What it does
Many everyday objects are two or three simple solids stuck together: an ice-cream cone with a scoop on top, a medicine capsule, a toy rocket, a tank with a domed roof, a block with a knob. This page draws five such shapes from the side and works out:
- the total surface area, adding the curved surface of each part and leaving out every face that is hidden where two parts join,
- the volume, which is simply the sum of the parts’ volumes,
- the slant height of any cone, l = √(r² + H²).
The hidden joins are drawn as a dashed red ring and listed with a line through them, so you can see what is not counted. This is the step most people get wrong in exams.
It follows the Class 10 Maths chapter surface areas and volumes (combinations of solids) and uses the cylinder, cone and sphere formulas from Class 9.
The circle pieces come from circle sectors, and the cone’s slant height is Pythagoras’ theorem in disguise.
How to use it
- Pick a shape from the five buttons. Each one loads tidy starting sizes.
- Slide the sizes or type them. Only the sizes the shape needs are shown: r (radius of the round parts), h (height of the cylinder), H (height of the cone) and a (side of the cube). All are in centimetres.
- Choose π = 22/7, 3.14 or exact. School answers usually use 22/7.
- Read the Surface area card: one line per visible surface with your numbers filled in, then the hidden faces crossed out, then the total.
- Read the Volume card, part by part and the total.
The five shapes
- Cone on a hemisphere (ice cream): area = πrl + 2πr², volume = ⅓πr²H + ⅔πr³.
- Capsule: a cylinder with a hemisphere on each end. Area = 2πrh + 4πr², volume = πr²h + ⁴⁄₃πr³.
- Cylinder with a cone (rocket or tent), base included: area = πr² + 2πrh + πrl.
- Cylinder with a dome: area = πr² + 2πrh + 2πr².
- Cube with a dome: the dome adds 2πr² but covers a circle of πr² on the top face, so area = 6a² − πr² + 2πr² = 6a² + πr². The dome must fit, so 2r can be at most a.
Worked example
The first shape starts with r = 3.5 cm and H = 12 cm, with π = 22/7. The slant height is √(3.5² + 12²) = 12.5 cm. The cone’s curved surface is 22/7 × 3.5 × 12.5 = 137.5 cm² and the hemisphere’s is 2 × 22/7 × 3.5² = 77 cm². The flat circle where they meet is inside, so the total surface area is 214.5 cm². The volume is ⅓ × 22/7 × 3.5² × 12 = 154 cm³ for the cone plus ⅔ × 22/7 × 3.5³ = 89.83 cm³ for the hemisphere, 243.83 cm³ in all.
Tips
- Area: add only what you could paint from outside. Volume: add everything, because nothing is hidden from volume.
- Check your units: areas in cm², volumes in cm³.
- Make the capsule’s h = 0 and you get a sphere, a good check on the formulas.
FAQ
Why is the cube’s total not 6a² + 2πr²? The dome sits on part of the top face, which is now covered. You add the dome’s curved surface (2πr²) and take away the covered circle (πr²).
Does the volume change when parts are joined? No. Joining does not remove any material, so the volumes just add.
Is this from the textbook? No. The shapes and numbers are made up for practice and follow the topics of the Class 10 syllabus; fibub is not part of NCERT.
Is anything saved? No. Everything is worked out in your browser and nothing is stored or sent.
Where the numbers come from
- Standard mensuration formulas: curved surface of a cylinder 2πrh, of a cone πrl with l = √(r² + h²), of a hemisphere 2πr²; volumes πr²h, ⅓πr²h, ⅔πr³ and a³. Written for fibub.
- Checked against hand-worked answers (cone r 3.5, H 12 on a hemisphere with π = 22/7: area 214.5 cm², volume 243.83 cm³) and against a separate Python calculation of every shape.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.