Volume of cuboids and prisms questions with answers

Questions

1. Question 1Volume of a cuboid

   The diagram shows a shoebox in the shape of a cuboid. Find its volume.

   Show the worked solution

   1. Step 1: Volume of a cuboid = length × breadth × height$V = 9 \times 4 \times 7$$= 252$ cm³
2. Question 2Triangular prisms

   The diagram shows a wedge in the shape of a triangular prism. The dashed line is perpendicular to the base of the triangle. Find its volume.

   Show the worked solution

   1. Step 1: Area of the triangle (the cross-section): use the perpendicular height$\text{Area} = \dfrac{1}{2} \times 20 \times 16$$= 160$ m²
   2. Step 2: Volume = area of cross-section × length$V = 160 \times 20$$= 3200$ m³
3. Question 3Trapezium prisms

   The diagram shows a swimming pool whose cross-section is a trapezium. The dashed line is perpendicular to the parallel sides. Find its volume.

   Show the worked solution

   1. Step 1: Area of the trapezium (the cross-section)$\text{Area} = \dfrac{1}{2} \times (3 + 11) \times 6$$= 42$ cm²
   2. Step 2: Volume = area of cross-section × length$V = 42 \times 8$$= 336$ cm³
4. Question 4L-shaped prisms

   The diagram shows a prism with an L-shaped cross-section. All the corners of its cross-section are right angles. Find its volume.

   Show the worked solution

   1. Step 1: Area of the L-shaped cross-section: the big rectangle minus the corner cut away$\text{Area} = 8 \times 6 - 5 \times 4$$= 28$ cm²
   2. Step 2: Volume = area of cross-section × length$V = 28 \times 10$$= 280$ cm³
5. Question 5A missing length from the volume

   The volume of the triangular prism is $504$ cm³. Find the value of $x$.

   Show the worked solution

   1. Step 1: Area of the cross-section (the triangle)$\text{Area} = \dfrac{1}{2} \times 9 \times 7$$= 31.5$ cm²
   2. Step 2: Volume = area of cross-section × length$31.5x = 504$$x = \dfrac{504}{31.5}$$x = 16$
6. Question 6Unit cubes made into a cuboid

   $72$ identical plastic 1 cm cubes are put together to make a solid cuboid. The top face of the cuboid has a perimeter of $14$ cm, and every edge of the cuboid is longer than $1$ cm. Find the height of the cuboid.

   Show the worked solution

   1. Step 1: The top face: 2 × (length + width) = perimeter$2(\text{length} + \text{width}) = 14$, so $\text{length} + \text{width} = 7$
   2. Step 2: Volume = length × width × height = 72: try each length and width that add up to 7$6 \times 1$: height $= \dfrac{72}{6}$$= 12$ (an edge is not longer than $1$ cm)$5 \times 2$: $10$ doesn't divide into $72$$4 \times 3$: height $= \dfrac{72}{12}$$= 6$ ✓
   3. Step 3: The height$\text{Height} = \dfrac{72}{4 \times 3}$$\phantom{\text{Height}} = 6$ cm