Triangle height calculator

What is triangle height?

Triangle height, sometimes referred to as the altitude, is a line segment perpendicular to a triangle’s base and extending to the opposite vertex. Height plays a critical role in solving geometrical problems and calculations related to triangles because it helps in determining the area of a triangle. Depending on the triangle type, the known variables, and required calculation, the way to determine height varies.

Calculating height in different types of triangles

Understanding how to calculate the height in various triangles starts with knowing which values are given and the type of triangle you are dealing with. Let’s explore how to determine the height for ordinary, right, isosceles, and equilateral triangles using specific formulas and methods.

Ordinary triangle

In an ordinary triangle with sides $a$, $b$, and $c$:

1. Using area and base:

   • If the area $A$ and base $b$ are known, height $h$ can be calculated as: $$h = \frac{2A}{b}$$

2. Using sides:

   • The height $h$ dropped to side $b$ of a triangle with known sides $a$, $b$ and $c$ can be expressed through a single formula as follows: $$h = \frac{2}{b} \cdot \sqrt{p(p-a)(p-b)(p-c)}$$ where $p$ is the semi-perimeter of the triangle: $$p = \frac{a + b + c}{2}$$

Right triangle

In a right triangle, with legs $a$ and $b$, and hypotenuse $c$, knowing the legs and hypotenuse, the height drawn from the vertex of the right angle to the hypotenuse can be calculated by the formula:

$$h = \frac{a \cdot b}{c}$$

Isosceles triangle

In an isosceles triangle, with two equal sides $a$, base $b$, and vertex angle $\beta$, the height can be calculated using:

$$h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$$

Equilateral triangle

For an equilateral triangle, where each side is $a$, the height can be calculated using:

$$h = \frac{a \sqrt{3}}{2}$$

Examples

Example 1: Height in an ordinary triangle

Consider a triangle with a known area of 36 square units and a base of 12 units. To find the height:

$$h = \frac{2 \cdot 36}{12} = 6 \text{ units}$$

Example 2: Height in an equilateral triangle

For an equilateral triangle with side length 8 units:

$$h = \frac{8 \cdot \sqrt{3}}{2} \approx 6.93 \text{ units}$$

Example 3: Height in a right triangle

In a right triangle with a hypotenuse of 13 units and legs of 5 and 12 units:

$$h = \frac{5 \cdot 12}{13} = \frac{60}{13} \approx 4.62 \text{ units}$$

Notes

• Always ensure angles are in the correct measure, such as degrees or radians, when performing trigonometric calculations.
• The ground line of measurement is critical; ensure it’s perpendicular when considering height and base.
• Familiarity with primary trigonometric functions (sine, cosine, tangent) is essential for applying formulas accurately.

Frequently asked questions

How to find the height of a triangle if area is 50 and base is 10?

The formula is $h = \frac{2 \times \text{A}}{\text{b}}$. Using the values:

$$h = \frac{2 \times 50}{10} = 10 \text{ units}$$

What is the height of an equilateral triangle with a side of 7 units?

Use the formula $h = \frac{a \sqrt{3}}{2}$:

$$h = \frac{7 \sqrt{3}}{2} \approx 6.06 \text{ units}$$

What if the isosceles triangle has sides 5 units and base 6 units?

Utilize $h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$:

$$h = \sqrt{5^2 - \left(\frac{6}{2}\right)^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ units}$$

If you need to find the height of an isosceles triangle dropped from the vertex angle to the base, use the isosceles triangle height calculator

How does the height of a right triangle change with different angles?

The height depends on the sine of the angle when calculated relative to the hypotenuse. If the angle increases or decreases, the sine value changes, altering the height.

Is height always perpendicular to the base in triangles?

Yes, by definition, height (altitude) must be perpendicular to the base of the triangle, making it one of the essential segments in a triangle’s geometric study.