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3D Shapes: Surface Area and Volume
Toolkit 82
3D Shapes: Surface Area and Volume
Total Surface Area (TSA) and Volume
Cube
T
S
A
=
6
a
2
\mathrm{TSA}=6a^2
TSA
=
6
a
2
V
=
a
3
V=a^3
V
=
a
3
Rectangular Prism (Cuboid)
T
S
A
=
2
(
a
b
+
a
c
+
b
c
)
\mathrm{TSA}=2(ab+ac+bc)
TSA
=
2
(
ab
+
a
c
+
b
c
)
V
=
a
b
c
V=abc
V
=
ab
c
Prism
V
=
h
⋅
Area
base
V=h\cdot \text{Area}_{\text{base}}
V
=
h
⋅
Area
base
Sphere
T
S
A
=
4
π
r
2
\mathrm{TSA}=4\pi r^2
TSA
=
4
π
r
2
V
=
4
3
π
r
3
V=\frac{4}{3}\pi r^3
V
=
3
4
π
r
3
Cylinder
T
S
A
=
2
π
r
(
r
+
h
)
\mathrm{TSA}=2\pi r(r+h)
TSA
=
2
π
r
(
r
+
h
)
Lateral Area
=
2
π
r
h
\text{Lateral Area}=2\pi rh
Lateral Area
=
2
π
r
h
V
=
π
r
2
h
V=\pi r^2h
V
=
π
r
2
h
Cone
T
S
A
=
π
r
(
r
+
l
)
\mathrm{TSA}=\pi r(r+l)
TSA
=
π
r
(
r
+
l
)
Lateral Area
=
π
r
l
\text{Lateral Area}=\pi rl
Lateral Area
=
π
r
l
l
=
r
2
+
h
2
l=\sqrt{r^2+h^2}
l
=
r
2
+
h
2
V
=
1
3
π
r
2
h
V=\frac{1}{3}\pi r^2h
V
=
3
1
π
r
2
h
Pyramid
V
=
1
3
Area
base
⋅
h
V=\frac{1}{3}\text{Area}_{\text{base}}\cdot h
V
=
3
1
Area
base
⋅
h
Regular Tetrahedron
T
S
A
=
a
2
3
\mathrm{TSA}=a^2\sqrt{3}
TSA
=
a
2
3
h
=
a
2
3
h=a\sqrt{\dfrac{2}{3}}
h
=
a
3
2
V
=
a
3
2
12
V=\frac{a^3\sqrt{2}}{12}
V
=
12
a
3
2
Related Problems
Core
MATHCOUNTS State Team 2025 (Problem 9)
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