Isosceles Triangle

Isosceles Triangle

A triangle with two sides that are the same length. Formally, an isosceles triangle is a triangle with at least two congruent sides.

Isosceles triangle with two equal sides labeled "leg" and bottom side labeled "base

See also

Scalene triangle, equilateral triangle

Key Formula

A = \frac{b}{4}\sqrt{4a^2 - b^2}

Where:

$A$ = Area of the isosceles triangle
$a$ = Length of each of the two equal sides (legs)
$b$ = Length of the base (the unequal side)

Worked Example

Problem: An isosceles triangle has two equal sides of length 10 cm each and a base of 12 cm. Find its area and the measure of the vertex angle.

Step 1: Identify the known values. The two equal sides (legs) are each 10 cm, and the base is 12 cm.

a = 10, \quad b = 12

Step 2: Find the height by dropping a perpendicular from the vertex angle to the base. This bisects the base into two segments of 6 cm each. Use the Pythagorean theorem.

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

Step 3: Calculate the area using the base and height.

A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 12 \times 8 = 48 \text{ cm}^2

Step 4: Find the base angles using cosine. In the right triangle formed by the height, half the base, and one leg:

\cos(\alpha) = \frac{6}{10} = 0.6 \implies \alpha = \cos^{-1}(0.6) \approx 53.13°

Step 5: Find the vertex angle. Since all angles sum to 180°:

\theta = 180° - 2(53.13°) = 180° - 106.26° \approx 73.74°

Answer: The area is 48 cm² and the vertex angle is approximately 73.74°.

Another Example

This example starts from angle measures instead of side lengths, showing how to work backward using the Law of Sines to find missing sides — a common variation in geometry and trigonometry courses.

Problem: An isosceles triangle has base angles of 70° each and a base of 8 cm. Find the length of each equal side and the perimeter.

Step 1: Find the vertex angle. The three angles must sum to 180°.

\theta = 180° - 2(70°) = 40°

Step 2: Use the Law of Sines to find the leg length. In this triangle, the base (8 cm) is opposite the vertex angle (40°), and each leg is opposite a base angle (70°).

\frac{a}{\sin 70°} = \frac{8}{\sin 40°}

Step 3: Solve for the leg length a.

a = \frac{8 \times \sin 70°}{\sin 40°} = \frac{8 \times 0.9397}{0.6428} \approx 11.69 \text{ cm}

Step 4: Calculate the perimeter by adding the base and both equal legs.

P = 2a + b = 2(11.69) + 8 \approx 31.38 \text{ cm}

Answer: Each equal side is approximately 11.69 cm and the perimeter is approximately 31.38 cm.

Frequently Asked Questions

What is the difference between an isosceles triangle and an equilateral triangle?

An isosceles triangle has at least two equal sides, while an equilateral triangle has all three sides equal. Every equilateral triangle is technically also isosceles (it satisfies the 'at least two' condition), but not every isosceles triangle is equilateral. In an equilateral triangle, all angles are 60°; in an isosceles triangle that is not equilateral, only two angles are equal.

Are the base angles of an isosceles triangle always equal?

Yes. This is guaranteed by the Isosceles Triangle Theorem (sometimes called the Base Angles Theorem). It states that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent. The converse is also true: if two angles are equal, the sides opposite them are equal.

How do you find the height of an isosceles triangle?

Drop a perpendicular from the vertex angle (the angle between the two equal sides) to the base. This line bisects the base into two equal halves. Then use the Pythagorean theorem:

h = \sqrt{a^2 - (b/2)^2}

, where

a

is the leg length and

b

is the base length.

Isosceles Triangle vs. Scalene Triangle

	Isosceles Triangle	Scalene Triangle
Definition	At least two sides are congruent	All three sides have different lengths
Equal angles	At least two angles are equal (base angles)	All three angles are different
Lines of symmetry	At least one line of symmetry	No lines of symmetry
Special properties	Height from vertex angle bisects the base	No altitude bisects any side (in general)
Examples	Roof trusses, pennants, road signs	Irregularly shaped land plots

Why It Matters

Isosceles triangles appear constantly in geometry courses, from basic angle-chasing problems to proofs involving congruence and symmetry. They also show up in real-world structures like bridge supports, roof frames, and architectural arches, where symmetry provides structural balance. Understanding their properties — especially the base angle theorem and the perpendicular bisector from the vertex — is essential for coordinate geometry, trigonometry, and standardized test problems.

Common Mistakes

Mistake: Assuming the base is always the bottom side of the triangle.

Correction: The base of an isosceles triangle is the side that is not equal to the other two, regardless of the triangle's orientation. When the triangle is rotated, the base might appear on top or on a side. Always identify the base as the unequal side.

Mistake: Forgetting that an equilateral triangle counts as isosceles.

Correction: The formal definition says 'at least two congruent sides,' not 'exactly two.' An equilateral triangle has three congruent sides, so it satisfies the definition. This matters in proofs and multiple-choice questions that ask you to classify triangles.

Related Terms

Triangle — General category that includes isosceles triangles
Equilateral Triangle — Special case with all three sides equal
Scalene Triangle — Triangle with no equal sides
Congruent — Describes the equal sides and base angles
Side of a Polygon — The legs and base are sides of the triangle
Pythagorean Theorem — Used to find the height from the vertex
Area of a Triangle — Area formula applied with base and height