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isosceles obtuse triangle — Definition, Formula & Examples

An isosceles obtuse triangle is a triangle that has two sides of equal length and one angle that measures more than 90°. The obtuse angle always sits between the two equal sides, while the two base angles are equal and each less than 45°.

A triangle classified as both isosceles and obtuse, meaning exactly two of its sides are congruent and the included angle between those congruent sides is strictly between 90° and 180°. The two base angles are congruent and each must be less than 45° so that the three interior angles sum to 180°.

Key Formula

A=12⋅a2⋅sin⁡(θ)A = \frac{1}{2} \cdot a^2 \cdot \sin(\theta)
Where:
  • AA = Area of the triangle
  • aa = Length of each equal side (leg)
  • θ\theta = The obtuse angle between the two equal sides (greater than 90°)

How It Works

To identify an isosceles obtuse triangle, check two things: are two sides the same length, and is one angle greater than 90°? The obtuse angle is always the one formed by the two equal sides (the legs). The two remaining base angles are equal to each other and always acute. You can find each base angle by subtracting the obtuse angle from 180° and dividing by 2. For example, if the obtuse angle is 120°, each base angle is (180°−120°)÷2=30°(180° - 120°) \div 2 = 30°.

Worked Example

Problem: An isosceles obtuse triangle has two equal sides of length 10 cm and an obtuse angle of 120° between them. Find the two base angles and the area.
Step 1: Find each base angle. Subtract the obtuse angle from 180° and divide by 2.
Base angle=180°−120°2=30°\text{Base angle} = \frac{180° - 120°}{2} = 30°
Step 2: Use the area formula with the two equal sides and the included obtuse angle.
A=12⋅102⋅sin⁡(120°)A = \frac{1}{2} \cdot 10^2 \cdot \sin(120°)
Step 3: Evaluate. Since sin(120°) = √3/2 ≈ 0.866, compute the area.
A=12⋅100⋅32=253≈43.3 cm2A = \frac{1}{2} \cdot 100 \cdot \frac{\sqrt{3}}{2} = 25\sqrt{3} \approx 43.3 \text{ cm}^2
Answer: Each base angle is 30°, and the area is 253≈43.325\sqrt{3} \approx 43.3 cm².

Another Example

Problem: An isosceles obtuse triangle has two equal sides of 6 cm and an obtuse angle of 100°. Find the base angles and the area.
Step 1: Find each base angle.
Base angle=180°−100°2=40°\text{Base angle} = \frac{180° - 100°}{2} = 40°
Step 2: Apply the area formula.
A=12⋅62⋅sin⁡(100°)A = \frac{1}{2} \cdot 6^2 \cdot \sin(100°)
Step 3: Evaluate using sin(100°) ≈ 0.985.
A=12⋅36⋅0.985≈17.7 cm2A = \frac{1}{2} \cdot 36 \cdot 0.985 \approx 17.7 \text{ cm}^2
Answer: Each base angle is 40°, and the area is approximately 17.7 cm².

Visualization

Why It Matters

Isosceles obtuse triangles appear in geometry courses when students classify triangles by both sides and angles. Recognizing this shape helps in roof design, kite construction, and tiling patterns where wide, symmetric triangles are needed. Understanding how obtuse angles constrain the base angles builds a foundation for trigonometry and polygon analysis.

Common Mistakes

Mistake: Thinking a base angle can be the obtuse angle.
Correction: Both base angles are equal, so if one were obtuse, their sum would already exceed 180°. The obtuse angle must be the vertex angle between the two equal sides.
Mistake: Assuming each base angle can be 45° or more.
Correction: Since the obtuse angle is greater than 90°, the two base angles must share less than 90° total, so each base angle is strictly less than 45°.

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