How this calculator works
All three base-height pairs describe the same area. Once area is known, dividing twice the area by the selected side gives its altitude. Three-sided mode obtains area from Heron’s formula. An obtuse triangle can have an altitude landing beyond the end of a side; the perpendicular distance is still positive. Area and one base determine only that base’s altitude, so other outputs are omitted in that mode.
Formula
K = ahₐ/2 = bhᵦ/2 = ch꜀/2; hₐ = 2K/a. From sides: s=(a+b+c)/2 and K=√[s(s-a)(s-b)(s-c)].
Worked example
Sides 3 m, 4 m and 5 m have area 6 m², so their altitudes are 4 m, 3 m and 2.4 m. A base of 10 cm and area 30 cm² give height 6 cm.
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Frequently asked questions
Is height always one of the sides?
No. In a right triangle the legs act as altitudes to each other, but the altitude to the hypotenuse is a separate segment.
Why is the shortest side paired with the longest height?
The product of each side and its altitude is the same: twice the area.
Can a slanted distance be entered as height?
Only if it is perpendicular to the base line. A slanted segment generally overstates the area.