How much does it hold, and how much paint does it need?

How to use this simulation

1. The simulation opens on a cone of radius 3 and height 4. It turns by itself; drag it, or press the arrow keys, to look from any side. The dashed lines are the measured lengths: r, h and the slant height l = 5.
2. Pick another solid with the buttons: cube, cuboid, cylinder, cone, sphere or hemisphere. Each has its own sliders (side, length, width, height or radius). The readings panel gives the volume, the curved (or lateral) surface area and the total surface area at once, and they change as you drag.
3. Tick “Wireframe” to see through the solid and count its faces and edges. Pause the animation to look at one angle for as long as you like.
4. Switch the view to “Unfold the net”. The faces fade into a flat pattern: for the cone, a circle and a sector of angle 216°; for the cylinder, two circles and a rectangle 18.85 wide. The areas of the flat pieces add up to the total surface area.
5. Switch the view to “Pour water”. Water from a cone fills a cylinder of the same radius and height, one third per pour. Choose the sphere experiment to see a sphere fill exactly two thirds of its circumscribing cylinder.
6. Use the pour slider to stop at any point, then read the filled fraction in the readings. Change r or h and watch the cylinder volume change while the fractions stay put.

Why does an ice-cream cone look so big and hold so little?

Take a paper cup and an ice-cream cone with the same opening and the same height. Fill the cone with water and pour it into the cup. The cup is not full. Pour a second cone: still not full. Only the third pour brings the water to the brim. The simulation does this for you with a cone of radius 3 and height 4: the cone holds 37.70 cubic units, the cylinder holds 113.10, and 113.10 ÷ 37.70 = 3.

That “one third” is no accident, and it is the most useful fact in this chapter. It means you only need to learn one formula for the cylinder and divide by three for the cone. This page lets you see it, then explains it, then uses it to solve the kind of problems that appear in exams and in real life.

Volume and surface area, from zero

Imagine a wooden block. Volume is how many unit cubes you could pack inside it. A block 4 long, 3 wide and 2 tall holds 4 × 3 × 2 = 24 unit cubes, so its volume is 24 cubic units. The unit is cubed because you multiplied three lengths.

Now imagine wrapping the same block in gift paper. The paper you need is the surface area: the area of every face added together. Each face is flat, so you work in squares: square centimetres, square metres. Surface area is a flat quantity living on the skin of a solid, volume is a roomy quantity living inside it.

Some solids have a curved skin. A can has a flat top, a flat bottom and one curved wall. The curved wall alone is the curved surface area (CSA). Everything together is the total surface area (TSA). For a cube or cuboid we usually say lateral surface area for the four walls, ignoring top and bottom.

Words you will meet

Keep this table at hand. Textbooks use slightly different names for the same ideas.

All six formulas in one table

Read across the table row by row. Notice how the cylinder is the “parent”: the cone takes a third of its volume, and the sphere and hemisphere are built from r alone.

Why a cone is one third of a cylinder

Three ways to see it, from the easy one to the deep one.

Unfolding the surface: nets

Surface area is easy when every piece is flat. A net is what a solid becomes when you cut along its edges and lay it down. Switch the simulation to the net view and the pieces fall flat.

Try it in the simulation

Six short experiments. Predict the answer first, then check it with the readings.

• Choose the cylinder and double r from 3 to 6. The volume grows 4 times, because r is squared. Now double h instead and the volume grows only 2 times.
• Choose the sphere and double its radius. The surface area grows 4 times and the volume 8 times. Lengths double, areas ×4, volumes ×8: that is true for every solid.
• Choose the cone and set r = 6, h = 8. The slant height becomes 10 and the volume 301.59. Every length is doubled, so the volume is 8 times larger than before.
• Choose the cuboid and make all three sliders equal. It turns into a cube, and l × w × h becomes a³.
• Open the cone net and push h very high. The sector gets narrower: a tall thin cone has a small sector angle. Lower h and the sector opens up toward a full circle.
• Run the sphere pour and check that the cylinder stops at the ⅔ mark, not at the top. The remaining third is exactly the volume of a cone of the same size.

Solved problems

Every number below is computed with the same formulas as the simulation.

Common mistakes

Almost every lost mark in this chapter is one of these.

• Mixing up CSA and TSA. For the default cone the CSA is 47.12, but the TSA is 75.40: the base circle is missing from the first. Read whether the question wants the “curved” or the “total” surface.
• Using h instead of l in the cone’s area. Using h = 4 instead of l = 5 in πrl gives 37.70, which is about 20% too small. Volume uses h, area uses l.
• Forgetting that 1 m³ is 1000 litres, not 100 litres. And 1 m³ is 1,000,000 cm³, not 1000 cm³.
• Using diameter where the formula wants radius. Halve the diameter first.
• Adding areas of faces that do not exist. A tank that is open on top has no lid to paint; a pipe has no end faces in its CSA.
• Mixing units in one formula, such as r in metres and h in centimetres. Convert everything first.

Where you will use this

Mensuration is one of the most practical chapters in school mathematics.

• Water tanks, silos and drums are cylinders. Engineers turn their volume into litres or tonnes.
• Tents, funnels, traffic cones and party hats are cones. Tailors cut the cloth as a sector, which is exactly the net you saw.
• Footballs, bubbles and planets are spheres. Domes and bowls are hemispheres.
• Room painting, tiling and packing all need surface area; cooking pots and concrete need volume.
• Scaling matters in biology: a small animal has more surface area per unit of volume, so it loses heat faster.

Exam corner: CBSE and school boards

Questions follow a handful of patterns: find a volume or area, combine two solids, melt and recast, or convert to litres. For board exams the value π = 22/7 is common when r is a multiple of 7.

Revision summary

Read this list once before every test.

• Volume is in cubic units, surface area in square units.
• Cylinder: V = πr²h, CSA = 2πrh, TSA = 2πr(r + h).
• Cone: V = ⅓πr²h, CSA = πrl, TSA = πr(r + l), with l = √(r² + h²).
• Sphere: V = ⁴⁄₃πr³, area 4πr². Hemisphere: V = ⅔πr³, CSA 2πr², TSA 3πr².
• Cube: V = a³, TSA = 6a². Cuboid: V = lwh, TSA = 2(lw + wh + lh).
• Cone = ⅓ cylinder; sphere = ⅔ of its circumscribing cylinder.
• Melting or recasting keeps the volume the same.
• 1 m³ = 1000 litres; 1000 cm³ = 1 litre.

Frequently asked questions