Volume of combined solids: Class 10 practice
The volume of a solid made of simple shapes is the sum of their volumes. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Volumes of cone, cylinder, sphere
The idea behind volume of combined solids
The volume of a solid made of simple shapes is the sum of their volumes. Find each part's dimensions carefully — total height minus the hemisphere's radius gives the cone or cylinder height.
Step-by-step method
- Split the solid into cones, cylinders, hemispheres.
- Work out each part's dimensions.
- Add (or subtract) the volumes.
- Use the π the question asks for.
Worked example 1 (Basic)
A solid is a cylinder of radius 14 cm and height 6 cm with a hemisphere of the same radius on top. Find its volume (π = 22/7).
- Cylinder = 3696.00 cm³
- Hemisphere = 5749.33 cm³
- Total = 9445.33 cm³
- Answer: 9445.33 cm³
Worked example 2 (Standard)
A toy is a cone of radius 3.5 cm on a hemisphere of the same radius; total height 15.5 cm. Find its volume (π = 22/7).
- Cone height = 15.5 − 3.5 = 12 cm
- Cone = 154.00, hemisphere = 89.83
- Total = 243.83 cm³
- Answer: 243.83 cm³
Worked example 3 (Challenge)
A cone of radius 7 cm and height 4 cm is hollowed out of a cylinder of the same radius and height. Find the remaining volume (π = 22/7).
- Cylinder − cone = 616.00 − 205.33
- = 410.67 cm³
- Answer: 410.67 cm³
Common mistakes and misconceptions
- Using total height for the cone
- Adding surface areas of hidden faces
Practice questions
- (Basic) A solid is a cylinder of radius 7 cm and height 8 cm with a hemisphere of the same radius on top. Find its volume (π = 22/7).
- (Basic) A solid is a cylinder of radius 7 cm and height 10 cm with a hemisphere of the same radius on top. Find its volume (π = 22/7).
- (Standard) A toy is a cone of radius 7 cm on a hemisphere of the same radius; total height 15 cm. Find its volume (π = 22/7).
- (Standard) A toy is a cone of radius 7 cm on a hemisphere of the same radius; total height 19 cm. Find its volume (π = 22/7).
- (Challenge) A cone of radius 7 cm and height 12 cm is hollowed out of a cylinder of the same radius and height. Find the remaining volume (π = 22/7).
- (Challenge) A cone of radius 7 cm and height 8 cm is hollowed out of a cylinder of the same radius and height. Find the remaining volume (π = 22/7).
Answer key with full working
- Q1: Cylinder = 1232.00 cm³ → Hemisphere = 718.67 cm³ → Total = 1950.67 cm³ → Answer: 1950.67 cm³
- Q2: Cylinder = 1540.00 cm³ → Hemisphere = 718.67 cm³ → Total = 2258.67 cm³ → Answer: 2258.67 cm³
- Q3: Cone height = 15 − 7 = 8 cm → Cone = 410.67, hemisphere = 718.67 → Total = 1129.33 cm³ → Answer: 1129.33 cm³
- Q4: Cone height = 19 − 7 = 12 cm → Cone = 616.00, hemisphere = 718.67 → Total = 1334.67 cm³ → Answer: 1334.67 cm³
- Q5: Cylinder − cone = 1848.00 − 616.00 → = 1232.00 cm³ → Answer: 1232.00 cm³
- Q6: Cylinder − cone = 1232.00 − 410.67 → = 821.33 cm³ → Answer: 821.33 cm³
Syllabus note
This skill is listed under "Surface Areas and Volumes" for Class 10 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is volume of combined solids in?
"Surface Areas and Volumes", Class 10 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.