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Trigonometry & geometry in CALCULA

This field covers trigonometry (angles, circular functions, identities) and metric geometry (lengths, areas, volumes, vectors). CALCULA works in degrees or radians as you prefer — a full turn is 360° or 2π rad.

The unit circle

sin²θ + cos²θ = 1 · tan θ = sin θ / cos θ

These relations let you recover one function from another. The notable values (0, 30, 45, 60, 90°) are shown on the circle visualisation.

Example

If sin θ = 0.6 with θ acute, then cos θ = √(1 − 0.36) = 0.8 and tan θ = 0.75.

Areas and volumes

Disc: A = π·r² · Triangle: A = ½·base·height · Sphere: V = 4/3·π·r³

CALCULA offers the usual solids (cube, cylinder, cone, sphere, prism) and computes area, volume and sometimes the surface-to-volume ratio, useful in physics and biology.

Example

A ball of radius 1 cm has a volume of 4.19 cm³ and a surface of 12.57 cm². Doubling the radius multiplies the volume by 8 and the surface by 4.

Vectors and distances

‖v‖ = √(x² + y² + z²) · v·w = vₓ·wₓ + v_y·w_y + v_z·w_z

The dot product gives the angle between two vectors; the cross product, a vector perpendicular to both. The distance between two points is the norm of the vector joining them.

Typing it into CALCULA

Choose the sub-field (trigonometry, plane geometry, solids, vectors) in the “≡” menu, enter your values, and read the result with its figure. Set the angle unit in the document settings. See also physics and orbital mechanics.