1. **State the problem:** We need to calculate the volume of a cuboid with given dimensions: height = 4 cm, length = 19 cm, and a diagonal of one face = 21 cm. 2. **Recall the formula for the volume of a cuboid:** $$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$ 3. **Identify known values:** - Length $l = 19$ cm - Height $h = 4$ cm - Diagonal of one face $d = 21$ cm 4. **Determine which face the diagonal belongs to:** The diagonal of a face is related to the length and width by the Pythagorean theorem: $$d^2 = l^2 + w^2$$ where $w$ is the width. 5. **Calculate the width $w$:** $$w^2 = d^2 - l^2 = 21^2 - 19^2 = 441 - 361 = 80$$ $$w = \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \approx 8.944$$ cm 6. **Calculate the volume:** $$\text{Volume} = l \times w \times h = 19 \times 4\sqrt{5} \times 4 = 19 \times 4 \times 4\sqrt{5}$$ 7. **Simplify the volume expression:** $$= 19 \times 16 \sqrt{5} = 304 \sqrt{5} \approx 304 \times 2.236 = 679.744$$ cm$^3$ **Final answer:** $$\boxed{304 \sqrt{5} \text{ cm}^3 \approx 679.74 \text{ cm}^3}$$