Pyramid, cone and sphere

What is it about?

For pyramids, cones and spheres you cannot simply compute volume and surface area as “length times width times height”. The simulation shows where the formulas come from: you can rotate the solid, pour water and unfold the net.

Building blocks

• Height $h$: perpendicular distance of the apex from the base.
• Slant height $h_a$ (pyramid): height of a lateral face, from the midpoint of a base edge to the apex.
• Slant height $s$ (cone) or lateral edge $s$ (pyramid): line from the apex to the edge of the base.
• Radius $r$: for cone and sphere, the distance from the centre.

These lengths are linked by right triangles (Pythagoras’ theorem). For the square pyramid with base diagonal $d = a\sqrt{2}$:

For the cone, the slant height satisfies $s^2 = h^2 + r^2$. Careful: the slant height $h_a$ and the lateral edge $s$ of a pyramid are different lines. $h_a$ meets the base edge at its midpoint at a right angle, $s$ ends at a corner.

Pyramid

With base area $G$ and height $h$:

For the square pyramid, $G = a^2$ and $M = 4 \cdot \tfrac{1}{2} \cdot a \cdot h_a$. The water from three pyramids exactly fills the prism with the same base and height.

Cone

The lateral surface unrolls into a circular sector with radius $s$ and arc length $2\pi r$. Its angle is $\alpha = 360^\circ \cdot \tfrac{r}{s}$.

Sphere

• Archimedes: cylinder (height $r$, radius $r$) = cone + hemisphere. The cone holds $\tfrac{1}{3}\pi r^3$, the cylinder $\pi r^3$, so the hemisphere gets $\tfrac{2}{3}\pi r^3$ and the whole sphere $\tfrac{4}{3}\pi r^3$.
• From pyramids: if you cut the sphere into many pyramids of height $r$, then $V = \tfrac{1}{3} \cdot O \cdot r$. It follows that $O = 3V : r = 4\pi r^2$.

Try it

1. Choose the square pyramid and switch between the helper triangles. Which line is always the longest? Load pyramid
2. Pour the pyramid three times. How much of the prism does one pyramid fill? Pouring
3. Try the same with the cone. What changes when you change $r$ or $h$? Pour a cone
4. Unroll the lateral surface of the cone. Change $h$ and watch the angle $\alpha$. Cone net
5. Pour after Archimedes: first the cone, then the hemisphere. Archimedes
6. Split the sphere into pyramids and rotate it. Sphere made of pyramids

Tasks

Task 1: Square pyramid

A square pyramid has base edge $a = 4\,\text{cm}$ and height $h = 5\,\text{cm}$. Calculate the slant height, volume, lateral surface and surface area. Load task

Task 2: Cone

A cone has radius $r = 3\,\text{cm}$ and height $h = 4\,\text{cm}$. Find $s$, $V$, $M$ and $O$. Load task

Task 3: Sphere

A sphere has radius $r = 2.5\,\text{cm}$. Calculate its volume and surface area. What percentage of the enclosing cylinder (height $2r$) does the sphere fill? Load task

Ideas for the classroom

• Use: about grades 9/10, calculating solids. Suitable as an introduction (pouring experiment), for practice (Pythagoras in solids) and for going further (Archimedes).
• Model: the pyramid has a regular $n$-gon as its base, with the apex directly above the centre. When pouring, the amount of water is calculated exactly from the volume; the drawing only approximates it.
• Common mistakes:
  • The factor $\tfrac{1}{3}$ is forgotten – correct: pyramid and cone fill only a third.
  • $r$ and $d$ are mixed up – the formulas use the radius.
  • Height $h$ is confused with the slant height $h_a$ or $s$ – $s$ is always longer than $h$.