Surface Areas and Volumes · Class 10 Mathematics: quick revision, 34 bites

The whole method

No formula of its own

A combined solid is two familiar solids joined together.

Takeaway. Add what you can touch.

What the join hides

Two circles vanish

When a cone sits on a cylinder, the cone's base and the cylinder's top are pressed together.

Takeaway. Never add the two TOTAL surface areas.

A consequence

Joining always shrinks it

Two faces disappear, so the combined solid has less surface than the two parts had apart.

Takeaway. If your answer went up, something is wrong.

Curved and total

The cylinder

Takeaway. The total adds the two flat circles.

Curved and total

The cone

Takeaway. l is the SLANT height.

Getting l right

Slant, not vertical

Using $h$ in $\pi r l$ is the second commonest error in the chapter.

Takeaway. Pythagoras first, formula second.

Curved and total

The sphere and hemisphere

Takeaway. A sphere has no flat face, so its two figures agree.

2πr² or 3πr²

Which hemisphere figure?

A hemisphere STUCK to something shows only its dome: $2\pi r^2$.

Takeaway. Ask whether the flat face is visible.

Cylinder with two domes

The capsule

Takeaway. Two hemispheres make exactly one sphere's surface.

Cone on a cylinder

The tent

$l = \sqrt{49 + 576} = 25$ m.

Takeaway. The floor is not canvas.

The cavity case

Scooping adds surface

Hollowing a hemisphere out of a face LOSES a circle, $\pi r^2$, and GAINS a bowl, $2\pi r^2$.

Takeaway. Removing material can make the surface larger.

The face after a cavity

Reduced, not removed

The flat face loses only a CIRCLE, not the whole face.

Takeaway. Subtract πr², not the entire top.

Before any formula

Three questions first

Takeaway. The words 'open' and 'hollow' change the answer.

A reliable slip

Halve the diameter

Questions often give a diameter, and every formula here wants the radius.

Takeaway. An ugly fraction usually means this was missed.

The unit test

Square units

Surface area is always in cm², m² and so on.

Takeaway. Check the unit before you stop.

Multiples of 7 and 3.5

Why the numbers are nice

$\frac{22}{7}$ cancels only for such radii, so questions choose them.

Takeaway. Ugly arithmetic is usually a misread radius.

Least surface

Why bubbles are round

A sphere has the least surface area of any solid holding a given volume.

Takeaway. Surface tension finds it automatically.

Volumes simply add

Nothing hides inside

A join has no thickness, so no space is lost when two solids are stuck together.

Takeaway. Where surfaces subtract, volumes add.

Base area times height

The cylinder

Takeaway. The simplest of the six.

A third of its cylinder

The cone

Takeaway. The 1/3 is the most-forgotten symbol in the chapter.

Pour it and see

Why a third?

Fill a cone with water and empty it into the cylinder on the same base with the same height.

Takeaway. The fraction is a fact, not a convention.

Four thirds and two thirds

Sphere and hemisphere

Takeaway. Cubed, not squared.

The easy two

Cube and cuboid

Takeaway. No π anywhere.

Which height

Vertical, not slant

Takeaway. Slant for the side, height for the inside.

The classic ratio

One, two, three

On the same base and of the same height, cone : hemisphere : cylinder is exactly $1 : 2 : 3$.

Takeaway. From the fractions 1/3, 2/3 and 1.

Cylinder plus a sphere

The capsule

Takeaway. Two hemispheres make one sphere, here as well.

Cone on a cylinder

The tent

Takeaway. The cone used height 24, not slant 25.

Subtract

When it is hollowed out

Takeaway. A cavity lowers the volume but RAISES the surface area.

A bowl in a 7 cm cube

Wood remaining

Takeaway. The cube minus the hemisphere.

The two conversions

Capacity in litres

Takeaway. Dividing by 100 makes the answer ten times too big.

Thick walls

Material or capacity?

Given an outer radius $R$ and an inner $r$, the MATERIAL is $\pi h(R^2 - r^2)$.

Takeaway. Two different questions about one container.

Cube the right number

Radius, not diameter

A sphere's volume cubes the RADIUS; using the diameter makes the answer eight times too large.

Takeaway. Halve first, cube second.

The unit test

Cubic units

Volume is cm³ or m³; capacity is litres.

Takeaway. Check before you stop.

What differs

The two topics side by side

Takeaway. Most errors are one topic's rule used in the other.