Composite volumes
Find the total volume of composite solids formed from rectangular prisms and cylinders, and solve practical problems involving solids with cylindrical sections removed.
Worked examples
Total volume of a rectangular prism and cylinder
Straightforward
Problem
A trophy base consists of a rectangular prism (length cm, width cm, height cm) with a cylinder (radius cm, height cm) on top. Find the total volume, correct to 2 decimal places.
1
Find the volume of the rectangular prism.
cm³
2
Find the volume of the cylinder.
cm³
3
Add the two volumes to find the total.
cm³
Answer
cm³
Volume of a solid with a cylindrical hole removed
Moderate
Problem
A metal block has dimensions cm by cm by cm. A cylindrical hole of radius cm is drilled completely through it (depth cm). Find the remaining volume, correct to 2 decimal places.
1
Find the volume of the full rectangular block.
cm³
2
Find the volume of the cylindrical hole.
cm³
3
Subtract the hole volume from the block volume.
cm³
Answer
cm³
Volume of metal in a hollow cylindrical pipe
Challenging
Problem
A hollow cylindrical pipe has an outer radius of cm and an inner radius of cm. The pipe is cm long. Find the volume of metal in the pipe, correct to 2 decimal places.
1
Find the area of the annular (ring-shaped) cross-section by subtracting the inner circle area from the outer circle area.
cm²
2
Multiply the cross-sectional area by the length of the pipe.
3
Calculate to 2 decimal places.
cm³
Answer
cm³
Practise
Q1·Straightforward
A composite solid is formed by placing a cylinder on top of a rectangular prism. The prism has length cm, width cm and height cm. The cylinder has radius cm and height cm. Find the total volume of the solid, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q2·Straightforward
A model is made by joining a cylinder (radius cm, height cm) to the top of a rectangular prism (length cm, width cm, height cm). Find the total volume, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q3·Straightforward
A cylinder with radius cm and height cm sits on top of a rectangular prism with length cm, width cm and height cm. Find the total volume of the composite solid, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q4·Straightforward
A decoration consists of a rectangular prism (length cm, width cm, height cm) with a cylinder (radius cm, height cm) on top. Find the total volume, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q5·Moderate
A rectangular block has dimensions cm by cm by cm. A cylindrical hole of radius cm is drilled completely through it (depth cm). Find the volume of the remaining solid, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q6·Moderate
A concrete slab measures cm by cm by cm. A cylindrical hole of radius cm is drilled completely through it. Find the remaining volume, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q7·Moderate
A rectangular block is cm long, cm wide and cm thick. A cylindrical hole of radius cm and depth cm is drilled through it. Find the volume of the remaining solid, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q8·Moderate
A metal block has dimensions cm by cm by cm. A cylindrical cavity of radius cm and depth cm is drilled into it. Find the remaining volume, correct to 2 decimal places.
Explanation
cm³; cm³; cm³
Q9·Challenging
A hollow cylindrical pipe has an outer radius of cm and an inner radius of cm. The pipe is cm long. Find the volume of metal in the pipe, correct to 2 decimal places.
Explanation
cm²; cm³
Q10·Challenging
A rectangular fish tank is cm long, cm wide and cm deep. A solid cylindrical ornament with radius cm and height cm stands inside the tank. How many litres of water can the tank hold when full? Give your answer correct to 2 decimal places. ( litre cm³)
Explanation
cm³; cm³; cm³ litres
Q11·Challenging
Two cylinders each have a height of cm. Cylinder A has radius cm and Cylinder B has radius cm. Find the difference in their volumes, correct to 2 decimal places.
Explanation
cm³; cm³; Difference cm³
Q12·Challenging
A planter box is made from a rectangular prism cm long, cm wide and cm deep. A cylindrical section of radius cm and depth cm is removed from the centre. Find the volume of the remaining solid, correct to 2 decimal places.
Explanation
cm³; cm³; cm³