Learn on PengiBig Ideas Math, Course 2Chapter 8: Circles and Area

Lesson 3: Areas of Circles

In this Grade 7 lesson from Big Ideas Math Course 2, students learn how to find the area of a circle using the formula A = πr², including how to apply it when given either the radius or the diameter. The lesson covers estimating circle area by comparing it to a surrounding square, deriving the formula by rearranging sectors into a parallelogram, and extending the concept to find the area of semicircles. Real-world problem-solving contexts, such as calculating the area of a semicircular orchestra pit, help students apply the formula in practical situations.

Section 1

Area of a circle

Property

The formula for area of a circle is:

A=πr2A = \pi r^2

Examples

  • A circular sandbox has a radius of 2.5 feet. Its area is A=πr2≈3.14(2.5)2=3.14(6.25)=19.625A = \pi r^2 \approx 3.14(2.5)^2 = 3.14(6.25) = 19.625 square feet.
  • The lid of a paint bucket has a radius of 7 inches. Using π≈227\pi \approx \frac{22}{7}, the area is A≈227(7)2=227(49)=154A \approx \frac{22}{7}(7)^2 = \frac{22}{7}(49) = 154 square inches.

Section 2

Calculating Area from Diameter: The Diameter Trap

Property

The area formula

A=πr2A = \pi r^2
ONLY works with the radius. If a problem gives you the diameter (d), you MUST divide it by 2 to find the radius before doing anything else.

Examples

  • Circle with diameter 10 cm: First, find the radius (10 / 2 = 5 cm). Then, A=π(52)=25πA = \pi(5^2) = 25\pi square cm.
  • Circle with diameter 14 inches: First, find the radius (14 / 2 = 7 inches). Then, A=π(72)=49πA = \pi(7^2) = 49\pi square inches.
  • Circle with diameter 6.8 meters: Radius is 3.4 meters. A=π(3.42)=11.56πA = \pi(3.4^2) = 11.56\pi square meters.

Explanation

This is the most common mistake in geometry! Since the diameter is twice as long as the radius, if you accidentally plug the full diameter into the A=πr2A = \pi r^2 formula, your final answer won't just be double—it will be FOUR times too big! Always ask yourself: "Did they give me the full slice across, or just from the center?" If it's the full slice, chop it in half first.

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Chapter 8: Circles and Area

  1. Lesson 1

    Lesson 1: Circles and Circumference

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  2. Lesson 2

    Lesson 2: Perimeters of Composite Figures

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  3. Lesson 3Current

    Lesson 3: Areas of Circles

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  4. Lesson 4

    Lesson 4: Areas of Composite Figures

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Lesson overview

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Section 1

Area of a circle

Property

The formula for area of a circle is:

A=πr2A = \pi r^2

Examples

  • A circular sandbox has a radius of 2.5 feet. Its area is A=πr2≈3.14(2.5)2=3.14(6.25)=19.625A = \pi r^2 \approx 3.14(2.5)^2 = 3.14(6.25) = 19.625 square feet.
  • The lid of a paint bucket has a radius of 7 inches. Using π≈227\pi \approx \frac{22}{7}, the area is A≈227(7)2=227(49)=154A \approx \frac{22}{7}(7)^2 = \frac{22}{7}(49) = 154 square inches.

Section 2

Calculating Area from Diameter: The Diameter Trap

Property

The area formula

A=πr2A = \pi r^2
ONLY works with the radius. If a problem gives you the diameter (d), you MUST divide it by 2 to find the radius before doing anything else.

Examples

  • Circle with diameter 10 cm: First, find the radius (10 / 2 = 5 cm). Then, A=π(52)=25πA = \pi(5^2) = 25\pi square cm.
  • Circle with diameter 14 inches: First, find the radius (14 / 2 = 7 inches). Then, A=π(72)=49πA = \pi(7^2) = 49\pi square inches.
  • Circle with diameter 6.8 meters: Radius is 3.4 meters. A=π(3.42)=11.56πA = \pi(3.4^2) = 11.56\pi square meters.

Explanation

This is the most common mistake in geometry! Since the diameter is twice as long as the radius, if you accidentally plug the full diameter into the A=πr2A = \pi r^2 formula, your final answer won't just be double—it will be FOUR times too big! Always ask yourself: "Did they give me the full slice across, or just from the center?" If it's the full slice, chop it in half first.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 8: Circles and Area

  1. Lesson 1

    Lesson 1: Circles and Circumference

    LearnPractice
  2. Lesson 2

    Lesson 2: Perimeters of Composite Figures

    LearnPractice
  3. Lesson 3Current

    Lesson 3: Areas of Circles

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  4. Lesson 4

    Lesson 4: Areas of Composite Figures

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