Cone and frustum lateral area both come from the side surface, not the circular ends. For a cone, the lateral area is πrs. For a conical frustum, the lateral area is π(R + r)s.
Formula Summary
| Shape | Lateral Area | Total Surface Area |
|---|---|---|
| Cone | πrs | πrs + πr2 |
| Conical frustum | π(R + r)s | π(R + r)s + πR2 + πr2 |
What the Variables Mean
R = frustum bottom radius
h = vertical height
s = slant height, measured along the side surface
Cone Lateral Area from a Net
When the side of a cone is unrolled, it becomes a circular sector. The sector arc length equals the base circumference, 2πr. The sector radius is the slant height, s. That gives the cone lateral area πrs.
Total cone surface area: SA = πrs + πr2 = πr(s + r)
Frustum Lateral Area Formula
A conical frustum is a cone with the top cut off parallel to the base. Its side surface spreads into a ring-shaped sector. The compact formula is:
Total frustum surface area: SA = π(R + r)s + πR2 + πr2
Find Slant Height First
If the problem gives vertical height instead of slant height, calculate slant height before using the surface area formula.
For a full cone, this becomes s = sqrt(h2 + r2).
Worked Example: Frustum Lateral Area
s = sqrt(102 + (8 - 4)2) = sqrt(116) = 10.77 cm
L = π(8 + 4)(10.77)
L = π(129.24)
L = 406.00 cm2
Worked Example: Total Frustum Surface Area
Lateral area = 406.00 cm2
Bottom area = π(82) = 201.06 cm2
Top area = π(42) = 50.27 cm2
Total SA = 657.33 cm2
Common Mistakes
- Using height as slant height: use
s, not verticalh, in the lateral area formula. - Adding caps when only lateral area is needed: labels, wrapping, and side coating usually use lateral area only.
- Swapping radius and diameter: if the diameter is given, divide by 2 first.
- Forgetting both circular ends: total frustum surface area includes the top and bottom circles unless the object is open.
Use the Calculators
Use the Conical Frustum Surface Area Calculator for frustums and the Cone Surface Area Calculator for full cones. For related checks, compare surface area vs lateral area, cylinder surface area from nets, and composite shape surface area.