Triangle Height Calculator - Altitudes from Sides or Area

The triangle height calculator finds the perpendicular altitude of any triangle from three side lengths, two sides and the included angle, or a base length and the triangle area. Use it when you need the altitude for a roof rafter, a survey stake, a gusset, or a homework problem.

• • Roof and rafter layout: Convert the measured rafter and base of a triangular gable into the perpendicular height.
• • Classroom geometry: Check the height in a textbook problem and pull the area and angles from the same input.
• • Survey and stake work: Recover the height to a baseline from tape-measured side lengths.
• • Reverse solving: Recover the height from the area and base when the original side measurements are gone.

Every triangle has three altitudes, one from each vertex perpendicular to the opposite side. A scalene triangle usually has three different altitudes; an isosceles or equilateral triangle shares one or two of them. When the recovered altitude needs to feed a separate area, perimeter, or surface workflow, the Triangle Area Calculator keeps the same Heron's formula approach but returns area, perimeter, and the three interior angles as the primary outputs.

The calculator uses three formulas depending on which inputs you supply. The Three Sides method runs Heron's formula to recover the area, then divides twice the area by each side. The Two Sides and Angle method uses the law of cosines and the sine of the included angle. The Area and Base method uses the rearranged area equation directly.

• side a, b, c: the three side lengths in the same length unit; used as denominators
• Area: enclosed area, recovered from Heron's formula when only the sides are known
• semiperimeter s: half of the perimeter, s = (a + b + c) / 2, the working variable inside Heron's formula
• included angle (C): the angle between the two known sides in the two-sides-and-angle method, in degrees

According to Wolfram MathWorld, the area of a triangle with sides a, b, c is Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.

The same three side lengths that feed this calculator also feed the SSS Triangle Calculator, which uses the law of cosines three times to recover the three interior angles as the primary outputs.

These four terms decide whether the altitude the calculator returns matches the side you want to use as a base.

A common error is to confuse the altitude with one of the sides. The altitude is always perpendicular to a side, while the sides themselves are the boundaries of the triangle. The altitude is the shortest path from a vertex to the line containing the opposite side, which is why it appears in the area formula as 2*area divided by that side.

The 3-4-5 worked example shows that the legs of a right triangle are already altitudes to each other, and the Right Triangle Calculator covers the rest of the right-triangle formulas, including the altitude to the hypotenuse.

Pick the input method that matches the measurements you already have, then read the altitude to whichever side you plan to use as a base.

1. 1 Pick the calculation method: Choose Three Sides, Two Sides and Included Angle, or Area and Base.
2. 2 Enter the side lengths or area: Type the three sides, the two sides and the included angle, or the base and the area, all in consistent units.
3. 3 Read the height to each side: Use Height to side a, b, and c as the perpendicular drop from the opposite vertex. Pick the one that matches the side you want to use as a base.
4. 4 Check the supporting outputs: Area, perimeter, and the three interior angles are returned alongside the altitudes.
5. 5 Sanity check with the shortest side: The shortest side has the longest altitude and the longest side has the shortest altitude. If the calculator returns the opposite pattern, double-check the inputs.

A roofer measures a triangular gable with rafters of 6 feet, a base of 17 feet, and a side of 14 feet. Picking Three Sides and entering side a = 6, side b = 14, side c = 17 returns height to side a = 13.17 feet, height to side b = 5.64 feet, height to side c = 4.65 feet, and area = 39.51 square feet.

When only two sides and the included angle are measured, the SAS Triangle Calculator is the matching peer that focuses on recovering the missing side and the two opposite angles before any altitude is computed.

A triangle height calculator that accepts three input shapes and returns all three altitudes plus the area, perimeter, and angles is more useful than a single-formula tool.

• • All three altitudes at once: The Height to side a, b, and c outputs come back together so you can pick the altitude that matches the chosen base without re-entering inputs.
• • Three input methods: Use Three Sides, Two Sides and Included Angle, or Area and Base depending on which two or three values you already have.
• • Area, perimeter, and angles included: Heron's formula gives the area, the law of cosines gives the three angles, and the sum of the sides gives the perimeter.
• • Cross-check friendly: Compute the height from three sides, then re-enter the recovered base angle into the Two Sides and Angle method to confirm.
• • Decimal precision: Decimal inputs work for scaled drawings, and the calculator returns up to four decimals on each altitude.

The three input shapes match the way triangles are described in the field. Surveyors usually have three side lengths, blueprinters usually have two sides and an included angle, and contractors often have an area and a base from a takeoff. The three methods keep the calculation close to the data you actually have.

For a scalene triangle where all three sides are different and only the area matters, the Scalene Triangle Area Calculator applies the same Heron's formula in a single-purpose layout with worked examples.

A few measurement and shape decisions affect whether the altitudes the calculator returns match the real object.

• • The Area and Base method returns only the height to the entered base. The other two altitudes and the three side lengths cannot be recovered from area and base alone.
• • The calculator assumes a planar (flat) triangle. Real objects with thickness or curved sides may need a small allowance beyond the geometric altitude.
• • Rounded output can differ by a few hundredths from a hand calculation that rounds after each step.

The altitude is independent of the base only when the shape is fixed by two or three independent measurements: three side lengths, two sides and the included angle, or one side and the area.

According to Wolfram MathWorld, the area of any triangle is one-half of the base multiplied by the corresponding altitude, so the altitude equals 2*area/base.

According to Wolfram MathWorld, the law of cosines relates a triangle's three sides and one included angle as c^2 = a^2 + b^2 - 2ab*cos(C), and the area is 0.5*a*b*sin(C) when two sides and the included angle are known.

When the triangle is isosceles and only the altitude to the base is needed, the Isosceles Triangle Height returns that altitude in one step from the two equal sides and the base.