Formulae of shapes

This article gives some formulae of some shapes, using words rather than symbols.

2-dimensional space

Perimeter

Circle/spherical digon/spherical henagon

Perimeter = π × diameter.

Polygons

Perimeter = The sum of the lengths of the sides.

Area

Circle/spherical digon/spherical henagon

Area = π × radius²

Ellipse

Area = Half of short axis × half of long axis × π

Triangle

Equilateral triangle

Area = ${\sqrt {3}}$ × side² ÷ 4

Right-angled

Area = One leg × the other leg ÷ 2

General

Area = Base × height ÷ 2

Numerous other formulas are given here

Quadrilateral

Square

Area = Side²

Rectangle

Area = One side × the other side

Parallelogram

Area = One side × height from that side

Trapezoid

Area = Sum of the lengths of the parallel sides × height between the parallel sides ÷ 2

Kite or other quadrilateral with perpendicular diagonals

Area = One diagonal × the other diagonal ÷ 2

Bicentric quadrilateral (with both an incircle and a circumcircle)

Area = Square root of the product of the four sides

Regular polygon

Area = Perimeter × distance from the center to a side ÷ 2

3-dimensional space

Surface area

Sphere

Surface area = 4 × radius² × π

Cone

Surface area = π × (radius of base)² + π × (radius of base) × height of cone

Cylinder

Surface area = 2 × π × radius × height + 2 × π × radius²

Regular tetrahedron

Surface area = ${\sqrt {3}}$ × (edge length)²

Square-based pyramid

Surface area = (Side of base)² + 2 × (side of base) × slant height

Cube

Surface area = (Side)² × 6

Regular octahedron

Surface area = 2 × ${\sqrt {3}}$ × (edge length)²

Volume

Sphere

Volume =  ⁴⁄₃ × π × radius³

Cone

Volume =  ¹⁄₃ × π × (radius of base)² × height

Cylinder

Volume = π × radius² × height

Cube

Volume = Side³

Regular cuboid

Volume = The product of the three sides

4-dimensional space

Hypervolume

Tesseract

Hypervolume = Side⁴

References

• More Murderous Maths by Kjartan Poskitt
• Algebra And Geometry by Dan Green and Basher