Circle Formulas

Grade 7 · geometry · 100 practice problems · read aloud

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🔵 Circle Formulas: Your Quick Guide

Understanding circles is key in geometry! We use formulas to find the distance around a circle (circumference) and the space inside it (area). You'll see these concepts in everything from pizza sizes to bicycle wheels!

Key Formulas to Know

  • Circumference (Perimeter): C = 2πr or C = πd
  • Area: A = πr²

π (Pi) is a special number, approximately 3.14. r is the radius (center to edge). d is the diameter (edge to edge through the center). Remember: d = 2 × r!

Step-by-Step Problem Solving

  1. Identify what you're finding (circumference or area).
  2. Find the radius. If you have the diameter, divide it by 2.
  3. Choose the correct formula and plug in the numbers.
  4. Calculate, using 3.14 for π.
  5. Label your answer with units (e.g., cm, cm²).

Worked Examples

Example 1: Finding Circumference
A circle has a radius of 5 cm. What is its circumference?

  1. We are finding C. We have r = 5 cm.
  2. Use the formula: C = 2πr.
  3. Plug in: C = 2 × 3.14 × 5
  4. Calculate: C = 31.4 cm

Example 2: Finding Area
A circle has a diameter of 12 m. What is its area?

  1. We are finding A. We have d = 12 m.
  2. First, find the radius: r = d ÷ 2 = 12 ÷ 2 = 6 m.
  3. Use the formula: A = πr².
  4. Plug in: A = 3.14 × (6 × 6) = 3.14 × 36
  5. Calculate: A = 113.04 m²

⚠️ Common Mistakes to Avoid

  • Using diameter in the area formula: Always use the radius for A = πr². If you're given the diameter, divide by 2 first!
  • Forgetting to square the radius: A = πr² means π × (r × r). It's not π × r × 2!
  • Incorrect units: Circumference is in units (cm). Area is in square units (cm²).

💡 Tips & Tricks

  • Memory Aid: For area, remember "Pie are square?" It's a silly joke to remind you that the Area formula has the radius squared (A = πr²).
  • Check your work: The area should always be a larger number than the circumference for the same circle.
  • Estimate: 3 × r² gives you a rough estimate of the area. 3 × 2 × r gives you a rough estimate of the circumference.

Practice Makes Perfect! 🎯

Try these on your own:

  1. Find the circumference of a circle with a radius of 7 in.
  2. Find the area of a circle with a diameter of 10 km.
  3. A circular trampoline has a radius of 4 feet. What is its area?

Tip: Draw the circle and label the radius or diameter before you start calculating!

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 π × (14² - 7²) = ?

Hint: When dealing with the difference of two squared values, consider using the difference of squares formula to simplify the calculation before multiplying by the constant.

Show the answer

Answer: 147π

  1. Calculate the squares inside the parentheses. 14² = 14 × 14 = 196 7² = 7 × 7 = 49
  2. Subtract the squares. 196 - 49 = 147
  3. Multiply by π. π × 147 = 147π So the final answer is 147π. Explanation: We used the order of operations (parentheses first, then multiplication) and computed the squares before subtracting. The π remains as a symbol in the final answer.

Let's solve step by step. We have: π × (14² - 7²)

2 π × (15² - 9²) = ?

Hint: This involves finding the area between two concentric circles. First calculate the squares of the numbers, then find their difference, and finally multiply by pi.

Show the answer

Answer: 144π

  1. Calculate 15 squared: 15 × 15 = 225
  2. Calculate 9 squared: 9 × 9 = 81
  3. Find the difference: 225 - 81 = 144
  4. Multiply by pi: π × 144 = 144π

The answer is 144π.

3 π × (21² - 9²) = ?

Hint: This problem involves finding the area between two concentric circles. First, calculate each square separately, then subtract the smaller square from the larger square, and finally multiply by π.

Show the answer

Answer: 360π

  1. Calculate the square of the larger radius: 21² = 441
  2. Calculate the square of the smaller radius: 9² = 81
  3. Subtract the smaller square from the larger square: 441 - 81 = 360
  4. Multiply the result by π: 360 × π = 360π

The answer is 360π.

4 π × (22² - 8²) = ?

Hint: This problem involves finding the area of a ring between two circles. First, calculate each square separately, then subtract the smaller from the larger, and finally multiply by π.

Show the answer

Answer: 420π

  1. Calculate 22² = 484
  2. Calculate 8² = 64
  3. Find the difference: 484 - 64 = 420
  4. Multiply by π: 420 × π = 420π

The answer is 420π.

5 π × (20² - 5²) = ?

Hint: This problem involves finding the area between two concentric circles. First, calculate each square separately, then subtract the smaller square from the larger square. Finally, multiply the result by π.

Show the answer

Answer: 375π

  1. Calculate the square of the larger radius: 20² = 400
  2. Calculate the square of the smaller radius: 5² = 25
  3. Subtract the smaller square from the larger square: 400 - 25 = 375
  4. Multiply the result by π: 375 × π = 375π

The answer is 375π.

6 π × (18² - 12²) = ?

Hint: This involves finding the area between two concentric circles. Remember to calculate the squares first, then subtract, and finally multiply by π.

Show the answer

Answer: 180π

  1. Calculate the square of the larger radius: 18² = 324
  2. Calculate the square of the smaller radius: 12² = 144
  3. Subtract the smaller square from the larger square: 324 - 144 = 180
  4. Multiply the result by π: 180 × π = 180π
  5. The final answer is 180π.
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