Isosceles Triangle Calculator

Q: What is an isosceles triangle?

An isosceles triangle has exactly two sides of equal length (called legs) and one different side (called the base). The two angles at the base are equal (base angles), and the angle between the two equal sides is the apex angle. Special cases: equilateral triangle (all three sides equal = isosceles with apex 60°) and isosceles right triangle (apex 90°, base angles 45°).

Q: How do you find the height of an isosceles triangle?

The height (altitude) from the apex to the base = √(a² − (b/2)²), where a is the leg length and b is the base. This comes from the Pythagorean theorem: the altitude bisects the base creating two right triangles with hypotenuse a, base b/2, and height h. Example: leg = 10, base = 12 → h = √(100 − 36) = √64 = 8.

Q: What is the area of an isosceles triangle?

Area = ½ × base × height = ½ × b × √(a² − (b/2)²). Simplified: Area = (b/4) × √(4a² − b²). Example: leg = 5 cm, base = 6 cm → height = √(25 − 9) = 4 cm → Area = ½ × 6 × 4 = 12 cm². In terms of leg and apex angle A: Area = (a²/2) × sin(A).

Q: How do you find the apex angle of an isosceles triangle?

Apex angle = 2 × arcsin(b/(2a)) where b is the base and a is the leg. Example: leg = 8, base = 8 → apex = 2 × arcsin(8/16) = 2 × arcsin(0.5) = 2 × 30° = 60° (confirming it's equilateral). Alternatively, base angle = arccos((b/2)/a) = arccos(b/(2a)), then apex = 180° − 2 × base angle.

Q: What are the base angles of an isosceles triangle?

Base angles are always equal to each other. If apex angle is A°, each base angle = (180° − A°)/2. Example: apex = 40° → base angles = (180 − 40)/2 = 70° each. Sum check: 40 + 70 + 70 = 180° ✓. The two base angles and apex angle always sum to 180°.

Q: What is an isosceles right triangle?

An isosceles right triangle has apex angle = 90° and base angles = 45° each. The legs are equal, and hypotenuse = leg × √2. This is the 45-45-90 triangle. Example: legs = 5 → hypotenuse = 5√2 ≈ 7.071. Area = ½ × 5 × 5 = 12.5. It is one of the two special right triangles (the other being the 30-60-90 triangle).

Q: Can an isosceles triangle be obtuse?

Yes. An isosceles triangle is obtuse if either the apex angle is obtuse (> 90°), or one of the base angles is obtuse. Since base angles are always equal, both base angles would be obtuse if one is — but then they would sum to more than 180° alone, which is impossible. So obtuse isosceles triangles always have the obtuse angle at the apex, e.g., apex = 120°, base angles = 30° each.

Find all properties of an isosceles triangle from legs + base or legs + apex angle.

△ What is an Isosceles Triangle?

An isosceles triangle is a triangle with (at least) two sides of equal length, called the legs. The third side is called the base. By the isosceles triangle theorem, the two angles opposite the equal sides (the base angles) are always equal to each other. The angle between the two equal sides is the apex angle.

Isosceles triangles are among the most common shapes in geometry, architecture, and nature. In architecture, many gabled roofs, pointed arches, and decorative pediments have isosceles triangular cross-sections. In mathematics, the isosceles triangle theorem (“if two sides of a triangle are equal, the angles opposite them are equal”) is one of the foundational propositions of Euclidean geometry — it appears as Proposition 5 in Euclid’s Elements.

The altitude from the apex to the base has a special property: it bisects the base at a right angle and also bisects the apex angle. This creates two congruent right triangles, which is why the height formula uses the Pythagorean theorem: h = √(a² − (b/2)²). The altitude is simultaneously the median (connecting vertex to midpoint of opposite side) and the perpendicular bisector of the base — the single axis of symmetry of a non-equilateral isosceles triangle.

Special cases of isosceles triangles worth knowing: the equilateral triangle (all sides equal, all angles 60° — a special isosceles with apex = 60°); the isosceles right triangle (apex = 90°, base angles = 45°, the 45-45-90 triangle with hypotenuse = leg × √2); and the golden gnomon (apex = 36°, base angles = 72°) and golden triangle (apex = 108°, base angles = 36°) that appear in regular pentagons and the golden ratio.

📐 Formulas

Height h = √(a² − (b/2)²) Area = ½ × b × h

a = leg length (equal sides) · b = base length

Apex angle = 2 × arcsin(b/(2a)) · Base angles = (180° − apex)/2

Perimeter = 2a + b

From legs + apex angle A: base = 2a × sin(A/2) · height = a × cos(A/2)

Example (a=10, b=12): h = √(100−36) = 8 · Area = ½×12×8 = 48 · Apex = 2×arcsin(0.6) ≈ 73.74°

📖 How to Use This Calculator

Steps

Choose your mode: “Legs + Base” if you measured the two equal sides and the different base; or “Legs + Apex Angle” if you know the equal sides and the angle between them.

Enter the values. For Legs + Base: ensure base < 2 × leg. For Legs + Apex: apex must be between 0° and 180°.

Click Calculate to see leg, base, area, perimeter, and a note with height and all angles.

💡 Example Calculations

Example 1 — Standard Isosceles (Legs + Base)

Legs = 10 cm, base = 12 cm

Height = √(10² − 6²) = √(100 − 36) = √64 = 8 cm

Area = ½ × 12 × 8 = 48 cm². Perimeter = 20 + 12 = 32 cm.

Height = 8 cm, Area = 48 cm². Apex angle = 2 × arcsin(0.6) ≈ 73.74°. Base angles ≈ 53.13°.

