Question

SOLVE :-

$\frac{x}{360} \times\pi {r}^{2}$
​

Answer 1

Answer:

Explanation:

The first flower has 3 petals corresponding to the 3 corners of the triangle. The completed animation shows that each petal is a semicircle, so the perimeter of the flower is 3×π× radius =3×π× (side of triangle) =3πr.

The second flower has 4 petals. This time, each petal is a sector of a circle rather than a simple semicircle. The angle of this sector is 360 - (2 triangle corners) - (1 square corner) = 360−2×60− 90=150∘.

Therefore, the total perimeter of this petal is 4×150360×(2×π× radius )=103×π× (side of the square) =103πr.

In general, we need to know 3 key bits of data to work out the perimeter of the flower.

They are:

The number of sides of the central shape; we'll call this n.

The length of each side in the central shape; we'll call this r. (Note that this is equal to the radius of the petals).  

The angle at the centre of each petal. This can be derived from n:

Angle =360−2×60−( Corner of shape)

Angle =360−120−180(n−2)n

Angle =240−180−360n

Angle =60+360n

Given these data, we can proceed to work out a general formula:

 

Perimeter= (number of petals) × (perimeter of a full circle) ×angle at centre of petal360

Perimeter =n× 2×π×r×(60+360n)360

Perimeter =2×π×n×r×(16+1n)

Perimeter =2×π×n×r×6+n6n

Perimeter =π×r×6+n3

 

Using this formula, we find the following results:

n=3 (Triangle): Perimeter = π×r×93=3πr

n=4 (Square): Perimeter = π×r×103

n=5 (Pentagon): Perimeter = π×r×113

n=6 (Hexagon): Perimeter = π×r×123=4πr

n=7 (Heptagon): Perimeter = π×r×133

n=8 (Octagon):  Perimeter = π×r×143

...

n=100: Perimeter = π×r×1063

 

So a shape with 100 sides will produce a flower with a perimeter of π×r×1063.

If each edge of the central shape has a length of 1, the perimeter of the flower will be 35.333×π, which is 111.00 to two decimal places.