# Geometric Tutorial solid geometry

**Technical specification ****Geometrical work book/teaching guide "Nane"****“Шар” Teaching model "Ball"**

The proposed model extends and complements already well-known collection of educational models “NANE”, which consists of a set of pyramids, prisms, cylinders, cones. With these models and the model"Ball" it is possible to demonstrate the full course of stereometry.The given model allows showing various combinations of the ball with other geometric bodies and obviously demonstrating the solution of different spatial three T dimensional/3D-problems. Thus, the opportunity is given to learn the section of stereometry"Ball" efficiently.

**The complete model "Ball" includes the following components.**

**1. Three pairs of rings with different diameters.**

**2. The cluster of six identical, fixed in one point of the rods, whichhave telescopic mode of functioning.**

**3. Four-legged cluster.**

**4. Three-legged clusters (4 pieces).**

**5. Additional bar/rod to show the height of the inscribed pyramid.**

**6.***The base on which the whole model is consolidated.*The model should be used worthwhile in accordance with regular succession of the presentation of the course material on **stereometry** textbook.

The display of the ball model using two pairs of corresponding rings (Drawing/Fig. 1)

The first structure denoted in the Fig. 1 we mortise into the basement. Then the ring, which is located inside, is inserted into the mortise/groove.

We reinforce the middle rings into the deepness of the bigger ring meant for it (Fig. 2).

Fig.2 shows the cross-section areas: great circles and two other circles.

We fasten the cluster of six rods using the cramps (Fig. 3), and we obtain the center, the diameter, the radius, the segment, the sector of the ball.

With the help of three – legged clusters inside the ball we assemble the inscribed cube, rectangular parallelepiped (Fig. 4).

We assemble the inscribed tetragonal pyramid with the help of four – legged clusters (Fig. 5). In this construction is used such an auxiliary rod which we vertically fix with one ring to the top of the pyramid, and with the other ring we attach to the edge of the groove. After that we build the basis for what we use two three – legged clusters. Having constructed the basis we will see that from each cluster remains one free rod. We fix them in the middle of an auxiliary rod.

To obtain the triangle pyramid we closely attach one to each other. The basement is constructed similarly to the previous case (Fig. 6).

To display the inscribed cone the four – legged clusters is used (Fig. 7) and one middle ring.

With the help of inscribed into each other small rings it is possible to show two versions of the inscribed full-sphere – into a cube and pyramid (Fig. 8-9).

**Example of the task solution.**

The regular triangular pyramid with height of **h **and basement side of **a **is entered into the ball.

**Solution:**

On the model of Fig. 10

The radius of the ball** **OA=R,

Radius of the circle described around the basis О_{1}А=a/√3,

The height of the pyramid DO=h, OO_{1}=h-R.

For a rectangular triangle of OAO_{1} according to the Pythagorean theorem R^{2}=(h-R^{2})+a^{2}/3, R=(h^{2}+a^{2}/3)/ 2h

The possibilities of this model aren't limited with this. Lots of options of the model usage are opening in front of you, which stay in your imagination.