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3D Shapes: Surface Area and Volume

Lesson · Beginner

Geometry: Solid Geometry

Total Surface Area (TSA) and Volume

Cube

Cube with side length a
TSA=6a2\mathrm{TSA}=6a^2
V=a3V=a^3

Rectangular Prism (Cuboid)

Rectangular prism with side lengths a, b, and c
TSA=2(ab+ac+bc)\mathrm{TSA}=2(ab+ac+bc)
V=abcV=abc

Prism

Prism with height h
V=h⋅AreabaseV=h\cdot \mathrm{Area}_{\mathrm{base}}

Sphere

Sphere with radius r
TSA=4πr2\mathrm{TSA}=4\pi r^2
V=43πr3V=\frac{4}{3}\pi r^3

Cylinder

Cylinder with radius r and height h
TSA=2πr(r+h)\mathrm{TSA}=2\pi r(r+h)
Lateral Area=2πrh\text{Lateral Area}=2\pi rh
V=πr2hV=\pi r^2h

Cone

Cone with radius r, height h, and slant height l
TSA=πr(r+l)\mathrm{TSA}=\pi r(r+l)
Lateral Area=πrl\text{Lateral Area}=\pi rl
l=r2+h2l=\sqrt{r^2+h^2}
V=13πr2hV=\frac{1}{3}\pi r^2h

Generalized Cone

Generalized cone with base area B and height h
V=13BhV=\frac{1}{3}Bh

Pyramid

Pyramid with base area and height h
V=13Areabase⋅hV=\frac{1}{3}\mathrm{Area}_{\mathrm{base}}\cdot h

Regular Tetrahedron

Regular tetrahedron with side length a
TSA=a23\mathrm{TSA}=a^2\sqrt{3}
h=a23h=a\sqrt{\frac{2}{3}}
V=a3212V=\frac{a^3\sqrt{2}}{12}

Triangular Prism

Triangular prism
V=12Areabase×LV=\frac{1}{2}\mathrm{Area}_{\mathrm{base}}\times L
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