Find the slant height first. Always. Then apply the formula. Composite solids hide faces at the join — list what is exposed before calculating anything.
55–60 minMS-M1 — MEDIUM3 MC3 SALesson 8 of 22Free
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Choose how you work: type answers on screen, or work in your book.
Think First
A party hat is a cone with no base. A beach ball is a sphere. An Egyptian pyramid has a square base and triangular sides. Before learning any formula — what do you think would be the hardest surface to calculate the area of, and why?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Write your initial thinking in your book
Saved
Come back to this at the end of the lesson.
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Surface Area Formulas — This Lesson
$\text{SA} = b^2 + 2b\ell$
Square pyramid — $b$ = base side, $\ell$ = slant height$b^2$ = square base; $2b\ell$ = four triangular faces (each $\frac{1}{2}b\ell$, four of them)Slant height: $\ell^2 = h^2 + (b/2)^2$ (Pythagoras)
Critical distinction: $\ell$ = slant height (along the face) | $h$ = vertical height (straight up through centre)
| These are NEVER equal
Know
SA formulas for square pyramids, cones ($\pi r\ell + \pi r^2$), and spheres ($4\pi r^2$)
How slant height relates to vertical height via Pythagoras
How to handle composite solids with hidden faces
Understand
Why slant height $\ell \neq$ vertical height $h$ — and why using $h$ in the formula gives a wrong answer
Why a hemisphere SA = $3\pi r^2$ (not $2\pi r^2$) — the flat face must be included
Why hidden faces at composite joins must be excluded from SA
Can Do
Find slant height using Pythagoras before every cone or pyramid SA calculation
Calculate SA of any cone, pyramid, sphere, or hemisphere
List exposed faces of composite solids and calculate total SA correctly
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Key Vocabulary
Slant height ($\ell$)Distance measured along the face from the base edge to the apex — not the vertical height
Vertical height ($h$)Perpendicular distance from the base to the apex — straight up through the centre
ApexThe top point of a pyramid or cone
Lateral faceA triangular side face of a pyramid — not the base
Hidden faceA face at the join between two composite solids — not part of the outer surface
Misconceptions to Fix
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Wrong: The slant height of a cone is the same as its perpendicular height
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Right: The slant height ℓ = √(r² + h²), where h is the perpendicular (vertical) height. Using h instead of ℓ in the curved surface area formula πrℓ gives a wrong answer.
Core Content
01
The Critical Distinction: Slant Height vs Vertical Height
This distinction is the source of more marks lost in this topic than anything else. Before any formula — know which height you have and which you need.
Vertical height $h$
Straight line from apex to centre of base — perpendicular to base
Inside the solid — not visible on the surface
Volume formulas (L10)
Slant height $\ell$
Distance along the face — from midpoint of a base edge (pyramid) or base circle (cone) to apex
On the surface — the face you would paint
Surface area formulas — this lesson
Finding Slant Height with Pythagoras
The right-angled triangle inside the solid has: vertical height $h$ as one leg, half the base dimension as the other leg, and slant height $\ell$ as the hypotenuse.
Right triangle
$h$, $b/2$, $\ell$
$h$, $r$, $\ell$
Formula
$\ell = \sqrt{h^2 + (b/2)^2}$
$\ell = \sqrt{h^2 + r^2}$
h is inside the solid; ℓ is on the surface. SA formulas always use ℓ.
Make it a mandatory first step: Write "Step 1: find $\ell$" at the top of every cone or pyramid problem where $\ell$ is not given. Skipping this step and substituting $h$ directly gives a wrong answer that can still look plausible. The Pythagoras working earns a mark even if the SA calculation has an error.
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02
Surface Area of a Square Pyramid
$$\text{SA} = b^2 + 2b\ell$$
One square base ($b^2$) + four identical triangular lateral faces (each $\frac{1}{2}b\ell$, giving $4 \times \frac{1}{2}b\ell = 2b\ell$ total).
If the pyramid has no base (open at the bottom — like a tent or roof frame): $\text{SA} = 2b\ell$ (lateral faces only).
