how to find the height of an isosceles triangle

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31-03-2025

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Finding the height of an isosceles triangle depends on what information you already have. An isosceles triangle, remember, has two sides of equal length. Let's explore several methods. This guide will show you how to calculate the height regardless of whether you know the base, the equal sides, or a combination of angles and sides.

Method 1: Using the Pythagorean Theorem (When Base and Side Length are Known)

This is the most straightforward method if you know the length of the base and one of the equal sides.

What you need:

• Base (b): The length of the bottom side of the triangle.
• Side (a): The length of one of the equal sides.

Steps:

1. Draw an altitude: Draw a line from the top angle (vertex) straight down to the midpoint of the base. This line is the height (h) and it bisects the base into two equal segments.

2. Apply the Pythagorean Theorem: The altitude creates two right-angled triangles. The Pythagorean theorem states: a² = b² + c². In our case:

   a² = h² + (b/2)²

3. Solve for h: Rearrange the equation to solve for the height (h):

   h = √(a² - (b/2)²)

Example:

Let's say the base (b) is 8 cm and the equal side (a) is 5 cm.

h = √(5² - (8/2)²) = √(25 - 16) = √9 = 3 cm

The height of the isosceles triangle is 3 cm.

Method 2: Using Trigonometry (When Base and Angle are Known)

If you know the base and one of the base angles, trigonometry offers a solution.

What you need:

• Base (b): Length of the base.
• Base Angle (θ): One of the angles at the base.

Steps:

1. Identify the relevant trigonometric function: The height (h) is the side opposite the base angle (θ), and half the base (b/2) is the adjacent side. Therefore, we use the tangent function:

   tan(θ) = h / (b/2)

2. Solve for h: Rearrange the equation to solve for the height (h):

   h = (b/2) * tan(θ)

Example:

If the base (b) is 10 cm and the base angle (θ) is 30°, then:

h = (10/2) * tan(30°) ≈ 2.89 cm (using a calculator)

The height of the isosceles triangle is approximately 2.89 cm.

Method 3: Using Trigonometry (When Two Sides and Included Angle are Known)

If you know the length of the two equal sides and the angle between them, you can use trigonometry.

What you need:

• Side (a): Length of one of the equal sides.
• Vertex Angle (α): The angle at the top vertex of the triangle.

Steps:

1. Use sine function: The height (h) is opposite to half of the vertex angle (α/2) in the right-angled triangle formed by the altitude. Thus:

   sin(α/2) = h / a

2. Solve for h: Rearrange the equation:

   h = a * sin(α/2)

Example:

If the equal side (a) is 7cm and the vertex angle (α) is 40°, then:

h = 7 * sin(40°/2) = 7 * sin(20°) ≈ 2.39 cm

The height is approximately 2.39 cm.

Method 4: Using Heron's Formula (When All Sides are Known)

Heron's formula is useful when you know the lengths of all three sides. It's a bit more involved.

What you need:

• a: Length of one equal side
• b: Length of the other equal side (same as 'a' in an isosceles triangle)
• c: Length of the base

Steps:

1. Calculate the semi-perimeter (s): s = (a + b + c) / 2

2. Apply Heron's formula to find the area (A): A = √[s(s-a)(s-b)(s-c)]

3. Calculate the height (h): Area is also given by A = (1/2) * base * height. Therefore: h = 2A / c

Example:

Let's say a = 5cm, b = 5cm, and c = 6cm

1. s = (5 + 5 + 6) / 2 = 8 cm
2. A = √[8(8-5)(8-5)(8-6)] = √(8 * 3 * 3 * 2) = √144 = 12 cm²
3. h = (2 * 12) / 6 = 4 cm

The height is 4 cm.

Remember to always double-check your calculations and use a calculator for trigonometric functions. The best method to use depends entirely on the information given to you. Choosing the right method ensures efficient and accurate calculation of the isosceles triangle's height.