SME
Grade 10 Math · Problem type
A level cut of a cone
Use this for “A cone has radius 6 cm and height 9 cm. What is the area of a level cut 3 cm up?”
Diagram
The cut is a circle of radius 4 cm: A = π × 4² = 16π ≈ 50.27 cm².
Level cuts of a cone are circles that shrink toward the tip.
This is an example. Type your own values to start a new problem.
ρ (cm)
=
r
×
(
h
−
z
)
÷
h
A (cm²)
=
π
×
ρ
2
z is how high the cut is above the base, and ρ is the radius of the cut.
Formulas
ρ
=
r
×
h
−
z
h
4
=
6
×
9 − 3
9
A
=
π
×
ρ
2
16π
=
π
×
4
2
Assumptions
•
A cut parallel to the base is a circle.
•
Its radius shrinks evenly to 0 at the tip: ρ = r(h − z) ÷ h at height z.
•
The small cone above the cut is similar to the whole cone.
Step-by-step
WE KNOW
r = 6 cm, h = 9 cm, z = 3 cm
FIND
radius of the cut (ρ), area of the cut (A)
Step 1 · Find radius of the cut (ρ)
Use ρ = r × (h − z) ÷ h
The radius left at that height: similar cones.
ρ
=
6
×
9 − 3
9
ρ
=
6
×
6
9
ρ
=
36
9
ρ
=
4
cm
Step 2 · Find area of the cut (A)
Use A = π × ρ²
The area of the circle.
A
=
π
×
4
2
A
=
16
×
π
A
=
16π
cm²
(≈
50.27
cm²)
Check
Put the numbers back in. Both sides should match.
4
=
6
×
9 − 3
9
✓
16π
=
π
×
4
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Related lessons
Surface area and volume of solids, cross sections and Cavalieri's principle
Main lesson
Surface area of a cylinder
Use this for “Find the surface area of a can with radius 4 cm and height 7 cm.”
Volume of a cone
Use this for “A cone has radius 3 cm and height 4 cm. Find its volume.”
Surface area of a cone
Use this for “A cone has radius 3 cm and height 4 cm. Find its slant height and surface area.”
Volume of a pyramid
Use this for “A square pyramid has base side 6 cm and height 10 cm. Find its volume.”
Surface area and volume of a sphere
Use this for “A ball has radius 5 cm. Find its surface area and volume.”
Surface area of a pyramid
Use this for “A square pyramid has base side 6 cm and height 4 cm. Find its slant height and surface area.”
What shape is the cut?
Use this for “A plane cuts a cube through three corners. What shape is the cross section?”
Turn a shape: which solid?
Use this for “A right triangle is turned about one leg. What solid does it sweep out?”