Geo Tools

Thirty calculators for shapes and the space they take

The area and the edge of eight flat shapes with a sketch of each, a regular polygon of any number of sides, the area of any outline at all from its corners, and a tiling of the three shapes that fill a plane on their own. Then the solids: what seven of them hold and what wraps them, with the curved surface told from the ends — which is the distinction every question about a tin or a tank turns on.

Triangles, where three measurements fix one in some combinations and not in others: the solver says which case it was handed, refuses the ones with no answer, and says when the measurements fix *two* triangles, which nearly every calculator prints one of. Circles, and the arc, the sector and the segment an angle cuts out of one — three pieces that are constantly confused for each other.

Points and the lines between them, in two dimensions or three: distance, midpoint, slope, the angle between two vectors, the same point in four coordinate systems, and the foot of the perpendicular from a point to a line or a plane. Then proportion — aspect ratios, scale and the golden ratio, with no claim about beauty — and the three geometric questions that come up on a building site: a mitre cut, a roof pitch and a flight of stairs.

Flat shapes

Area
The area of eight flat shapes, with the formula and a sketch.

Perimeter
The distance round the edge, and where there is no exact one.

Regular polygon
Any number of equal sides: area, angles and both radii.

Area from coordinates
Any shape given by its corners, by the shoelace formula.

Ellipse
Area, focuses, eccentricity — and the perimeter nobody has a formula for.

Trapezium
Four sides fix it, and the height follows from them.

Geometric pattern
A tiling of the three regular shapes that tile a plane alone.

Solids

Volume
What seven solids hold, with the formula each of them uses.

Surface area
What wraps a solid, with the side told from the ends.

Sphere
A ball, and the cap a flat cut takes off it.

Cylinder
A tube, and the pipe a wall thickness makes of it.

Cone
A cone or a bucket, and the fan its surface unrolls into.

Triangles

Triangle solver
Any three measurements — and it says when they fix two triangles, or none.

Right triangle
Any two parts give the rest, because the right angle is the third.

Pythagoras
The missing side, and whether three numbers make a triple.

Circles

Circle
Any one of the four measurements gives the other three.

Arc and sector
The arc, the sector and the segment — which are not the same shape.

Unit circle
The three functions at any angle, with the table of the ones worth knowing.

Points and lines

Distance between points
How far apart, where the middle is, and at what slope — in two dimensions or three.

Slope
A line's slope, its angle and its equation — three numbers people confuse.

Midpoint and ratio
The middle of a segment, or the point dividing it in any ratio.

Angle between
Between two vectors or two slopes, in degrees and in radians.

Coordinate systems
The same point in Cartesian, polar, cylindrical and spherical.

Point to line or plane
The shortest distance, and the foot of the perpendicular.

Proportion

Aspect ratio
The missing dimension, and what rounding to whole pixels costs.

Scale
Model to real and back, with what the area and the volume do.

Golden ratio
The division, a scale built from it, and no claim about beauty.

On site

Mitre angle
The cut for a corner, and the number the saw actually wants.

Roof pitch
The three ways a pitch is written, none of which is either of the others.

Stairs
A whole number of risers, and whether the flight is comfortable to climb.

Questions

Why does a triangle sometimes have two answers?: Because two sides and an angle *opposite* one of them is the one combination that does not fix a triangle. The sine rule gives two angles with the same sine — one acute and one obtuse — and where both leave the three angles adding to a half turn, both close. Nearly every calculator prints one of the two without saying the other exists; this one prints both and says which is which.
Why is the surface of a cone not πrh?: Because the curved surface is wrapped along the *slant*, not the height, so it is πrl with l = √(r² + h²). Using the height is the commonest mistake in the subject and always gives a number that is too small. The tool prints the slant height beside the answer so the two cannot be confused.
Why do a roof's pitch, grade and angle disagree?: Because they are three different numbers for the same roof. A pitch of 6 : 12 is fifty per cent and 26.57 degrees; a roofer means the first, a surveyor the second and a calculator the third. The tool prints all three together, and the rafter factor beside them, so that nothing has to be converted by hand.
Is anything I type uploaded?: No. The tools are compiled to WebAssembly and run inside the browser tab. Nothing you type is sent to a server, and you can watch your browser's network panel while you use one to see that nothing leaves the page.

This tool failed to start