Find the altitude using area, sides, or angles
When you know the triangle's area and base
When you know all three side lengths
When you know base, adjacent side, and angle
Select a method and enter values to calculate
Height
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Area
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Base
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Perimeter
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The height (or altitude) of a triangle is the perpendicular distance from a vertex to the opposite side (called the base). Every triangle has three different heights, one for each base. Understanding how to calculate triangle height is fundamental for finding area, analyzing geometric properties, and solving real-world problems in engineering and design.
The height of a triangle is always measured along a line perpendicular (at 90°) to the base. This perpendicular line extends from the opposite vertex down to the base (or the base extended, in the case of obtuse triangles). The key property is that this line forms a right angle with the base. For a deeper exploration of these concepts, you can learn about triangle altitude properties at Wolfram MathWorld.
Since any side can serve as the base, every triangle has three different heights. However, these heights are related to each other through the triangle's area - the area remains constant regardless of which side you choose as the base. You can also review altitude formulas for different triangles for additional calculation methods.
This is the most direct method. If you know the triangle's area and have chosen a base, the height is:
height = 2 × Area ÷ base
This formula comes directly from rearranging the triangle area formula: Area = (base × height) / 2. It's the quickest method when area is already known.
When you know all three sides but not the area, first calculate the area using Heron's formula:
s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]
Then use the area-base method above to find the height to any chosen base. This is particularly useful when you've measured the triangle's sides directly and want to find the height without measuring angles.
If you know a base, an adjacent side, and the angle between them, you can find the height directly using trigonometry:
height = adjacent side × sin(included angle)
This method is common in surveying and engineering applications where angles can be measured with theodolites or other instruments.
Different triangle types have special height properties:
Triangle height calculations are essential in various fields:
Our Triangle Height Calculator supports three different input methods to match your available data:
The calculator automatically determines which method you're using based on your inputs and provides the height along with related properties like area and perimeter. The visual diagram shows the height as a dashed line perpendicular to the base, with a right angle indicator to clarify the perpendicular relationship.
Use the formula: height = (2 × area) / base. This comes from rearranging the area formula (Area = base × height / 2). For example, if a triangle has an area of 24 square units and a base of 6 units, the height is (2 × 24) / 6 = 8 units. This is the most direct method when area is known.
Yes, every triangle has three different heights - one corresponding to each side that can serve as a base. While all three heights are different lengths (unless the triangle is equilateral), they're related through the triangle's constant area. The three heights always meet at a single point called the orthocenter.
Heron's formula calculates a triangle's area using only the three side lengths: Area = √[s(s-a)(s-b)(s-c)], where s is the semiperimeter. Once you have the area from Heron's formula, you can find the height to any base using height = 2 × area / base. This two-step process lets you find height from just the three sides without needing to measure any angles.
The height's location depends on the triangle type. In acute triangles, all three heights fall inside the triangle. In right triangles, two heights are the legs themselves, and the third falls inside. In obtuse triangles, two of the three heights fall outside the triangle, requiring you to extend the base to meet the perpendicular line from the opposite vertex.
If you know a base, an adjacent side, and the angle between them, use: height = adjacent side × sin(included angle). The sine function gives you the ratio of the height to the adjacent side. For example, with a base of 10, an adjacent side of 8, and an included angle of 30°, the height is 8 × sin(30°) = 8 × 0.5 = 4 units.
The height is defined as the perpendicular (90°) distance from a vertex to the opposite side because this gives the shortest distance and creates the proper geometric relationship for area calculation. Any non-perpendicular line from the vertex to the base would be longer and wouldn't correctly represent the true height needed for the area formula Area = ½ × base × height.
Deepen your understanding of triangles and trigonometry with our guides and tutorials:
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