The exact area of a circle inscribed in a square with a side of 14 cm is 49π square centimeters.
Understanding Inscribed Circles in Squares
When a circle is inscribed within a square, it means that the circle fits perfectly inside, touching all four sides of the square at their midpoints. This geometric relationship is crucial for determining the circle's dimensions based on the square's side length.
Key Relationship: The diameter of a circle inscribed in a square is always equal to the side length of the square.
Calculating the Area of the Inscribed Circle
To find the area of the circle, we follow these steps:
1. Determine the Circle's Diameter
Since the circle is inscribed in a square with a side length of 14 cm, its diameter is directly equal to that side.
- Side of the square: 14 cm
- Diameter of the circle (d): 14 cm
2. Calculate the Circle's Radius
The radius (r) of a circle is half of its diameter.
- Radius (r): Diameter / 2 = 14 cm / 2 = 7 cm
3. Apply the Area Formula
The area (A) of any circle is calculated using the formula: A = πr², where π (pi) is a mathematical constant approximately equal to 3.14159.
- *A = π (7 cm)²**
- *A = π 49 cm²**
- A = 49π cm²
This result, 49π cm², represents the exact area of the inscribed circle. If an approximate numerical value is needed, one would substitute a value for π (e.g., 3.14 or 22/7).
Here’s a summary of the calculations:
| Property | Value | Calculation |
|---|---|---|
| Side of Square | 14 cm | Given |
| Diameter of Circle | 14 cm | Equal to square's side |
| Radius of Circle | 7 cm | Diameter ÷ 2 |
| Area of Circle | 49π cm² | π × (Radius)² |
For more detailed information on calculating the area of a circle, you can explore resources on circle geometry.
Practical Insight: Understanding the "Shaded Region"
In many geometry problems, you might encounter a "shaded region" which refers to the area of the square that is not covered by the inscribed circle. This provides an excellent real-world application of both square and circle area calculations.
-
Area of the Square:
The area of the square is calculated as side × side:
Area of Square = (14 cm)² = 196 cm². -
Area of the Inscribed Circle:
As calculated above, the area of the circle is 49π cm². -
Area of the Shaded Region (Square - Circle):
To find the area of the region between the square and the circle, subtract the circle's area from the square's area:
Area of Shaded Region = Area of Square - Area of Circle
Area of Shaded Region = 196 cm² - 49π cm².Using the common approximation of π ≈ 22/7 for calculation simplicity in many geometry problems:
Area of Shaded Region ≈ 196 cm² - (49 × 22/7) cm²
Area of Shaded Region ≈ 196 cm² - (7 × 22) cm²
Area of Shaded Region ≈ 196 cm² - 154 cm²
Hence, the area of shaded region, which represents the four corner sections of the square not occupied by the circle, is 42 cm². This demonstrates how the areas of combined shapes can be determined.
Key Takeaways
- The relationship between an inscribed circle and its square is fundamental: the circle's diameter matches the square's side.
- Always find the radius first, then apply the area formula A = πr².
- An answer in terms of 'π' is exact, while using a numerical approximation of 'π' yields an approximate value.
- Understanding these concepts allows for solving more complex problems, such as finding the area of regions between shapes.
