Finding of diagonal of the correct prism is often used as the intermediate stage at the solution of more difficult tasks. The general formula is easily removed by consideration of two rectangular triangles.
Instruction
1. For finding of diagonal of the correct prism you need to understand only several definitions. A prism is called the polyhedron having as the bases two equal polygons (triangle, quadrangles, etc.) lying in the parallel planes, and as side sides - parallelograms. A direct prism is called the prism which has side sides rectangles. The correct prism is called the direct prism which bases are regular polygons (an equilateral triangle, a square, etc.) of ABCDA1B1C1D1 - the Correct quadrangular prism. AA1B1B - a side side of the correct quadrangular prism. All four side sides of this prism are equal. ABCD and A1B1C1D1 are the prism bases (the squares lying in the parallel planes). Diagonal of a polyhedron is called the piece connecting two of its not adjacent tops, top t.e which do not belong to one side. From the drawing it is visible that a point And yes a point C 1 do not belong to one side and therefore AC1 piece - the diagonal of this prism.
2. To find diagonal, prisms it is necessary to consider ACC1 triangle. This rectangular triangle. AC1 prism diagonal in the considered triangle will be a hypotenuse, and pieces the EXPERT and CC1 legs. Follows from Pythagorean theorem (the square of a hypotenuse is equal in a rectangular triangle to the sum of squares of legs) that: AC12 = AC2 + CC12 (1);
3. Further it is necessary to consider ACD triangle. ACD triangle too rectangular (since the prism basis - a square). For convenience it is possible to designate the party of the basis by a letter and. Thus on Pythagorean theorem: AC2 = a2 + a2, EXPERT = 2a (2);
4. If to designate prism height by letter h and to substitute expression (2) in expression (1), it will turn out: AC12 = 2a2+h2, AC1 = √ (2a^2+h^2), where and - the party of the basis, h - height. This formula is fair for any correct prism.