How this instrument works
Area = ½·base·height is usually read left to right: measure two lengths, get an area. Run the same identity the other way and it answers a different, equally practical question — you already know how much surface the triangle covers and how tall it stands, and you need the base that produced that pairing. Multiply the area by two and divide by the height: b = 2A ⁄ h. Nothing new is being proved here; it is the original relation undone in two algebraic steps, and it returns exactly the base that, paired with the height you supplied, would enclose precisely that area.
The relationship hides a rectangular hyperbola. For a fixed area, base and height are tied by b·h = 2A — a constant product — so halving the height doubles the base, and letting the height shrink toward zero sends the required base toward infinity while the enclosed area never moves. A triangle holding 24 square units of area can stand 8 wide and 6 tall, or 48 wide and 1 tall, or 4,800 wide and a hundredth of a unit tall: identical area, wildly different silhouette, with no floor on how thin the sliver is allowed to get.
The height entered must be the perpendicular distance from the chosen base line to the opposite vertex, never the length of a slanted side. Feed in a side that isn't perpendicular and the recovered base comes back wrong even though every digit was typed correctly. Worth remembering too: 'the' base is a choice, not a fixed feature of the shape. A triangle has three sides, each with its own matching height, so this instrument hands back whichever base pairs with the height you actually measured — not some single privileged dimension.
- Enter the triangle's known area into the Area field, in whatever square unit your figures already use.
- Enter the perpendicular height — measured straight from that same base line to the opposite vertex — into the Height field.
- Read the recovered length in the Base field; it updates the moment either input changes.
- If Base comes back as zero, check whether Area was left at zero — a real triangle needs positive area to have a positive base.
- If a warning appears instead, check Height for zero: the formula divides by it and cannot recover a base from no height at all.
Worked example — cutting a jib sail's foot
A sailmaker is cutting a small jib that must carry exactly 24 square metres of cloth. The luff — the sail's leading edge, hoisted up the forestay — fixes the height at 6 metres. The foot, the sail's bottom edge, is the base being solved for: base = 2 × 24 ⁄ 6 = 48 ⁄ 6 = 8 metres, the length to mark and cut before adding seam allowance.
Checking it the other way costs nothing: a triangle with an 8-metre foot and a 6-metre luff has area ½ × 8 × 6 = 24 square metres, matching the order exactly. The tradeoff is worth knowing before the shears come out — at this height, every 10 centimetres trimmed from the foot removes 0.3 square metres of sail area, since the height stays fixed while area moves in direct proportion to the base.
Questions
How do you find the base of a triangle from its area and height?
Double the area and divide by the height: b = 2A ⁄ h. It falls straight out of A = ½bh by multiplying both sides by two, then dividing by h. For area 24 and height 6, that is 2 × 24 ⁄ 6 = 8 — no angle or side length beyond those two figures is needed.
Which height do I use if the triangle isn't a right triangle?
Always the perpendicular distance from the base line to the opposite vertex, never a slanted side. In an obtuse triangle that perpendicular can land outside the shape itself; extend the base line until the foot of the perpendicular meets it, and measure to that point. A slanted side in place of the true perpendicular height returns an incorrect base.
Why does a tiny height give such a huge base for the same area?
Because base and height trade off exactly: b·h always equals 2A for a fixed area, so as h shrinks toward zero, b must grow without bound to hold that product steady. A 24-square-unit triangle only 0.01 units tall needs a base of 4,800 units — an extremely thin sliver, but a perfectly legitimate triangle.
Can the recovered base be longer than the triangle's other two sides?
Yes, easily. The base is simply whichever side the height was measured against, and nothing in b = 2A ⁄ h weighs it against the remaining two sides. A short, tall triangle can carry a base far shorter than its legs; a low, wide one can carry a base far longer than either.
What happens if I enter zero for the height?
The formula divides by height, so zero height is undefined — a triangle cannot stand at zero height and still enclose a nonzero area. This sheet requires a height above a small positive minimum; a genuinely zero height means the triangle has collapsed to a flat line with no area left to recover a base from.
How is this different from the ordinary triangle-area formula?
Same identity, opposite direction. The area formula starts from base and height and halves their product; this sheet starts from a known area and height and asks what base produced that pairing, so it doubles the area and divides by the height instead of multiplying and halving.