a circular pizza is cut along four diameters into eight identical sectors. If the total perimeter if each sector is 10 inches what is area of the whole pizza? express answer as a decimal to nearest tenth
To find the area of the whole pizza, we first need to find the radius of each sector.
Each sector has a total perimeter of 10 inches, which means each sector's arc length is \( \frac{10}{8} = 1.25 \) inches.
Since each arc length represents a fraction of the circumference, we can use the formula for the circumference of a circle to find the radius.
The formula for the circumference of a circle is \( C = 2\pi r \), where \( C \) is the circumference and \( r \) is the radius.
Substituting the given values, we have:
\[ 1.25 = 2\pi r \]
Now, we can solve for \( r \):
\[ r = \frac{1.25}{2\pi} \]
Using the value of \( \pi \) as 3.14, we can calculate the radius:
\[ r = \frac{1.25}{2 \times 3.14} \approx 0.199 \, \text{inches} \]
Next, we'll find the area of one sector using the formula for the area of a sector:
\[ A = \frac{1}{2} r^2 \theta \]
In this case, since the sector is 1/8th of the whole circle, the angle \( \theta \) is \( \frac{360}{8} = 45 \) degrees.
Substituting the values, we get:
\[ A = \frac{1}{2} \times 0.199^2 \times 45 \]
Calculating this expression, we find:
\[ A \approx 0.7 \, \text{square inches} \]
Since there are eight identical sectors in the whole pizza, we can multiply the area of one sector by eight to find the area of the whole pizza:
\[ \text{Area of whole pizza} \approx 0.7 \times 8 = 5.6 \, \text{square inches} \]
Therefore, the area of the whole pizza is approximately 5.6 square inches.
To find the area of the whole pizza, we need to determine the area of one sector and then multiply it by the total number of sectors.
First, let's find the circumference of the pizza. The total perimeter of each sector is given as 10 inches. Since each sector is identical, the circumference of the pizza is equal to the sum of the perimeters of all eight sectors, which is 10 inches per sector multiplied by 8 sectors, equaling 80 inches.
We know that the circumference of a circle (perimeter) is given by the formula C = 2πr, where C is the circumference and r is the radius of the circle. Rearranging the formula, we can solve for the radius:
C = 2πr
80 = 2πr
40 = πr
Now, let's solve for the radius:
r = 40/π
Next, we can use the formula for the area of a circle (A = πr^2) to find the area of one sector:
A_sector = (πr^2) / 8
Substituting the value of r:
A_sector = (π(40/π)^2) / 8
= (40^2/π) / 8
= (1600/π) / 8
= 200/π
Finally, to find the area of the whole pizza, we need to calculate the area of one sector and then multiply it by the total number of sectors:
A_pizza = A_sector * 8
= (200/π) * 8
Using a calculator, we can evaluate this expression:
A_pizza ≈ 639.3 square inches
Therefore, the area of the entire pizza is approximately 639.3 square inches, rounded to the nearest tenth.
Let r=radius of pizza.
Each sector has a radius of
2 times radius plus 1/8 of the circumference, or
10"=2r+(2πr)/8
Solve for r.
Then calculate πr²=area of whole pizza.