9. In The Figure Below, ABCD Is A Square And Semicircle O Has A Radius Of 6. What Is The Area Of The Figure? 1) 36+6x 2) 36+18 3) 144+ 18 4) 144 +36
Understanding the Problem
In the figure below, ABCD is a square and semicircle O has a radius of 6. The problem requires us to find the area of the figure.
Visualizing the Figure
[Insert figure of square ABCD with semicircle O]
Breaking Down the Problem
To find the area of the figure, we need to calculate the area of the square and the area of the semicircle, and then add them together.
Calculating the Area of the Square
The area of a square is given by the formula:
Area = side^2
In this case, the side length of the square is equal to the diameter of the semicircle, which is twice the radius. Therefore, the side length of the square is 2 x 6 = 12.
Area of square = 12^2 = 144
Calculating the Area of the Semicircle
The area of a semicircle is given by the formula:
Area = (Ï€r^2)/2
where r is the radius of the semicircle.
Area of semicircle = (Ï€(6)^2)/2 = 18Ï€
Approximating the Value of π
To find the numerical value of the area of the semicircle, we need to approximate the value of π. A commonly used approximation of π is 3.14.
Area of semicircle ≈ (3.14 x 6^2)/2 = 18 x 3.14 = 56.52
Rounding the Value of π
For the purpose of this problem, we can round the value of π to 3.14.
Calculating the Total Area
The total area of the figure is the sum of the area of the square and the area of the semicircle.
Total area = area of square + area of semicircle = 144 + 56.52 ≈ 200.52
Rounding the Total Area
For the purpose of this problem, we can round the total area to 200.
Conclusion
The area of the figure is approximately 200.
Answer
The correct answer is:
- 144 + 36
Note: The answer is not exactly 200, but rather 144 + 36, which is the closest option to the calculated value.
Discussion
This problem requires us to apply the formulas for the area of a square and a semicircle, and then add them together to find the total area. The problem also requires us to approximate the value of π, which is a fundamental constant in mathematics.
Key Concepts
- Area of a square
- Area of a semicircle
- Approximation of π
Related Problems
- Finding the area of a circle
- Finding the area of a rectangle
- Finding the area of a triangle
Q&A: Solving the Area of a Figure with a Square and Semicircle ===========================================================
Frequently Asked Questions
Q: What is the formula for the area of a square?
A: The formula for the area of a square is:
Area = side^2
Q: What is the formula for the area of a semicircle?
A: The formula for the area of a semicircle is:
Area = (Ï€r^2)/2
Q: What is the value of π used in the problem?
A: In the problem, we used the approximation of π as 3.14.
Q: How do we calculate the side length of the square?
A: The side length of the square is equal to the diameter of the semicircle, which is twice the radius. Therefore, the side length of the square is 2 x 6 = 12.
Q: How do we calculate the area of the semicircle?
A: To calculate the area of the semicircle, we use the formula:
Area = (Ï€r^2)/2
where r is the radius of the semicircle.
Q: Why do we need to approximate the value of π?
A: We need to approximate the value of π because it is an irrational number and cannot be expressed exactly as a finite decimal or fraction.
Q: What is the total area of the figure?
A: The total area of the figure is the sum of the area of the square and the area of the semicircle.
Q: Why is the answer not exactly 200?
A: The answer is not exactly 200 because we used an approximation of π as 3.14, which is not exact.
Q: What is the correct answer?
A: The correct answer is:
- 144 + 36
Q: What are some related problems that we can solve?
A: Some related problems that we can solve include:
- Finding the area of a circle
- Finding the area of a rectangle
- Finding the area of a triangle
Q: What are some key concepts that we need to understand?
A: Some key concepts that we need to understand include:
- Area of a square
- Area of a semicircle
- Approximation of π
Q: How can we apply these concepts to real-world problems?
A: We can apply these concepts to real-world problems such as:
- Finding the area of a room or a building
- Finding the area of a circular garden or a pond
- Finding the area of a triangular roof or a triangular piece of land
Conclusion
In this Q&A article, we have discussed the solution to the problem of finding the area of a figure with a square and a semicircle. We have also covered some frequently asked questions and provided some related problems and key concepts that we can apply to real-world problems.