Foundations — Surface area and volume of all above 3D shapes

This page assumes nothing. We name every letter, sign, and squiggle the parent note uses, draw the picture it stands for, and say why the topic can't move without it. Read it top to bottom — each idea is a rung the next one stands on.

1. A length, and the letters we call it

The very first thing any solid needs is a distance — how far it is from one point to another along a straight line.

Those letters are just nicknames for numbers we don't know yet. Writing $a$ instead of "5 cm" lets one formula work for every cube, not only the 5 cm one.

• $a$ — the edge of a cube (all edges are equal, so one letter is enough).
• $l,w,h$ — length, width, height of a box: the three directions a box can stretch.
• $r$ — radius, explained in section 5.

Look at the picture: the box has three arrows, one per direction. Each arrow is a length, each gets its own letter. The topic needs these letters because a formula is a recipe with blank slots, and the letters are the slots.

2. Units and the little raised number (powers)

Every measurement carries a unit — the ruler mark we counted in, like cm (centimetre). But look at the parent note: volume is in ${\text{cm}}^3$ and area in ${\text{cm}}^2$. What is that raised number?

Why those names? Because of what they look like:

• $a^2$ is the area of a square of side $a$ — literally $a$ rows of $a$ little unit-squares. That is why area units carry a ${}^2$: they count squares.
• $a^3$ is the volume of a cube of side $a$ — $a$ layers, each an $a\times a$ square. That is why volume units carry a ${}^3$: they count little cubes.

3. Area — the amount of flat space

The parent note leans on three area facts constantly, so pin them down now:

These come from Area of2D Shapes. Why does the topic need area before volume? Because of the core principle:

4. The symbol $\times$, the dot $\cdot$, and invisible multiplication

Three notations mean the exact same thing — multiply:

• $l\times w$ — the school "times" cross.
• $l\cdot w$ — a raised dot, used when $\times$ might be confused with the letter $x$.
• $lw$ — two letters touching, with nothing between them, also means multiply.

So $lwh$, $l\times w\times h$, and $l\cdot w\cdot h$ are one and the same. The topic switches between them freely, so recognise all three.

5. The circle's kit: radius $r$, diameter $d$, and $\pi$

Circles power the cylinder, cone, sphere, and hemisphere. Their symbols:

Now the famous one:

6. The square-root sign  ​ and the right triangle

The cone, pyramid, and sphere all use  ​. What is it?

Why does the topic need it? Because of the right triangle — a triangle with one square corner ($90^{\circ }$). Whenever a solid has a slanted edge (a cone's slope, a pyramid's face), that slope, the height, and a base length form a right triangle. The Pythagorean Theorem connects their lengths:

This is exactly how the parent finds a cone's slant height $l=\sqrt{r^2+h^2}$: the radius $r$ and height $h$ are the two legs, the slant $l$ is the hypotenuse. The  ​ appears because we know $l^2$ and want $l$ itself — we must undo the square.

7. The fraction $\frac{1}{3}$ and the integral $\int$

Two last symbols the parent leans on.

You do not need to compute integrals to use this topic — the final formulas are given. But knowing $\int$ means "sum of all the slices" makes the sphere derivation readable. The full machinery lives in Integration and Calculus.

Prerequisite map

Each foundation on the left feeds into the parent topic on the right. If any left-hand box is shaky, the topic will feel like memorising magic instead of understanding it.

Equipment checklist

Test yourself — cover the right side and answer out loud.

Once every checklist item is instant, jump back to Surface area and volume of all above 3D shapes and watch each formula assemble itself from these pieces. Related uses live in Density and Mass, Similar Solids, and Real-World Applications.