What is the area, the angle or the distance?

It gives the area of any triangle from its three side lengths alone: compute the semiperimeter s = (a + b + c) ÷ 2, then take √(s(s−a)(s−b)(s−c)). Its value is that it needs no height and no angle, so it works on a plot of land you can only measure along the edges. It is attributed to Heron of Alexandria in the first century, though Archimedes may well have known it earlier.

Each side of a triangle has to be shorter than the sum of the other two — otherwise the two shorter sides cannot reach across the longest one and no triangle closes. Sides of 2, 3 and 9 are impossible. In Heron's formula, violating the inequality makes one of the (s − side) factors negative, so the product under the square root goes negative and the area is not a real number. That is the algebra reporting the geometry honestly.

Compare the square of the longest side against the sum of the squares of the other two. If they are equal, the triangle is right-angled — that is the converse of the Pythagorean theorem, and it is why 3, 4, 5 works. If the longest side squared is smaller, every angle is acute; if larger, one angle is obtuse. The test costs three multiplications and needs no trigonometry at all.

V = 4/3 × π × r³, and the surface area is S = 4 × π × r². Both take the radius, not the diameter, which is where most errors come from: putting a diameter in the radius slot inflates the volume by a factor of eight. A useful relationship is that the surface area is exactly the derivative of the volume with respect to the radius, which is the geometric statement that growing a sphere adds a thin shell.

Because it is 3 ÷ r, so it falls as the sphere grows. Small things have enormous surface relative to their bulk and exchange heat, water and gas with their surroundings very fast; large things cannot. It is why mice must eat constantly and elephants must shed heat through their ears, why crushed ice melts faster than a block, and why big storage tanks need less steel per liter than small ones.

Distance = √((x₂ − x₁)² + (y₂ − y₁)²). It is the Pythagorean theorem with the horizontal and vertical gaps as the two legs and the distance as the hypotenuse. From (3, 4) to (5, 12) the gaps are 2 and 8, so the distance is √(4 + 64) = √68 ≈ 8.246. The same construction extends to three dimensions by adding a Δz² term under the root.

Not directly, and doing so is a classic source of wrong mileages. Degrees of latitude are roughly constant at about 111 km, but degrees of longitude shrink from 111 km at the equator to nothing at the poles, so treating a lat/long pair as flat coordinates overstates east-west distances badly outside the tropics. The haversine formula, which works on the sphere, is the right tool for that job.

Degrees are an arbitrary division of a circle into 360 parts, inherited from Babylonian arithmetic and convenient because 360 has so many divisors. A radian is defined by the circle itself: it is the angle whose arc length equals the radius, so a full turn is exactly 2π radians. That definition is what makes calculus work cleanly — the derivative of sin x equals cos x only when x is measured in radians.

Multiply by 180 ÷ π, which is 57.29577951308232. Going the other way, multiply degrees by π ÷ 180. The anchors worth remembering are π rad = 180°, π/2 = 90°, π/3 = 60°, π/4 = 45° and π/6 = 30°. Since a radian is a ratio of two lengths it carries no physical dimension, which is why it can be introduced or dropped in a formula without breaking dimensional analysis.

On a unit circle, the cosine of an angle is the x coordinate of the point at that angle and the sine is the y coordinate — which is why sin²θ + cos²θ = 1 is just the Pythagorean theorem on a radius of length one. In a right triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent, which also makes tangent equal to sine divided by cosine.

Because the tangent is sine divided by cosine, and at 90° the cosine is exactly zero. Dividing by zero has no value, so the tangent is undefined there rather than infinite: approaching 90° from below the tangent grows without bound, and from above it comes back from negative infinity. Geometrically, the tangent measures a slope, and at 90° the line is vertical and has no slope.

Five angles cover most of what you will ever need. sin 0° = 0, sin 30° = 0.5, sin 45° = √2/2 ≈ 0.7071, sin 60° = √3/2 ≈ 0.8660 and sin 90° = 1. Cosine is the same list read backwards, since cos θ = sin(90° − θ). Tangent follows from dividing them, giving tan 30° ≈ 0.5774, tan 45° = 1 and tan 60° ≈ 1.7321.