Isosceles triangle height calculator

What is the height of an isosceles triangle

The height of an isosceles triangle is a perpendicular line drawn from the vertex (the point where the two equal sides meet) to the base, or extension of the base, of the triangle. In an isosceles triangle, two sides are equal in length (known as the lateral sides), while the third side is the base. The height from the vertex to the base bisects the base, creating two equal segments, and acts as the angle bisector at the vertex. You can use our isosceles triangle calculator to calculate its area and perimeter.

Characteristics of heights in an isosceles triangle

In an isosceles triangle, the height drawn from the vertex to the base has several noteworthy characteristics:

• It divides the base into two equal parts.
• It acts as the median of the triangle.
• It is the angle bisector at the vertex.
• It is perpendicular to the base.

The height from one base angle to a lateral side has its own characteristics:

• It is equal to the height from the opposite base angle.
• It forms a right angle with the lateral side.
• It divides the lateral side into unequal segments.

Formulas for calculating heights

Height from the vertex (h₁)

1. Using the lateral side and the base: $h_1 = \sqrt{a^2 - \frac{b^2}{4}}$

2. Using the area and the base: $h_1 = \frac{2A}{b}$

3. Using the base angle and the lateral side: $h_1 = a \sin{\alpha}$

Height from the base angle (h₂)

1. Using the vertex angle and the lateral side: $h_2 = a \sin{\beta}$

2. Using the lateral side and the base. To start, we’ll use the formula for the height from the vertex: $h_2 = a \sin{\beta}$ where the calculation for angle $\beta$ is conducted as: $\beta = 180° - 2\alpha$, with $\alpha=\arccos{\left(\frac{b}{2a}\right)}$

3. Using the area and the lateral side: $h_2 = \frac{2A}{a}$

Example calculations

Example 1

Given: Lateral side $a = 10$ cm, base $b = 12$ cm. Find: Height from the vertex $h_1$

Solution: $h_1 = \sqrt{a^2 - \frac{b^2}{4}} = \sqrt{100 - \frac{144}{4}} = \sqrt{100 - 36} = \sqrt{64} = 8$ cm

Example 2

Given: Area $A = 60 \text{ cm}^2$, base $b = 10 \text{ cm}$ Find: Height from the vertex $h_1$

Solution: $h_1 = \frac{2A}{b} = \frac{2 \times 60}{10} = 12$ cm

Example 3

Given: Vertex angle $\beta = 36°$, lateral side $a = 15 \text{ cm}$ Find: Height from the vertex $h_2$

Solution: $h_2 = a \sin{\beta} = 15 \sin{36°} = 15 \times 0.5878 \approx 8.817 \text{ cm}$

Example 4

Given: Area $A = 40 \text{ cm}^2$, lateral side $a = 13 \text{ cm}$ Find: Height from the base angle $h_2$

Solution: $h_2 = \frac{2A}{a} = \frac{2 \times 40}{13} \approx 6.15 \text{ cm}$

Important notes

1. When calculating the height, remember that in an isosceles triangle:

   • The lateral sides are equal.
   • The base angles are equal.
   • The sum of all angles equals 180°.

2. Consider the relationships among the triangle’s elements:

   • If $\alpha$ is a base angle, then $\beta = 180° - 2\alpha$
   • If $\beta$ is the vertex angle, then $\alpha = \frac{180° - \beta}{2}$

3. The height can be drawn either inside or outside the triangle, depending on the angles:

   • If the vertex angle is acute, the height is inside the triangle.
   • If the vertex angle is obtuse, the height is outside the triangle.
   • If the vertex angle is right, the height coincides with the lateral side.

Frequently asked questions

How to find the height of an isosceles triangle if the lateral side is $a = 17 \text{ cm}$ and the base angle is $\alpha = 42°$?

$h_1 = a \sin{\alpha} = 17 \sin{42°} = 17 \times 0.669 \approx 11.37 \text{ cm}$

What’s the difference between the height from the vertex and the height from the base angle?

The height from the vertex is measured to the base and bisects the vertex angle, while the height from a base angle is measured to a lateral side and does not have special properties other than being perpendicular to the side.

Can the height of an isosceles triangle be greater than its lateral side?

No, the height is always less than the lateral side since it acts as a leg of a right triangle where the lateral side is the hypotenuse.

How does the height of the triangle change if the base is increased while the lateral sides remain constant?

Increasing the base length will decrease the height from the vertex, while the height from a base angle will initially increase and then decrease.

How to find the height of an isosceles triangle if the area is $A = 48 \text{ cm}^2$ and the base is $b = 16 \text{ cm}$?

$h_1 = \frac{2A}{b} = \frac{2 \times 48}{16} = 6 \text{ cm}$

What is the height of an isosceles triangle when its lateral sides are equal to its base?

In such a case, the triangle is equilateral, and the height is calculated as: $h_1 = \frac{a\sqrt{3}}{2}$ where $a$ is the side length of the triangle.