I need to find the area of the shaded portion of a circle with a square in the middle all corners touching the circle. My square has a line measurement of 10 cm. I know that the area of the square is 100 cm(10 x 10). But I am not sure what to do with the circle. Is the diameter 10 cm thus I take 10 x 3.14 minus the square area? I just do not understand.
None of the sides of the square is the diameter. The diameter is the line connecting two opposite corners -- in other words the hypotenuse of the two triangles formed. Use the Pythagorean theorem to find this diagonal line.
a^2 + b^2 = c^2
10^2 + 10^2 = c^2
100 + 100 = 200
The square root of 200 = 14.142
To find the area of the shaded portion, you need to subtract the area of the square from the area of the circle. Here's how you can do it step-by-step:
1. Calculate the area of the square: Since you mentioned that the side length of the square is 10 cm, you are correct that the area of the square is 10 cm × 10 cm = 100 cm².
2. Calculate the area of the circle: To find the area of a circle, you need to know either the radius or the diameter. In this case, if the square is inscribed inside the circle (meaning all four corners of the square touch the circle), the diameter of the circle is equal to the side length of the square. So, the diameter of the circle is also 10 cm.
3. Calculate the radius of the circle: The radius is half of the diameter. In this case, the radius is 10 cm ÷ 2 = 5 cm.
4. Calculate the area of the circle: The formula to find the area of a circle is A = πr², where A represents the area and r represents the radius. Plugging in the values, we get A = π × (5 cm)² = 25π cm².
5. Subtract the area of the square from the area of the circle: Finally, subtract the area of the square (100 cm²) from the area of the circle (25π cm²) to find the area of the shaded portion.
The area of the shaded portion = 25π cm² - 100 cm².
To find the area of the shaded portion, you need to calculate the area of the circle and subtract the area of the square.
First, let's find the area of the square. You mentioned that the square has a line measurement of 10 cm. Since all sides of a square are equal, the length of each side of the square is 10 cm.
To find the area of a square, you need to multiply the length of one side by itself. So, in this case, the area of the square is 10 cm multiplied by 10 cm, which equals 100 cm².
Now, let's move on to finding the area of the circle. To do this, you need to know the radius or diameter of the circle. If the square touches the circle at all four corners, then the diagonal of the square is equal to the diameter of the circle.
Since you mentioned that the length of one side of the square is 10 cm, you can use the Pythagorean theorem to find the length of the diagonal (diameter). The formula is a² + b² = c², where a and b are the lengths of the sides of the square, and c is the diagonal (diameter) of the square.
In this case, a = b = 10 cm. Plugging these values into the formula, we get:
10² + 10² = c²
100 + 100 = c²
200 = c²
Taking the square root of both sides, we find that c ≈ 14.14 cm (rounded to two decimal places). So, the diameter of the circle is approximately 14.14 cm.
Now that we have the diameter of the circle, we can calculate its radius. The radius is half of the diameter, so the radius would be 14.14 cm divided by 2, which is approximately 7.07 cm (rounded to two decimal places).
Finally, we can find the area of the circle using the formula A = πr², where A represents the area and r represents the radius.
Plugging in the radius we found, we get:
A = π(7.07 cm)²
A = 3.14 × (7.07 cm)²
A ≈ 3.14 × 49.99 cm²
A ≈ 157.06 cm²
Now that we have the area of the circle, which is approximately 157.06 cm², and the area of the square, which is 100 cm², we can find the area of the shaded portion by subtracting the area of the square from the area of the circle:
Area of shaded portion = Area of circle - Area of square
Area of shaded portion = 157.06 cm² - 100 cm²
Area of shaded portion ≈ 57.06 cm²
So, the approximate area of the shaded portion is 57.06 cm².