7. Diameter = 11 cm

🔑 Solutions to Practice Quiz

1. $12.57$ cm$^2$
2. $113.10$ inches$^2$
3. $63.62$ m$^2$
4. $176.71$ cm$^2$
5. $314.16$ inches$^2$
6. $28.27$ m$^2$
7. $95.03$ cm$^2$

⭐ Conclusion

Calculating the area of a circle from its diameter is a fundamental skill in geometry. By understanding the relationship between diameter, radius, and $\pi$, you can easily solve these problems and apply them in various real-world contexts.

📚 Understanding Circle Area and Diameter

In geometry, a circle's area is the amount of space inside the circle. The diameter is the distance across the circle through the center. Knowing the diameter makes finding the area simple!

📜 History and Background

The study of circles dates back to ancient civilizations like the Egyptians and Greeks. Mathematicians like Archimedes developed methods to approximate the value of pi ($\pi$), which is crucial for calculating circle area. The formula we use today is a result of centuries of mathematical exploration.

🔑 Key Principles

The formula to calculate the area of a circle using the diameter involves a few key steps:

• 📏Find the Radius: The radius ($r$) is half of the diameter ($d$). So, $r = \frac{d}{2}$.
• 🧮Square the Radius: Calculate $r^2$, which means $r * r$.
• ➗Multiply by Pi: Multiply the squared radius by pi ($\pi$), which is approximately 3.14159. The formula is: $A = \pi r^2$.
• ✅Final Formula (Diameter): Substituting $r = \frac{d}{2}$ into the area formula gives $A = \pi (\frac{d}{2})^2 = \frac{\pi d^2}{4}$.

➕ Example 1: Finding Area from Diameter

Let's say a circle has a diameter of 10 cm. Here’s how to find the area:

• 📏Find the Radius: $r = \frac{10}{2} = 5$ cm.
• 🧮Square the Radius: $5^2 = 25$ cm².
• ➗Multiply by Pi: $A = \pi * 25 \approx 3.14159 * 25 \approx 78.54$ cm².

➕ Example 2: Using the Formula Directly

If the diameter is 8 inches, we can use the formula $A = \frac{\pi d^2}{4}$:

• 📝Substitute: $A = \frac{\pi * 8^2}{4} = \frac{\pi * 64}{4} = 16\pi$.
• 🔢Calculate: $A \approx 16 * 3.14159 \approx 50.27$ square inches.

➕ Example 3: Real-World Application

Imagine a circular pizza with a diameter of 12 inches. What is the area of the pizza?

• 🍕Find the Radius: $r = \frac{12}{2} = 6$ inches.
• 🧮Square the Radius: $6^2 = 36$ square inches.
• 🥧Multiply by Pi: $A = \pi * 36 \approx 3.14159 * 36 \approx 113.10$ square inches.

💡 Practice Quiz

Calculate the area of the circles with the following diameters:

1. Diameter = 4 cm
2. Diameter = 7 inches
3. Diameter = 14 m

✅ Answers to Quiz

1. Area ≈ 12.57 cm²
2. Area ≈ 38.48 in²
3. Area ≈ 153.94 m²

🔑 Conclusion

Calculating the area of a circle from its diameter is straightforward once you understand the relationship between diameter, radius, and pi. This skill is useful in various real-world scenarios, from cooking to construction. Keep practicing, and you’ll master it in no time!