The circumference and area of a circle s inscribed positive N deformation.

Solution: 1Ask for the girth:

Positive n deformation can be divided into n holoisosceles triangle waist length is r, let one of them be abc, a is the vertex, then the point a is ae perpendicular bc to e, we can know the angle bae=2 2n, so be=r*sin(2 2n)=r*sin(n), so n deformation perimeter = 2*n*r*sin( n);

2.Find the area:

Using the triangle area formula s= is the angle of the ab side) we can know that the triangle abc area = in this problem, such a triangle has n so the area of the positive n variable edge is: s=

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Consider a triangle formed by each side and the center of the circle.

Let the side length of the regular n-sided be x, and the corresponding angle of this side is 360 nx = root number(r 2+r 2 - 2r*r*cos(360 n)) = r*root number(2-2cos(360 n)).

Circumference nx = nr*root number(2-2cos(360 n)) A triangle with an area of r*r*sin(360 n) 2 regular n-sided area nrr*sin(360 n) 2

Let the regular n-sided call answer, the radius r of the circle is r, and the circumference is l and the area is the central angle of the circle of the megachain song s side pair = n 360

Side length = 2rsin (n 720).

Circumference l=2nr*sin(n 720).

Area s=(1 2)*r 2*n*sin(n 360).

The line connecting all vertices from the center of the circle to the regular n-sided is the radius and the length is rThese lines divide the regular n-sided into n congruent isosceles triangles. In this way, the apex angle of each triangle is 2 n, the waist length is r, the side length of the regular polygon is x, and the perpendicular line on the bottom edge of the isosceles triangle is made through the center of the circle, and the trigonometric function is used in the divided right triangle

sin((2π/n)/2)=(x/2)/rx=2rsin(π/n).

Let the centroid distance be y, y=rcos(n).

The area of each isosceles triangle = side length and side centroid distance 2rcos( n)*2rsin( n) 2r*rsin( n)cos( n).

r*rsin(2π/n)/2

The area of a regular polygon.

r*rsin(2π/n)/2 × n

nr*rsin(2π/n)/2

For the inscribed n-sides of a circle, each of their vertices is connected to the center of the circle, and the n-sides are assigned to n isosceles triangles of the same area.

The apex angle of each triangle is (360 n) degrees, then the area of each triangle = 1 2)r*r*sin(360 n)So: the total area of the n-sided square is :(n 2)(r 2)sin(360 n).

The circumference of the inscribed regular n-sided shape is: 2nrsin (180° suspicious bond n).

Let the n-sided shape have the radius r of the circle, the circumference of l and the area of s

The central angle of the pair of edges = n 360

Side length = 2rsin (n 720).

Circumference l=2nr*sin(n 720).

Area s=(1 2)*r 2*n*sin(n 360).

The central angle of the sides of a regular n-sided shape is: 2 n

So the side length is: 2rsin( n). The distance between the edge and the center of the circle is: rcos( n)

So the circumference is: 2nrsin( n)

The area is: NR 2SIN (2 N) 2

Regular hexagon:

Connect the center of the circle and the adjacent vertices respectively to form a triangle with a regular triangle, and the side length is 2rsin30° with a radius of 2rsin30°.

then the perimeter is 6r, and the area is the sum of the area of 6 regular triangles = 3r 2 4*6 = 3 3r 2 2 regular dodecagons:

Connect the center of the circle with the two adjacent vertices respectively to form a triangle with an isosceles triangle with a base angle of 75°. The apex angle is 30°

The side length is 2rsin15° of circular radius.

Then the circumference is 24rsin15°, and the area is the sum of the area of 12 isosceles triangles = r 2sin15° cos15°*12 = 3r 2

Regular twenty-quarters:

Connect the center of the circle with the two adjacent vertices respectively to form a triangle with an isosceles triangle and a base angle. The apex angle is 15°

then the side length is the radius of the circle.

then the circumference is 48rsin15° and the area is the sum of the areas of 24 isosceles triangles = r

Regularity: For regular n-sided shapes.

The circumference is: 2nrsin (180° n).

Area: 1 2nr 2sin (360° n) method can be deduced as I did above.

I wish my studies to improve every day, and if you don't understand, you can continue to ask me.

Hello classmates, this doesn't need graphics to memorize, you just need to remember the rules, this is a very simple law.

The side length of the circumscribed regular n-sided shape of the circle is an =2*r*sin(180° n), and the circumference is 2n*r*sin(180° n).

The area is n*r*sin(360° n)2

o The center angle of the inscribed regular n-sided is 360° n. Let one side of it be ab, (ab=an), connect oaob, and do od ab in d, then in rt oad, oa=r ad=an 2, od=rn, (rn is the side centr), aod=180° n. So an=2rsin180° n; rn=rcos180°/n.

s oab=1 2anrn=r sin180° ncos180° n, so the circumference of the regular n-sided =2nrsin180° n. Area s=nr sin180° ncos180° n.