Calculate Triangle Height: Formulas and Examples

To calculate a triangle’s height, divide its area by its base and multiply by 2: height = 2 × area ÷ base. Enter the area and the length of the base below to get the perpendicular height in the same unit as the base.

Formula

The standard formula is h = 2A ÷ b, rearranged from the triangle area formula A = (b × h) ÷ 2.

• h is the perpendicular height, measured in metres (m) here.
• A is the triangle’s area, in square metres (m²) here.
• b is the chosen base length, in metres (m) here.

The base and height must correspond: the height is the shortest, perpendicular distance from the chosen base line to the opposite vertex. If you change which side is the base, use the corresponding height. The same formula works in feet and square feet, or in centimetres and square centimetres, provided the units are consistent.

Worked Example

Using the calculator’s defaults, the area is 24 m² and the base is 8 m.

1. Start with h = 2A ÷ b.
2. Substitute the values: h = 2 × 24 m² ÷ 8 m.
3. Calculate: h = 48 ÷ 8 = 6 m.

The triangle’s perpendicular height is 6 m. The units reduce correctly: square metres divided by metres gives metres.

What Triangle Height Means

A triangle’s height, also called its altitude, is not necessarily one of its sides. It is a perpendicular segment from a vertex to the line containing the opposite side. In an acute triangle, that segment usually falls inside the triangle. In an obtuse triangle, it can fall outside, so the opposite side may need to be extended to meet the perpendicular.

Every triangle has three possible base-and-height pairings, one for each side chosen as the base. Each pairing produces the same area when used consistently. A triangle that looks tall or flat in a drawing may be misleading: the height is determined by perpendicular distance, not by the page’s vertical direction or the apparent slant of a side.

Using the Calculator and Choosing Inputs

Use the calculator when you know a triangle’s area and one base length but need the associated height. Enter a nonnegative area and a positive base. The height cannot be determined from the area alone, or from the base alone; there are many triangles with different heights that share either one of those measurements.

• Choose a side as the base and measure its full length.
• Use the area of the same triangle, not the area of a larger shape that contains it.
• Keep units compatible. For example, if the base is 8 cm, convert an area in m² to cm² before calculating, or convert the base to metres.
• Read the result in the unit implied by the area and base. For example, ft² divided by ft gives ft.

If the area is 24 ft² and the base is 8 ft, the height is 2 × 24 ÷ 8 = 6 ft. In metric units, if the area is 24 m² and the base is 8 m, the result is 6 m. The numerical values match only because the measurements in these examples have the same numerical inputs; feet and metres are not interchangeable.

Finding Height from Other Measurements

You may not always know the area directly. Other triangle measurements can help, but the correct formula depends on what is given.

• Base and perpendicular height: Find area with A = bh ÷ 2. This is the same relationship rearranged.
• Three side lengths: First calculate the area using Heron’s formula. If the sides are a, b, and c, let s = (a + b + c) ÷ 2; then A = √[s(s − a)(s − b)(s − c)]. For height to side b, calculate h = 2A ÷ b.
• Two sides and the included angle: If side lengths b and c meet at angle θ, the area is A = (bc sin θ) ÷ 2. The height to base b is therefore h = c sin θ.
• Coordinates: If the vertices are known, calculate the perpendicular distance from the chosen vertex to the line through the base endpoints. This is a line-to-point distance, not usually the length of a coordinate difference alone.

For a right triangle, one leg can be the base and the other leg is its height. The hypotenuse is not automatically the height: a height must be perpendicular to the selected base.

Common Mistakes and Limits

The most frequent error is using a sloping side as the height. A side is the height only when it meets the selected base at a right angle. Another common error is forgetting the factor of 2: area divided by base is half the height, because a triangle occupies half the area of a parallelogram with the same base and perpendicular height.

• Mixing units: Dividing 24 m² by 8 cm without conversion produces a meaningless result. Convert both measurements to compatible units first.
• Rounding too early: Keep extra digits in intermediate calculations, then round the final height to a precision justified by the measurements.
• Using an approximate area: The computed height is only as reliable as the area and base values entered.
• Assuming the altitude must be inside: For an obtuse triangle, an altitude may meet an extension of the opposite side.

The calculation assumes a flat, Euclidean triangle and a known area and base. It does not infer missing measurements or correct inaccurate input. The result is rounded to two decimal places by the calculator, so a displayed value may differ slightly from an exact or more precise measurement.

Frequently Asked Questions

What is the formula for the height of a triangle?
Use h = 2A ÷ b, where A is the area and b is the corresponding base. The answer has the same length unit as the base when units are consistent.

How do you find a triangle’s height without its area?
You need other suitable measurements, such as two sides and their included angle, all three side lengths, or coordinates. The base alone does not determine a unique height.

Is the height of a triangle always inside it?
No. In an obtuse triangle, an altitude from an acute vertex can meet the extension of the opposite side outside the triangle. It is still the perpendicular height to that base line.

Can a sloping side be the height?
Yes, but only when that side is perpendicular to the chosen base. Otherwise, the height is a separate perpendicular distance and may not be one of the triangle’s sides.

How do you calculate height from the three side lengths?
Use Heron’s formula to find the area, then calculate height to the desired side using h = 2A ÷ b. The side used as b must be the base for that height.

Related

• Cylinder Volume: Formula and Calculator
• Calculate Convergence Angle: Formula and Examples
• Trapezoid Area: Formula, Example, and Calculator

in a | Area | m² | 24 | 0
in b | Base | m | 8 | 0
out h | Triangle height | m | 2 * a / b | 2