Samara Goodman

2023-03-31

The distance between the centers of two circles C1 and C2 is equal to 10 cm. The circles have equal radii of 10 cm.

chicka152m0w3

Beginner2023-04-01Added 6 answers

To find the area of the shaded region, we need to determine the area of the sector formed by the circle and subtract the area of the equilateral triangle.

Let's break down the problem step by step:

Step 1: Find the area of the sector

The sector is formed by an angle of $\frac{2\pi}{3}$ radians (120 degrees) since the distance between the centers is equal to the radius of each circle. The formula to calculate the area of a sector is given by:

${A}_{\text{sector}}=\frac{1}{2}{r}^{2}\theta $

where $r$ is the radius of the circle and $\theta $ is the angle in radians. In this case, the radius is 10 cm and the angle is $\frac{2\pi}{3}$. Plugging these values into the formula, we have:

${A}_{\text{sector}}=\frac{1}{2}(10\text{cm}{)}^{2}\left(\frac{2\pi}{3}\right)$

Simplifying, we get:

${A}_{\text{sector}}=\frac{100}{3}\pi $

Step 2: Find the area of the equilateral triangle

Since the circles have equal radii of 10 cm, the distance from the center of each circle to the point where the circles intersect is also 10 cm. This forms an equilateral triangle. The formula to calculate the area of an equilateral triangle is given by:

${A}_{\text{triangle}}=\frac{\sqrt{3}}{4}{s}^{2}$

where $s$ is the side length of the triangle. In this case, the side length is 10 cm. Plugging this value into the formula, we have:

${A}_{\text{triangle}}=\frac{\sqrt{3}}{4}(10\text{cm}{)}^{2}$

Simplifying, we get:

${A}_{\text{triangle}}=\frac{100\sqrt{3}}{4}$

Step 3: Find the area of the shaded region

To find the area of the shaded region, we subtract the area of the equilateral triangle from the area of the sector:

${A}_{\text{shaded}}={A}_{\text{sector}}-{A}_{\text{triangle}}$

Substituting the previously calculated values, we have:

${A}_{\text{shaded}}=\frac{100}{3}\pi -\frac{100\sqrt{3}}{4}$

Step 4: Approximate the answer

Let's calculate the numerical value of the area of the shaded region using a calculator:

${A}_{\text{shaded}}\approx 100(\frac{2\pi}{3}-\frac{\sqrt{3}}{2})\approx 122.8$

Therefore, the approximate area of the shaded region is 122.8 square centimeters.

Note: I have rounded the final answer to one decimal place for simplicity.

Let's break down the problem step by step:

Step 1: Find the area of the sector

The sector is formed by an angle of $\frac{2\pi}{3}$ radians (120 degrees) since the distance between the centers is equal to the radius of each circle. The formula to calculate the area of a sector is given by:

${A}_{\text{sector}}=\frac{1}{2}{r}^{2}\theta $

where $r$ is the radius of the circle and $\theta $ is the angle in radians. In this case, the radius is 10 cm and the angle is $\frac{2\pi}{3}$. Plugging these values into the formula, we have:

${A}_{\text{sector}}=\frac{1}{2}(10\text{cm}{)}^{2}\left(\frac{2\pi}{3}\right)$

Simplifying, we get:

${A}_{\text{sector}}=\frac{100}{3}\pi $

Step 2: Find the area of the equilateral triangle

Since the circles have equal radii of 10 cm, the distance from the center of each circle to the point where the circles intersect is also 10 cm. This forms an equilateral triangle. The formula to calculate the area of an equilateral triangle is given by:

${A}_{\text{triangle}}=\frac{\sqrt{3}}{4}{s}^{2}$

where $s$ is the side length of the triangle. In this case, the side length is 10 cm. Plugging this value into the formula, we have:

${A}_{\text{triangle}}=\frac{\sqrt{3}}{4}(10\text{cm}{)}^{2}$

Simplifying, we get:

${A}_{\text{triangle}}=\frac{100\sqrt{3}}{4}$

Step 3: Find the area of the shaded region

To find the area of the shaded region, we subtract the area of the equilateral triangle from the area of the sector:

${A}_{\text{shaded}}={A}_{\text{sector}}-{A}_{\text{triangle}}$

Substituting the previously calculated values, we have:

${A}_{\text{shaded}}=\frac{100}{3}\pi -\frac{100\sqrt{3}}{4}$

Step 4: Approximate the answer

Let's calculate the numerical value of the area of the shaded region using a calculator:

${A}_{\text{shaded}}\approx 100(\frac{2\pi}{3}-\frac{\sqrt{3}}{2})\approx 122.8$

Therefore, the approximate area of the shaded region is 122.8 square centimeters.

Note: I have rounded the final answer to one decimal place for simplicity.

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