Diagrams and step-by-step solutions for 40 calculators

A number on its own is easy to doubt. Did I measure the radius or the diameter? Is that height the slanted side or straight up? Why is a cone exactly a third of a cylinder? Today 40 calculators answer those questions before you have to ask them: they draw the shape you are measuring, and they show the working with your own numbers.

A diagram that follows your inputs

Open the volume calculator and you will see a box under the result, labelled with your length, width and height. Switch the shape to a cylinder, cone, sphere, pyramid or hemisphere and the drawing changes with it. Type a new height and the box stretches to match; the proportions follow your numbers, within limits, so a very long, thin box still looks like a box.

The diagram and the form are connected:

• Click into a field and the matching dimension lights up, so you can see exactly which measurement goes where.
• Change a value and its dimension flashes once.
• Point at a label in the drawing and the input it belongs to is highlighted; click it to jump straight there.

Every label shows the value as you typed it, in the unit you picked. Answers that are themselves lengths, such as the hypotenuse on the right triangle calculator or the arc on the arc length calculator, are drawn in the accent colour.

The working, with your numbers

Under each of these calculators there is a new Step by step card. It does what a teacher writes on the board:

1. Formula: V = π × r² × h
2. Plug in: V = π × (3 ft)² × 4 ft
3. Work it out: V = π × 9 ft² × 4 ft = 36π ft³
4. Answer: V ≈ 113.1 ft³

A few details we care about:

• Units travel with the numbers. If you enter the radius in inches and the height in feet, the solution converts one of them and says so on the same line, for example (48 in = 4 ft).
• The answer always matches the result above it. If you asked for gallons rather than cubic feet, the last line shows both: V = 320 ft³ ≈ 2,393.77 gal. Every preset and worked example on these pages is checked by an automated test, so the steps and the result can never disagree.
• = or ≈. We write “=” only when a number is exact, and “≈” when it has been rounded.
• Your language, your number format. In German the same line reads V ≈ 113,1 ft³, and the step names are translated into all 16 languages of the site.

Which calculators have them

Solids: volume, surface area, cylinder volume, cone volume (including truncated cones, the shape of a bucket), sphere volume and rectangular prism volume.

Flat shapes and circles: area, perimeter, circle, circumference, area of a circle, radius, diameter, trapezoid area, hexagon, arc length and sector area.

Triangles: triangle area (base and height, three sides with Heron’s formula, or two sides and an angle), the triangle calculator, the right triangle calculator, the hypotenuse calculator and the Pythagorean theorem calculator, which draws a square on each side so you can see why a² + b² = c².

Coordinates and lines: slope (with the rise and run drawn on a grid), midpoint, distance formula (in 3D it becomes the diagonal of a box), endpoint, slope-intercept form, point-slope form and linear interpolation.

Percentages and algebra: the percentage calculator shows the part as a share of the whole, and percent change, percent increase, percent decrease, percent difference and percent error compare the two values side by side. The quadratic formula calculator plots the parabola and goes through the discriminant before giving the roots.

Home projects: square footage, cubic yards and concrete (slab, footing or post hole, with the waste allowance in the working), and roof pitch with the rise, run and angle drawn on the roof line.

You will find most of them in geometry, math and home and construction, and more calculators will get diagrams where a picture genuinely helps.

Keep exploring

The end of every calculator page has been redesigned as well. Goes well with shows the calculators we picked as companions, including ones from other categories, with the category named on each card: after the mortgage calculator you will find home affordability and the amortization schedule; after the volume calculator, the cubic yard calculator for ordering concrete or soil. More in this category lists the closest relatives first, with a link to the whole category, and the popular calculators are one row below.

One menu across the Keen family

There is a new menu button at the right end of the header. It opens a compact panel with every category, the full A–Z list, your history, this blog, the changelog, and the other free Keen tools: KeenTimer, KeenInvoice, KeenWheel, KeenText, KeenSignature and KeenResume. It is the same menu on every Keen site, so once you know one, you know them all.

It opens without moving anything on the page, closes with Esc or a tap outside, and works with a keyboard and screen readers. Languages stay where they were, at the bottom of every page.

Light by design

None of this slows the pages down. The diagrams are drawn as inline vector graphics, not images, and the first drawing and the worked solution are already in the page when it arrives. The few kilobytes of code that redraw them load only on the calculators that have them, after the page is ready. As always, your numbers stay in your browser.

If a diagram or a step reads oddly to you, or you would like one on a calculator that does not have it yet, tell us. The full list of changes is in the changelog.