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Example 2 — Isosceles Right Triangle (45-45-90)

Leg = 5, apex angle = 90°

Base = 2 × 5 × sin(45°) = 10 × (√2/2) = 5√2 ≈ 7.071

Height = 5 × cos(45°) = 5/√2 ≈ 3.536. Area = ½ × 5 × 5 = 12.5

The 45-45-90 triangle: leg 5, hypotenuse ≈ 7.071. Area = 12.5. Perimeter ≈ 17.071.

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Example 3 — Tall Isosceles (Apex angle = 30°)

Leg = 8 m, apex angle = 30°

Base = 2 × 8 × sin(15°) = 16 × 0.2588 ≈ 4.141 m

Height = 8 × cos(15°) ≈ 8 × 0.9659 ≈ 7.727 m. Area = ½ × 4.141 × 7.727 ≈ 16.00 m²

Tall, narrow triangle: base ≈ 4.141 m, height ≈ 7.727 m. Base angles = 75° each. Area ≈ 16.00 m².

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Example 4 — Equilateral Triangle (apex = 60°, leg = 6)

Leg = 6, apex angle = 60°

Base = 2 × 6 × sin(30°) = 12 × 0.5 = 6 (= leg, confirming equilateral)

Height = 6 × cos(30°) = 6 × (√3/2) = 3√3 ≈ 5.196. Area = ½ × 6 × 5.196 ≈ 15.59

All sides = 6, all angles = 60° — this is an equilateral triangle, confirming it is a special isosceles case.

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Frequently Asked Questions

What is an isosceles triangle?

An isosceles triangle has exactly two sides of equal length (called legs) and one different side (called the base). The two angles at the base are equal (base angles), and the angle between the two equal sides is the apex angle. Special cases: equilateral triangle (all three sides equal = isosceles with apex 60°) and isosceles right triangle (apex 90°, base angles 45°).

How do you find the height of an isosceles triangle?

The height (altitude) from the apex to the base = √(a² − (b/2)²), where a is the leg length and b is the base. This comes from the Pythagorean theorem: the altitude bisects the base creating two right triangles with hypotenuse a, base b/2, and height h. Example: leg = 10, base = 12 → h = √(100 − 36) = √64 = 8.

What is the area of an isosceles triangle?

Area = ½ × base × height = ½ × b × √(a² − (b/2)²). Simplified: Area = (b/4) × √(4a² − b²). Example: leg = 5 cm, base = 6 cm → height = √(25 − 9) = 4 cm → Area = ½ × 6 × 4 = 12 cm². In terms of leg and apex angle A: Area = (a²/2) × sin(A).

How do you find the apex angle of an isosceles triangle?

Apex angle = 2 × arcsin(b/(2a)) where b is the base and a is the leg. Example: leg = 8, base = 8 → apex = 2 × arcsin(8/16) = 2 × arcsin(0.5) = 2 × 30° = 60° (confirming it's equilateral). Alternatively, base angle = arccos((b/2)/a) = arccos(b/(2a)), then apex = 180° − 2 × base angle.

What are the base angles of an isosceles triangle?

Base angles are always equal to each other. If apex angle is A°, each base angle = (180° − A°)/2. Example: apex = 40° → base angles = (180 − 40)/2 = 70° each. Sum check: 40 + 70 + 70 = 180° ✓. The two base angles and apex angle always sum to 180°.

What is an isosceles right triangle?

An isosceles right triangle has apex angle = 90° and base angles = 45° each. The legs are equal, and hypotenuse = leg × √2. This is the 45-45-90 triangle. Example: legs = 5 → hypotenuse = 5√2 ≈ 7.071. Area = ½ × 5 × 5 = 12.5. It is one of the two special right triangles (the other being the 30-60-90 triangle).

Can an isosceles triangle be obtuse?

Yes. An isosceles triangle is obtuse if either the apex angle is obtuse (> 90°), or one of the base angles is obtuse. Since base angles are always equal, both base angles would be obtuse if one is — but then they would sum to more than 180° alone, which is impossible. So obtuse isosceles triangles always have the obtuse angle at the apex, e.g., apex = 120°, base angles = 30° each.

What is the perimeter of an isosceles triangle?

Perimeter = 2 × leg + base = 2a + b. Example: leg = 7 cm, base = 5 cm → Perimeter = 14 + 5 = 19 cm. If you know the apex angle instead: base = 2a × sin(apex/2), so Perimeter = 2a + 2a × sin(apex/2) = 2a(1 + sin(apex/2)).

How is an isosceles triangle different from a scalene triangle?

Isosceles: at least two sides equal → at least two angles equal. Scalene: all three sides different → all three angles different. Equilateral: all three sides equal (special case of isosceles). All triangles are exactly one of: equilateral, isosceles (but not equilateral), or scalene. The isosceles property gives lines of symmetry that scalene triangles lack.

What is the axis of symmetry of an isosceles triangle?

An isosceles triangle (that is not equilateral) has exactly one axis of symmetry: the altitude drawn from the apex to the midpoint of the base. This line is simultaneously the altitude, median, angle bisector, and perpendicular bisector of the base. An equilateral triangle has three axes of symmetry; a scalene triangle has none.

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📌 Quick Tips

💡In an isosceles triangle, the two base angles are always equal. If the apex angle is A°, each base angle is (180−A)/2.

💡The height (altitude) from the apex to the base bisects the base at 90°, creating two congruent right triangles. Height = √(leg² − (base/2)²).

💡An equilateral triangle is a special case of an isosceles triangle with apex angle = 60° and base = leg.

💡The triangle inequality requires: base < 2 × leg (the base must be shorter than twice the leg). This ensures the two sides can actually meet at the apex.

💡In 'legs + apex angle' mode, as the apex angle approaches 0°, the base shrinks to zero and the triangle collapses to a line. As apex approaches 180°, the base approaches 2 × leg.