Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB\equal{}a$ and $ CD\equal{}c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.
1. **Identify the given elements and their relationships:**
   - Trapezoid \(ABCD\) with \(AB = a\) and \(CD = c\) as parallel sides.
   - Diagonals \(AC\) and \(BD\) are perpendicular.
   - \(AB\) and \(CD\) are diameters of circles \(K_a\) and \(K_b\) respectively.
   - Circle \(K\) is the circumscribed circle of the trapezoid.

2. **Determine the center \(O\) and the intersection point \(P\) of the diagonals:**
   - Since \(AC\) and \(BD\) are perpendicular, \(P\) is the orthocenter of the trapezoid.
   - \(O\) is the center of the circumscribed circle \(K\).

3. **Calculate the radius of the circumscribed circle \(K\):**
   - The radius \(R\) of circle \(K\) is the distance from \(O\) to any vertex of the trapezoid.
   - Since \(AB\) and \(CD\) are diameters of circles \(K_a\) and \(K_b\), the radii of these circles are \(\frac{a}{2}\) and \(\frac{c}{2}\) respectively.

4. **Calculate the area of the sector \(BOC\):**
   - The angle \(\angle BOC\) is \(90^\circ\) (since diagonals are perpendicular).
   - The area of sector \(BOC\) is \(\frac{1}{4} \pi R^2\).

5. **Calculate the areas of sectors \(EBP\) and \(CFP\):**
   - \(E\) and \(F\) are midpoints of \(AB\) and \(CD\) respectively.
   - The areas of sectors \(EBP\) and \(CFP\) are \(\frac{1}{4} \pi \left(\frac{a}{2}\right)^2\) and \(\frac{1}{4} \pi \left(\frac{c}{2}\right)^2\) respectively.
   - Their sum is \(\frac{1}{4} \pi \left(\left(\frac{a}{2}\right)^2 + \left(\frac{c}{2}\right)^2\right)\).

6. **Calculate the area of the part inside circle \(K\) but outside circles \(K_a\) and \(K_b\):**
   - The area of the part inside circle \(K\) but outside circles \(K_a\) and \(K_b\) is the area of the trapezoid minus the areas of the sectors \(EBP\) and \(CFP\).
   - The area of the trapezoid is \(\frac{1}{2} \times a \times c\).

7. **Calculate the perimeter of the part inside circle \(K\) but outside circles \(K_a\) and \(K_b\):**
   - The arc length of \(BC\) is \(\frac{1}{2} \pi R\).
   - The arc lengths of \(BP\) and \(CP\) are \(\frac{1}{2} \pi \left(\frac{a}{2}\right)\) and \(\frac{1}{2} \pi \left(\frac{c}{2}\right)\) respectively.
   - The total perimeter is \(\pi \left(\frac{a}{2} + \frac{c}{2} + \frac{\sqrt{a^2 + c^2}}{2}\right)\).

The final answer is \(\boxed{\frac{ac}{2}}\) for the area and \(\boxed{\pi \left(\frac{a}{2} + \frac{c}{2} + \frac{\sqrt{a^2 + c^2}}{2}\right)}\) for the perimeter.