You will need

- a pair of compasses;
- - the range;
- pencil.

Instruction

1

To start building, draw an arbitrary circle and label its center O. Then draw the radius of this circle in any direction. The point of intersection of a radius with a circumference indicate by the letter A. then move the compass to point a and guide circle or arc of the same radius as the original circle (OA). This arc will intersect the original circle at two points. Designate them by the letters b and C.

2

Connect the two obtained points. The segment BC will intersect the radius OA. The point of their intersection denote by the letter D. the resultant segments BD and DC are equal and each of them will be approximately equal to the right side of

**the heptagon**but that can be inscribed in the original circle.3

The compass, measure the distance BD (or DC) and starting from any point on the circle, put it away six times. Then connect all seven points. So you will get

**the heptagon**, which is a small margin of error can be called correct. All its sides and angles will be approximately equal.4

There is another way to build a correct

**heptagon**. To start, draw a random circle and spend two mutually perpendicular diameter of the circle. Call them AB and CD. Then one of the diameters (e.g., AB) divide into seven equal parts. For example, if the length of your diameter is 14 cm, the length of each part will be equal to 2, see the result on the given diameter, you should receive the six marks.5

Then, move the compass to one of the ends of this diameter (for example,) and from this point, draw an arc, the radius of which is equal to the diameter of the original circle (AB). Then extend the second diameter (CD) to the intersection with the constructed arc. The resulting point label with the letter E.

6

Now from point e, draw straight, passing through only the even or odd division on the diameter AB. For example, through the second, fourth and sixth division. The point of intersection of these lines with the circle are three of the seven vertices of your polygon. Label them F, G and H. the Fourth vertex is point a (in that case, if you held straight through the even-numbered level) or point To (if one of the lines passed through the nearest to the point And cut-off).

7

To find the fifth, sixth and seventh peaks, guide of the points F, G and H are straight, strictly perpendicular to the diameter AB. The points at which these lines cross the opposite side of the circle are three required vertices. To complete the build you will need to connect all seven peaks.

# Advice 2 : How to find the diagonal of a square

A square is a regular quadrilateral or a rhombus, in which all sides are equal and form between themselves angles of 90 degrees. The diagonal of a square - cut that connects two opposite corners of a square.

To find the diagonal of a square is easy enough

To find the diagonal of a square is easy enough

Instruction

1

So, you should start with the fact that around the square, describe a circle, the diagonal of which is exactly equal to the diagonal of the square. In order to calculate the radius of the circumscribed circle, use the formula:

R = (√2*a)/2, where a is square side.

Also in the square and inscribed circle. The circle at the points of tangency with the sides of a square divides them in half. The formula, which can be used to calculate the radius of the inscribed circle looks like this:

r = a/2

If, when solving the problem, the known radius of the circle inscribed in this square, it is possible to Express the side of the square, the value of which is required for finding the diagonal of a square:

a = 2*r

R = (√2*a)/2, where a is square side.

Also in the square and inscribed circle. The circle at the points of tangency with the sides of a square divides them in half. The formula, which can be used to calculate the radius of the inscribed circle looks like this:

r = a/2

If, when solving the problem, the known radius of the circle inscribed in this square, it is possible to Express the side of the square, the value of which is required for finding the diagonal of a square:

a = 2*r

2

The length of the radius of the circle equal to half the length of its diagonal. Thus, the length of the diagonal of the circumscribed circle, and, hence, the length of the diagonal of a square can be calculated by the formula:

d = √2*a

d = √2*a

3

For clarity, we can consider a small example:

Given a square with a side length of 9 cm, it is required to find the length of its diagonal.

Solution: in order to calculate its length, you will need to use the formula above:

d = √2*9

d = √162 cm

Answer: the length of the diagonal of a square with side 9 cm is equal to √162 cm or approximately 14.73 cm

Given a square with a side length of 9 cm, it is required to find the length of its diagonal.

Solution: in order to calculate its length, you will need to use the formula above:

d = √2*9

d = √162 cm

Answer: the length of the diagonal of a square with side 9 cm is equal to √162 cm or approximately 14.73 cm

Note

It is worth considering that the diagonal of a square divides it into 2 right-angled triangles which, after all, still are isosceles.

The fact that the diagonal is the hypotenuse of one of the two right triangles. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2 = b2 + c2). Then the length of the diagonal a can be found using the formula:

a = √(b2 + c2), where b = c, since they are equal sides of a square.

The fact that the diagonal is the hypotenuse of one of the two right triangles. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2 = b2 + c2). Then the length of the diagonal a can be found using the formula:

a = √(b2 + c2), where b = c, since they are equal sides of a square.

Useful advice

We must not forget that the square is a unique geometric figure. The square has five axes of symmetry, four of which are reflective axes and one rotational.

# Advice 3 : How to fit a square into a circle

To inscribe a square in a circle is easy with the help of drawing instruments. But this problem can be solved even if you complete their absence. You only need to remember some properties of a square.

You will need

- compass
- pencil
- -gon
- -scissors

Instruction

1

Draw the sketch for the challenge. It is obvious that the diameter of the circle is the diagonal of the inscribed into this square circle. Remember a well-known property of the square: its diagonals are mutually perpendicular. Use this relationship of the diagonals when building a given square.

2

Draw the circle diameter. From the center by means of a square guide, a second diameter at an angle of 90 degrees to the first. Connect the points of intersection of the perpendicular diameters of the circle and get inscribed in this circle a square.

3

If the drawing tools you have only a compass, draw a circle. Mark on the circle an arbitrary point and swipe through her diameter with any object with a straight edge. Now you need with the help of a compass to divide a semi-circle between the ends of the diameter into two equal parts. From the points of intersection of the diameter with the circumference make two notches, keeping unchanged the solution of the compass. Through the point of intersection of these notches and the center of the circle do a second diameter. It is obvious that it will be perpendicular to the first.

4

If the drawing tools you have, scissors cut paper circle, outlined with a given circumference. Fold cut out a shape exactly in half. Retry the operation. To combine the ends of the bend line, then the curved portions coincide with no additional effort. Fix line addition. Now expand the circle. The fold lines are clearly visible. Fold the circle segments between points of intersection of the bend lines with the circle and cut these segments. Cut lines are the sides of the required square. Place the cut square in a given circle, aligning its center with the point of intersection of the fold lines of the circle. The vertices of the square will be lying on the circle that was required to perform.