A right-angled triangular prism has a cross-section that is a right triangle with perpendicular sides 21 m and 15 m, and the prism is 37 m long. Find its volume.

Step-by-step solution

1. 1

   Use the general prism rule, not a memorised special case. For any prism,

   $V=(\text{area of cross-section})\times(\text{length})$

   because every slice perpendicular to the length is the same shape. This means the triangle's area must be found first, in isolation.

2. 2

   Compute the area of the right-triangular cross-section. In a right triangle the two perpendicular sides are the base and the height, so no extra trigonometry is needed:

   $A=\frac12\times 21\times 15=\frac{315}{2}=157.5\ \text{m}^2$

   Forgetting the factor $\tfrac12$ here — treating the face as a $21\times 15$ rectangle — doubles the final answer.

3. 3

   Multiply by the length of the prism.

   $V=157.5\times 37$

4. 4

   Carry out the multiplication. Split $37$ into $30+7$:

   $157.5\times 30=4725,\qquad 157.5\times 7=1102.5$

   $V=4725+1102.5=5827.5\ \text{m}^3$

   $\boxed{5827.5\ \text{m}^3}$

5. 5

   Check the units and the size. Two metre lengths gave an area in $\text{m}^2$, and multiplying by a third length gives $\text{m}^3$ — correct for a volume. As a bound, the prism fits inside a $21\times 15\times 37=11655\ \text{m}^3$ box and fills exactly half of it, which matches $5827.5$.