Surface Area & Volume: Cones, Spheres & Pyramids · Knowledge Organiser

Higher · grade 6–7 · calculator

Key formulas

Cone volume $V=\tfrac13\pi r^2 h$$r$ = base radius, $h$ = PERPENDICULAR height (not the slant)

Cone surface $\pi r l + \pi r^2$curved part $\pi r l$ (slant $l$) + base circle $\pi r^2$; curved only if open

Sphere $V=\tfrac43\pi r^3,\quad A=4\pi r^2$a hemisphere is half: curved $2\pi r^2$, plus base $\pi r^2$ if solid

Pyramid volume $V=\tfrac13\times\text{base area}\times h$works for any pyramid; $h$ is the perpendicular height

Slant height $l^2=r^2+h^2$Pythagoras – find the slant $l$ BEFORE a cone surface-area question

Worked examples

Cone volume · $r=3,\ h=7$

$V=\tfrac13\pi(3)^2(7)=21\pi$

$=66.0$ cm$^3$ (3 s.f.)

Sphere volume · $r=6$

$V=\tfrac43\pi(6)^3=288\pi$

$=905$ cm$^3$ (3 s.f.)

Slant height · $r=5,\ h=12$

$l^2=5^2+12^2=169$

$l=13$ cm

Key words

Perpendicular height: the straight-up height $h$ from the base to the apex – used in the volume formulas.

Slant height: the sloping distance $l$ up the surface of a cone; $l^2=r^2+h^2$.

Curved surface area: the sloping/round surface only ($\pi r l$ for a cone, $2\pi r^2$ for a hemisphere).

Hemisphere: half a sphere; a SOLID one has the flat base circle as well.

Composite solid: two solids joined (e.g. a cone on a hemisphere) – add the volumes.

Common mistakes

✗ Using the slant height $l$ in the volume formula

✓ volume needs the perpendicular height $h$; find $h$ (or $l$) with Pythagoras.

✗ Cone surface $=\pi r l$ only, for a SOLID cone

✓ a solid cone also has the base: $\pi r l+\pi r^2$.

✗ $\tfrac43\pi r^3$ with $r^2$

✓ the SPHERE volume uses $r^3$ (surface area uses $r^2$).

✗ Rounding $\pi$ to $3.14$ too early

✓ use the $\pi$ button and round only at the end.

Key facts

• Cone and pyramid volumes have the $\tfrac13$; a cylinder/prism does not.

• Use the PERPENDICULAR height for volume, the SLANT height for cone surface area.

• Leave answers in terms of $\pi$ for an exact value, or round to 3 s.f. for a decimal.

Remember

• Write the formula first, then substitute – it stops you mixing up $r^2$ and $r^3$.

• For a cone surface-area question, do the Pythagoras slant step first.

• For a composite solid, work out each part separately, then add (or subtract).

Retrieval starter · Surface Area & Volume: Cones, Spheres & Pyramids

Fill it in from memory, then check

Cover the organiser. Fill in as much as you can from memory, then turn it over to check and correct in a different colour.

A · Write each formula

Cone volume

Cone surface

Sphere

Pyramid volume

Slant height

B · Define each key word

Perpendicular height

Slant height

Curved surface area

Hemisphere

Composite solid

C · Complete the facts

The volume of a cone is $\tfrac13\pi r^2 h$, where $h$ is the height.

The volume of a sphere is $\tfrac43\pi r^{\square}$ – the power is .

The curved surface area of a cone is $\pi r l$, where $l$ is the height.

You find the slant height using $l^2=r^2+h^2$, which is theorem.