In a arcade game, the "monster" is the shaded sector of a circle of radius 3 cm, as shown in the figure. The missing piece (the mouth) has a central angle of 60 degrees. What is the area of the monster in cm^2
since 60 is 1/6 of 360, the mouth is 1/6 of the circle.
You now have just 5/6 of the circle.
I'm sure you know how to find the area of 5/6 of a circle, no?
To find the area of the monster, we need to calculate the area of the shaded sector and subtract the area of the missing piece (mouth).
1. Start by calculating the area of the shaded sector:
- The formula to find the area of a sector is A = (θ/360) * π * r^2, where θ is the central angle and r is the radius.
2. Substitute the given values into the formula:
- θ = 360 - 60 (since the missing piece has a central angle of 60 degrees and we want the shaded sector)
θ = 300 degrees
- r = 3 cm (given radius)
3. Plug the values into the formula:
A = (300/360) * π * 3^2
4. Simplify the equation:
A = (5/6) * π * 9
A = 45/2 * π
5. Calculate the numerical value by multiplying π (pi) with 45/2:
A ≈ 70.69 cm^2 (rounded to two decimal places).
Therefore, the area of the shaded sector (monster) is approximately 70.69 cm^2.
To find the area of the monster, we need to subtract the area of the missing piece (the mouth) from the total area of the shaded sector.
First, let's find the area of the whole shaded sector.
The formula to calculate the area of a sector is:
Area = (θ/360) * π * r^2
Where:
- θ is the central angle in degrees
- π is a mathematical constant approximately equal to 3.14159
- r is the radius of the circle
In this case, the central angle (θ) is given as 360 degrees - 60 degrees = 300 degrees.
Let's substitute the values into the formula:
Area of the whole shaded sector = (300/360) * π * 3^2
Now, let's find the area of the missing piece (the mouth). Since the missing piece is a sector with a central angle of 60 degrees, we can use the same formula to calculate its area.
Area of the missing piece (mouth) = (60/360) * π * 3^2
Finally, let's subtract the area of the missing piece from the area of the whole shaded sector to find the area of the monster:
Area of the monster = Area of the whole shaded sector - Area of the missing piece (mouth)
I'll calculate the actual values for you:
Area of the whole shaded sector = (300/360) * 3.14159 * 3^2 = 23.5619 cm^2
Area of the missing piece (mouth) = (60/360) * 3.14159 * 3^2 = 4.7124 cm^2
Area of the monster = 23.5619 cm^2 - 4.7124 cm^2 = 18.8495 cm^2
Therefore, the area of the monster is approximately 18.8495 cm^2.