3D Shapes Formulas

Key Terminology:

Volume: Space occupied by a 3D shape (measured in cubic units)
Total Surface Area (TSA): Total area of all faces including top and bottom
Curved/Lateral Surface Area (CSA/LSA): Area of curved or lateral faces only

1. Cube

Definition

A cube is a 3D shape with 6 equal square faces, 12 edges, and 8 vertices. All edges are of equal length.

Formulas

Volume: \(V = a^3\)
Total Surface Area: \(TSA = 6a^2\)
Lateral Surface Area: \(LSA = 4a^2\)
Diagonal of face: \(d_f = a\sqrt{2}\)
Space diagonal: \(d_s = a\sqrt{3}\)

where \(a\) = side length (edge)

2. Cuboid (Rectangular Prism)

Definition

A cuboid is a 3D shape with 6 rectangular faces, 12 edges, and 8 vertices. Opposite faces are equal and parallel.

Formulas

Volume: \(V = l \times b \times h\)
Total Surface Area: \(TSA = 2(lb + bh + hl)\)
Lateral Surface Area: \(LSA = 2h(l + b)\)
Diagonal: \(d = \sqrt{l^2 + b^2 + h^2}\)

where \(l\) = length, \(b\) = breadth (width), \(h\) = height

3. Cylinder

Definition

A cylinder is a 3D shape with two parallel circular bases connected by a curved surface.

Formulas

Volume: \(V = \pi r^2 h\)
Curved Surface Area: \(CSA = 2\pi rh\)
Total Surface Area: \(TSA = 2\pi r(r + h)\)
Base Area: \(A_{base} = \pi r^2\)

where \(r\) = radius of base, \(h\) = height

4. Cone

Definition

A cone is a 3D shape with a circular base and a curved surface tapering to a single point called apex or vertex.

Formulas

Volume: \(V = \frac{1}{3}\pi r^2 h\)
Curved Surface Area: \(CSA = \pi rl\)
Total Surface Area: \(TSA = \pi r(r + l)\)
Slant height: \(l = \sqrt{r^2 + h^2}\)
Base Area: \(A_{base} = \pi r^2\)

where \(r\) = radius of base, \(h\) = height, \(l\) = slant height

5. Sphere

Definition

A sphere is a perfectly round 3D shape where all points on the surface are equidistant from the center.

Formulas

Volume: \(V = \frac{4}{3}\pi r^3\)
Surface Area: \(SA = 4\pi r^2\)
Diameter: \(d = 2r\)

where \(r\) = radius

6. Hemisphere

Definition

A hemisphere is exactly half of a sphere, cut through its center.

Formulas

Volume: \(V = \frac{2}{3}\pi r^3\)
Curved Surface Area: \(CSA = 2\pi r^2\)
Total Surface Area: \(TSA = 3\pi r^2\)

where \(r\) = radius

Note: TSA includes the circular base

7. Frustum of a Cone

Definition

A frustum is the portion of a cone that remains after cutting off the top portion parallel to the base.

Formulas

Volume: \(V = \frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1r_2)\)
Curved Surface Area: \(CSA = \pi l(r_1 + r_2)\)
Total Surface Area: \(TSA = \pi[l(r_1 + r_2) + r_1^2 + r_2^2]\)
Slant height: \(l = \sqrt{h^2 + (r_1 - r_2)^2}\)

where \(r_1\) = radius of larger base, \(r_2\) = radius of smaller base, \(h\) = height, \(l\) = slant height

8. Prism (General)

Definition

A prism is a 3D shape with two identical polygon bases connected by rectangular lateral faces.

Formulas

Volume: \(V = \text{Base Area} \times h\)
Lateral Surface Area: \(LSA = \text{Perimeter of base} \times h\)
Total Surface Area: \(TSA = 2 \times \text{Base Area} + LSA\)

where \(h\) = height of prism

9. Pyramid (General)

Definition

A pyramid is a 3D shape with a polygon base and triangular faces meeting at a single apex point.

Formulas

Volume: \(V = \frac{1}{3} \times \text{Base Area} \times h\)
Lateral Surface Area: \(LSA = \frac{1}{2} \times \text{Perimeter} \times \text{Slant height}\)
Total Surface Area: \(TSA = \text{Base Area} + LSA\)

where \(h\) = perpendicular height from base to apex

10. Hollow Cylinder

Definition

A hollow cylinder is a cylinder with a cylindrical hole through its center.

Formulas

Volume: \(V = \pi h(R^2 - r^2)\)
Curved Surface Area: \(CSA = 2\pi h(R + r)\)
Total Surface Area: \(TSA = 2\pi(R + r)(h + R - r)\)

where \(R\) = outer radius, \(r\) = inner radius, \(h\) = height

11. Quick Reference Table

3D Shape	Volume Formula	Total Surface Area
Cube	\(a^3\)	\(6a^2\)
Cuboid	\(l \times b \times h\)	\(2(lb + bh + hl)\)
Cylinder	\(\pi r^2 h\)	\(2\pi r(r + h)\)
Cone	\(\frac{1}{3}\pi r^2 h\)	\(\pi r(r + l)\)
Sphere	\(\frac{4}{3}\pi r^3\)	\(4\pi r^2\)
Hemisphere	\(\frac{2}{3}\pi r^3\)	\(3\pi r^2\)
Frustum	\(\frac{1}{3}\pi h(r_1^2 + r_2^2 + r_1r_2)\)	\(\pi[l(r_1+r_2) + r_1^2 + r_2^2]\)

12. Volume Relationships

Key Volume Comparisons

Cone vs Cylinder: Volume of cone = ⅓ × Volume of cylinder (with same base and height)
Sphere vs Cylinder: Volume of sphere = ⅔ × Volume of cylinder (when diameter = height = 2r)
Hemisphere vs Sphere: Volume of hemisphere = ½ × Volume of sphere
Pyramid vs Prism: Volume of pyramid = ⅓ × Volume of prism (with same base and height)

13. Euler's Formula for Polyhedra

For any convex polyhedron:

\[F + V = E + 2\]

\(F\) = Number of Faces
\(V\) = Number of Vertices
\(E\) = Number of Edges

14. Faces, Edges, and Vertices

3D Shape	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Cylinder	3 (2 flat + 1 curved)	2	0
Cone	2 (1 flat + 1 curved)	1	1
Sphere	1 (curved)	0	0
Triangular Prism	5	9	6
Square Pyramid	5	8	5

15. Important Constants

Constant	Value	Used For
\(\pi\) (Pi)	3.14159... or \(\frac{22}{7}\)	Circular shapes (sphere, cylinder, cone)
\(\sqrt{2}\)	1.414...	Cube face diagonal
\(\sqrt{3}\)	1.732...	Cube space diagonal