Leg of an Isosceles Triangle

Q: How do you find the leg of an isosceles triangle given the base and area?

First, find the height using the area formula: $h = \frac{2A}{b}$. Then use the Pythagorean theorem to find the leg: $l = \sqrt{h^2 + (b/2)^2}$. The altitude always bisects the base in an isosceles triangle, which creates the right triangle you need.

Leg of an Isosceles Triangle

Either of the two congruent sides of an isosceles triangle.

Isosceles triangle with "leg" labeled on both equal sides and "base" labeled on the bottom side.

See also

Base of an isosceles triangle

Key Formula

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

Where:

$h$ = Height (altitude) drawn from the vertex angle perpendicular to the base
$l$ = Length of each leg of the isosceles triangle
$b$ = Length of the base of the isosceles triangle

Worked Example

Problem: An isosceles triangle has two equal legs of length 13 cm and a base of 10 cm. Find the height from the vertex angle to the base and the area of the triangle.

Step 1: Identify the legs and base. The two congruent sides (legs) each measure 13 cm, and the base measures 10 cm.

l = 13, \quad b = 10

Step 2: The altitude from the vertex angle bisects the base, creating two right triangles. Each right triangle has a hypotenuse equal to a leg and a horizontal side equal to half the base.

\frac{b}{2} = \frac{10}{2} = 5

Step 3: Apply the Pythagorean theorem to find the height.

h = \sqrt{l^2 - \left(\frac{b}{2}\right)^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm}

Step 4: Calculate the area using the base and height.

A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 10 \times 12 = 60 \text{ cm}^2

Answer: The height is 12 cm, and the area of the triangle is 60 cm².

Another Example

This example works in reverse — given the base and height, you solve for the leg length instead of finding the height from the legs.

Problem: An isosceles triangle has a base of 16 cm and a height of 6 cm drawn to the base. Find the length of each leg.

Step 1: Write down the known values. The base is 16 cm and the altitude to the base is 6 cm.

b = 16, \quad h = 6

Step 2: The altitude bisects the base, so each half measures 8 cm. This forms a right triangle where the leg of the isosceles triangle is the hypotenuse.

\frac{b}{2} = \frac{16}{2} = 8

Step 3: Use the Pythagorean theorem to solve for the leg length.

l = \sqrt{h^2 + \left(\frac{b}{2}\right)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}

Answer: Each leg of the isosceles triangle is 10 cm long.

Frequently Asked Questions

What is the difference between the leg and the base of an isosceles triangle?

The legs are the two sides that have equal length, while the base is the third side, which generally has a different length. The angle between the two legs is called the vertex angle, and the two angles at either end of the base are called base angles. In an equilateral triangle, all three sides are equal, so any side could be considered a leg or a base.

How many legs does an isosceles triangle have?

An isosceles triangle always has exactly two legs. These two sides are congruent by definition. The remaining side is the base. If all three sides happen to be equal, the triangle is equilateral, which is a special case of isosceles.

How do you find the leg of an isosceles triangle given the base and area?

First, find the height using the area formula:

h = \frac{2A}{b}

. Then use the Pythagorean theorem to find the leg:

l = \sqrt{h^2 + (b/2)^2}

. The altitude always bisects the base in an isosceles triangle, which creates the right triangle you need.

Leg of an Isosceles Triangle vs. Base of an Isosceles Triangle

	Leg of an Isosceles Triangle	Base of an Isosceles Triangle
Definition	Either of the two congruent sides	The third (non-congruent) side
How many	Always 2 legs	Always 1 base
Adjacent angles	One base angle and the vertex angle	Two equal base angles
Altitude from opposite vertex	Does not necessarily bisect the leg	The altitude from the vertex angle bisects the base
Role in formulas	Acts as the hypotenuse in the right triangle formed by the altitude	Gets bisected when calculating the height

Why It Matters

Understanding the legs of an isosceles triangle is essential when you apply the Pythagorean theorem to find missing measurements like height, area, or perimeter. These calculations appear frequently in geometry courses, standardized tests, and real-world problems involving symmetric structures like roof trusses and bridge supports. Recognizing which sides are legs also helps you correctly identify congruent angles and apply triangle congruence theorems such as SAS and SSS.

Common Mistakes

Mistake: Confusing the legs with the base and applying the altitude-bisection property to the wrong side.

Correction: The altitude from the vertex angle (the angle between the two legs) bisects the base — not the legs. Always drop the altitude to the base to create two congruent right triangles.

Mistake: Assuming that the legs are always the two shorter sides of the triangle.

Correction: The legs are the two congruent sides, regardless of whether they are longer or shorter than the base. In some isosceles triangles the base is actually the longest side, yet the two equal sides are still called legs.

Related Terms

Isosceles Triangle — The triangle type that has two equal legs
Base of an Isosceles Triangle — The third side that is not a leg
Congruent — The two legs are congruent to each other
Side of a Polygon — A leg is a specific side of a triangle
Base Angles of an Isosceles Triangle — Angles opposite the legs; they are always equal
Vertex Angle of an Isosceles Triangle — The angle formed between the two legs
Altitude of a Triangle — Height from vertex angle bisects the base
Pythagorean Theorem — Used to relate leg, base, and height