SME
Grade 10 Math · Problem type
Surface area and volume of a sphere
Use this for “A ball has radius 5 cm. Find its surface area and volume.”
Diagram
Net
Solid
A sphere has no flat net: its surface is as much as four of its great circles.
S = 4 × π × 5² = 100π ≈ 314.16 cm²
This is an example. Type your own values to start a new problem.
S (cm²)
=
4π
×
r
2
V (cm³)
=
4
3
×
π
×
r
3
Formulas
S
=
4
×
π
×
r
2
100π
=
4
×
π
×
5
2
V
=
4
×
π
×
r
3
3
500π/3
=
4
×
π
×
5
3
3
Assumptions
•
The surface area is four times a great circle: S = 4πr².
•
The volume is V = 4πr³ ÷ 3: two thirds of the cylinder that just holds the sphere.
•
Given S or V, work back to the radius first.
Step-by-step
WE KNOW
r = 5 cm
FIND
surface area (S), volume (V)
Step 1 · Find surface area (S)
Use S = 4 × π × r²
Four great circles, each πr².
S
=
4
×
π
×
5
2
S
=
4
×
25
×
π
S
=
100
×
π
S
=
100π
cm²
(≈
314.16
cm²)
Step 2 · Find volume (V)
Use V = 4 × π × r³ ÷ 3
Cavalieri:
a
hemisphere
matches
a
cylinder
with
a
cone
taken
out,
so
V
is
4π
r
3
3
.
V
=
4
×
π
×
5
3
3
V
=
4π
×
125
3
V
=
1,571
3
V
=
500π/3
cm³
(≈
523.6
cm³)
Check
Put the numbers back in. Both sides should match.
100π
=
4
×
π
×
5
2
✓
500π/3
=
4
×
π
×
5
3
3
✓
Topic check
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