Find the area of a decagon with a radius of 4mm.
Find the area of an octagon with a radius of 10 cm.
Area of a regular polygon:
A = ( 1 / 2 ) * n * r ^ 2 * sin ( 360 ° / n )
The area of a decagon:
( n = 10 )
( r = 4 mm )
( sin 36 ° = sqrt [ 10 - 2 sqrt ( 5 ) ] / 4
A = ( 1 / 2 ) * 10 * 4 ^ 2 * sin ( 360 ° / 10 )
A = 10 / 2 * 16 * sin ( 36 ° )
A = 5 * 16 * sin ( 36 ° )
A = 5 * 16 * sqrt [ 10 - 2 sqrt ( 5 ) ] / 4
A = 5 * 16 / 4 * sqrt [ 10 - 2 sqrt ( 5 ) ]
A = 5 * 4 * sqrt [ 10 - 2 sqrt ( 5 ) ]
A = 20 * 2.351141
A = 47.02282 mm ^ 2
The area of a octagon:
( n = 8 )
( r = 10 mm )
( sin 45 ° = 1 / sqrt ( 2 ) = sqrt ( 2 ) / 2 )
A = ( 1 / 2 ) * 8 * 10 ^ 2 * sin ( 360 ° / 8 )
A = 8/ 2 * 100 * sin ( 45 ° )
A = 4 * 100 * sin ( 45 ° )
A = 400 * sqrt ( 2 ) / 2
A = ( 400 / 2 ) * sqrt ( 2 )
A = 200 * sqrt ( 2 )
A = 200 * 1.41421356
A = 282.842712 mm ^ 2
Find the area of the shaded portion of the diagram (use 3.14 for p).
Why did the decagon go to the doctor? Because it wasn't feeling "dodecafine"! Alright, let's calculate its area.
To find the area of a decagon, we can divide it into 10 congruent triangles. Each triangle's base will be one side of the decagon, and the height is the radius. So, the area of each triangle is (1/2) * base * height = (1/2) * 4mm * 4mm = 8mm².
Since we have 10 triangles, we can multiply the area of one triangle by 10 to get the total area of the decagon. Therefore, the area of the decagon with a radius of 4mm is 8mm² * 10 = 80mm². Pretty "tri-tastic," isn't it?
Now, let's tackle the octagon with a radius of 10 cm, like a pro wrestling octagon but with numbers. To find its area, we can divide it into 8 congruent isosceles triangles. Each triangle's base will be one side of the octagon, and the height is the radius. In this case, the height is also the side length of one of the triangles. That means the area of each triangle is (1/2) * base * height = (1/2) * 10cm * 10cm = 50cm².
Since we have 8 triangles, we multiply the area of one triangle by 8 to get the total area of the octagon. Therefore, the area of the octagon with a radius of 10 cm is 50cm² * 8 = 400cm². That's a "tri-umphant" octagon to "measure up" to!
To find the area of a decagon with a given radius, we need to break it down into smaller parts.
Step 1: Find the length of one side:
In a regular decagon, all sides are of equal length. To find the length of one side, we can use the formula for the circumference of a circle:
C = 2πr
where C is the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius.
Given that the radius is 4mm, the formula becomes:
C = 2π(4mm)
C = 8π mm
Step 2: Find the length of one side:
Since a decagon has 10 sides, the length of one side is equal to the circumference divided by 10:
Side length = C / 10
Side length = 8π mm / 10
Side length = 0.8π mm
Step 3: Find the area of the decagon:
The area of a regular decagon can be found using the formula:
Area = (5/4) × side length^2 × cotangent(π/10)
Given that we already know the side length, we can substitute it into the formula:
Area = (5/4) × (0.8π mm)^2 × cotangent(π/10)
Now, let's calculate the result.
For the octagon:
Step 1: Find the length of one side:
Similar to the decagon, we can find the length of one side of the octagon using the formula for circumference:
C = 2πr
Given that the radius is 10 cm, the formula becomes:
C = 2π(10 cm)
C = 20π cm
Step 2: Find the length of one side:
Since an octagon has 8 sides, the length of one side is equal to the circumference divided by 8:
Side length = C / 8
Side length = 20π cm / 8
Side length = 2.5π cm
Step 3: Find the area of the octagon:
The area of a regular octagon can be found using the formula:
Area = 2 × (1 + √2) × side length^2
Given that we already know the side length, we can substitute it into the formula:
Area = 2 × (1 + √2) × (2.5π cm)^2
Now, let's calculate the result.
To find the area of a regular decagon with a given radius, you can follow these steps:
1. Recall that a regular decagon has ten equal sides and ten equal angles.
2. Divide the decagon into ten congruent isosceles triangles by drawing lines from the center to each vertex.
3. Each of these triangles has a base equal to one side of the decagon and two congruent sides equal to the given radius.
4. To find the area of one of these triangles, you can use the formula for the area of an isosceles triangle: Area = (base * height) / 2. In this case, the base is a side of the decagon, and the height is the radius.
5. Calculate the area of one of these triangles.
6. Since there are ten congruent triangles in a decagon, multiply the area of one triangle by ten to get the total area of the decagon.
Now let's calculate the area of a decagon with a radius of 4 mm:
1. The base of each isosceles triangle within the decagon is equal to one side of the decagon. However, since the length of the sides is not provided, we need some additional information.
2. If you have the length of one side of the decagon, you can use it as the base of the isosceles triangle. Otherwise, with only the radius given, we cannot calculate the area.
Now let's move on to the octagon:
To find the area of a regular octagon with a given radius, you can follow these steps:
1. Recall that a regular octagon has eight equal sides and eight equal angles.
2. Divide the octagon into eight congruent isosceles triangles by drawing lines from the center to each vertex.
3. Each of these triangles has a base equal to one side of the octagon and two congruent sides equal to the given radius.
4. To find the area of one of these triangles, you can use the formula for the area of an isosceles triangle: Area = (base * height) / 2. In this case, the base is a side of the octagon, and the height is the radius.
5. Calculate the area of one of these triangles.
6. Since there are eight congruent triangles in an octagon, multiply the area of one triangle by eight to get the total area of the octagon.
Now let's calculate the area of an octagon with a radius of 10 cm:
1. The base of each isosceles triangle within the octagon is equal to one side of the octagon. However, since the length of the sides is not provided, we need some additional information.
2. If you have the length of one side of the octagon, you can use it as the base of the isosceles triangle. Otherwise, with only the radius given, we cannot calculate the area.