SME
Grade 10 Math · Problem type
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.”
Diagram
Net
Solid
The net: the base circle and a sector of radius ℓ = 5 cm whose arc wraps the base, 2πr long.
The
sector's
angle
is
360
×
r
ℓ
=
360
×
3
5
=
216°.
S
=
π
×
3
2
+
π
×
3
×
5
S = 24π ≈ 75.4 cm²
This is an example. Type your own values to start a new problem.
r
2
+
h
2
=
ℓ
2
S (cm²)
=
π
×
r
2
+
π
×
r
×
ℓ
Formulas
r
2
+
h
2
=
ℓ
2
3
2
+
4
2
=
5
2
S
=
π
×
r
2
+
π
×
r
×
ℓ
24π
=
π
×
3
2
+
π
×
3
×
5
Assumptions
•
The slant height ℓ is the hypotenuse of the right triangle with legs r and h.
•
Unrolled, the side is a sector of radius ℓ whose arc wraps the base; its area is πrℓ.
•
So S = πr² + πrℓ.
Step-by-step
WE KNOW
r = 3 cm, h = 4 cm
FIND
slant height (ℓ), surface area (S)
Step 1 · Find slant height (ℓ)
Use r² + h² = ℓ²
The
slant
height
is
the
long
side
of
the
right
triangle
with
legs
r
and
h
.
ℓ
=
√
r
2
+
h
2
ℓ
=
√
3
2
+
4
2
ℓ
=
√
9 + 16
ℓ
=
√
25
ℓ
=
5
cm
Step 2 · Find surface area (S)
Use S = π × r² + π × r × ℓ
The base circle and the sector, whose area is πrℓ.
S
=
π
×
3
2
+
π
×
3
×
5
S
=
9
×
π
+
3
×
5
×
π
S
=
9π
+
15
×
π
S
=
9π
+
15π
S
=
24π
cm²
(≈
75.4
cm²)
Check
Put the numbers back in. Both sides should match.
3
2
+
4
2
=
5
2
✓
24π
=
π
×
3
2
+
π
×
3
×
5
✓
Topic check
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