ℓ = slant height — drawn from apex to midpoint of a base edge
Why $2b\ell$: Each of the four triangular faces has base $b$ and height $\ell$ (slant height). Area of each = $\frac{1}{2}b\ell$. Four of them = $4 \times \frac{1}{2}b\ell = 2b\ell$. This derivation is worth knowing — if you forget the formula, you can reconstruct it.
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Surface Area of Pyramids, Cones, and Spheres
For every pyramid or cone: show Pythagoras working for slant height first. For composite solids: list exposed faces before calculating.
03
Surface Area of a Cone
$$\text{SA} = \pi r\ell + \pi r^2$$
Curved lateral surface ($\pi r\ell$) + circular base ($\pi r^2$).
Factored: $\text{SA} = \pi r(\ell + r)$.
Formula
Column B
$\ell$ is the slant height — not $h$. If you are given $r$ and $h$ (vertical height), calculate $\ell = \sqrt{r^2 + h^2}$ before substituting. This step is not optional.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
04
Surface Area of a Sphere and Hemisphere
$$\text{SA}_\text{sphere} = 4\pi r^2$$
One of the most elegant results in geometry: the total surface area of a sphere equals exactly four circles of the same radius laid flat.
Most common hemisphere error: Writing $\text{SA} = \frac{1}{2}(4\pi r^2) = 2\pi r^2$ — halving the sphere's SA and forgetting the flat circular base. A hemisphere has a flat bottom face that appears when you cut the sphere. Both surfaces must be included. Think physically: if you dipped a hemisphere in paint, both the dome and the flat bottom would be coated.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
05
Composite Solids — Hidden Faces at Joins
When two solids are joined, the faces at the join become internal. They are hidden from the outside and are not part of the surface area.
The Rule
At any join between two solids, the joined faces are not included in the total SA. List which faces are exposed on the outer surface — then calculate only those.
Cone on Cylinder Example
Include?
✓ Yes
✓ Yes
✗ No
✓ Yes
✗ No
Why
Exposed on outside
Exposed on outside
Hidden at the join
Exposed on outside
Hidden at the join
Process: List all faces of both solids. Mark each as exposed or hidden. Calculate exposed faces only. This listing step takes 30 seconds and prevents multi-mark errors. Writing "cylinder top and cone base are hidden" at the top of your working signals clear method to the examiner.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
06
Common Mistakes
Mistake 1 — Using vertical height $h$ instead of slant height $\ell$
Substituting $h$ into $\text{SA} = b^2 + 2b\ell$ or $\text{SA} = \pi r\ell + \pi r^2$. Every time you see a cone or pyramid, write "Step 1: find $\ell$" before the formula.
Mistake 2 — Hemisphere SA = half of sphere SA
The flat circular base ($\pi r^2$) appears when the sphere is cut — it must be included. SA of hemisphere = $2\pi r^2 + \pi r^2 = 3\pi r^2$, not $2\pi r^2$.
Mistake 3 — Including hidden faces in composite SA
Adding all faces of all component solids without removing the joined faces. Before calculating, list faces as exposed or hidden. Anything at the join is hidden.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
Worked Examples
07
Worked Example 1
Square Pyramid SA
Problem
A square pyramid has base side 8 cm and vertical height 3 cm. Find the total surface area.
Half the sphere's surface gives the dome. The flat circular face appears when you cut the sphere — it is new surface that must be added. Students who write only $72\pi = 226.19$ cm² earn zero marks for part (b).
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
10
Worked Example 4
Composite SA — Cone on Cylinder
Problem
A cone (radius 4 cm, vertical height 3 cm) sits on top of a cylinder (radius 4 cm, height 6 cm). Find the total SA correct to 2 decimal places.
Step-by-Step Solution
1
List exposed faces ✓ Cylinder curved surface ✓ Cylinder base (bottom only) ✓ Cone curved surface ✗ Cylinder top — hidden at join ✗ Cone base — hidden at join
Write this list before calculating anything. It commits you to the correct set of faces and earns method marks.
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