Base of Isosceles Triangle: Formula & Calculator

This calculator finds the base of an isosceles triangle when you know the lengths of the equal sides and the angle between them (the vertex angle). Enter those two values, and you get the base length instantly, along with the triangle’s height and area. The base is the side that is not equal to the other two, and it is the side opposite the vertex angle.

Formula

The base of an isosceles triangle is found using the sine of half the vertex angle:

Base = 2 × a × sin(θ / 2)

Where:

• a = length of one of the two equal sides (cm, m, inches, etc.)
• θ = vertex angle (the angle between the two equal sides), in degrees
• sin = sine function (use degrees mode on your calculator)

The height (altitude) from the vertex to the base is:

Height = a × cos(θ / 2)

And the area is simply:

Area = ½ × base × height

These formulas come from splitting the isosceles triangle into two congruent right triangles. The base of the isosceles triangle is twice the length of the opposite side in one right triangle, and the height is the adjacent side.

Worked Example

Using the default values from the calculator: equal side a = 10 cm and vertex angle θ = 60°.

1. First, compute half the vertex angle: θ/2 = 60° / 2 = 30°.
2. Find sin(30°) = 0.5 and cos(30°) ≈ 0.8660.
3. Base = 2 × 10 × 0.5 = 10 cm.
4. Height = 10 × 0.8660 ≈ 8.66 cm.
5. Area = 0.5 × 10 × 8.66 ≈ 43.30 cm².

The calculator shows base = 10.00 cm, height = 8.66 cm, area = 43.30 cm² — exactly matching the manual calculation.

When to Use This Calculator

You need the base of an isosceles triangle in many practical situations:

• Roof trusses and gable ends where two rafters of equal length meet at a peak.
• Designing tents, awnings, or any structure with a triangular cross-section.
• Cutting fabric or sheet metal for a triangular piece where two sides are equal.
• Solving geometry homework problems that give the equal sides and the vertex angle.

You might also know the equal sides and the base angle (the angle at one of the base corners). In that case, the vertex angle is 180° − 2 × base angle. Enter that as θ and the calculator works the same.

How to Read the Results

The base length is always in the same unit you entered for the equal side. If you enter meters, the base comes out in meters; if you enter inches, you get inches. The height is the perpendicular distance from the vertex to the base, and the area is in square units (e.g., cm², m², in²).

Notice that as the vertex angle approaches 0°, the base approaches 0 and the triangle becomes very thin. As the vertex angle approaches 180°, the base approaches twice the equal side length (the triangle flattens into a straight line). The calculator handles any angle between 0 and 180 degrees.

Common Mistakes and Pitfalls

• Using radians instead of degrees: The formula uses the sine of half the angle. If your calculator is in radian mode, you’ll get a wrong answer. This page uses degrees, so make sure your own calculator matches.
• Confusing the base angle with the vertex angle: The base angle is at the bottom corners, not at the top. If you enter a base angle by mistake, the result will be far too large or too small.
• Forgetting to double the half-base: The sine of half the angle gives you only half of the base. You must multiply by 2.
• Rounding too early: If you round sin(θ/2) to 2 decimals before multiplying, the error can be significant. Keep full precision until the final step.
• Unit mix-ups: If you enter the equal side in centimeters and expect the area in square meters, you’ll be off by a factor of 10,000. Convert all inputs to the same unit first.

Limitations of the Formula

This formula assumes a perfect Euclidean triangle on a flat plane. It doesn’t apply to spherical triangles (like on the Earth’s surface) or to triangles with curved sides. Also, it requires the vertex angle to be strictly between 0° and 180°. At exactly 0° or 180°, the triangle degenerates into a line, and the formula gives 0 or 2a respectively, which are the limits but not real triangles.

If you know the base and the equal sides instead, you can find the vertex angle using the inverse sine: θ = 2 × asin(base / (2a)). But that’s a different calculation — this tool is specifically for finding the base from the angle.

Frequently Asked Questions

What if I only know the equal sides and the height?

Then you can find the base using the Pythagorean theorem: half the base = √(a² − h²), so base = 2 × √(a² − h²). You don’t need the angle at all. This works because the height splits the isosceles triangle into two right triangles with hypotenuse a and one leg h.

Can I use this for an equilateral triangle?

Yes. An equilateral triangle is a special isosceles triangle where all sides are equal. If you set the vertex angle to 60° and the equal side to any length, the calculator will return the same length for the base, which is correct because all sides are equal.

How do I find the base if I know the base angles?

Since the sum of angles in any triangle is 180°, the vertex angle = 180° − 2 × base angle. Just compute that and enter it as θ. For example, if each base angle is 70°, the vertex angle is 180° − 140° = 40°.

What is the relationship between the base and the equal sides?

The base is always shorter than the sum of the two equal sides (triangle inequality) and longer than the difference between them. In an isosceles triangle with equal sides a, the base b satisfies 0 < b < 2a. The exact value depends on the vertex angle, as shown by the formula.

Why does the base get longer when the vertex angle increases?

Because sin(θ/2) increases as θ goes from 0° to 180°. When the two equal sides spread apart more, the distance between their endpoints (the base) grows. At θ = 180°, the two equal sides lie in a straight line, and the base equals 2a.

Can I calculate the base without any angles?

Yes, if you know the area and the height. Since area = ½ × base × height, you can rearrange to get base = 2 × area / height. Or if you know the perimeter and the equal sides, base = perimeter − 2a.

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in a | Equal side length | cm | 10 | 0
in v | Vertex angle | degrees | 60 | 0
out base | Base length | cm | 2 * a * sind(v / 2) | 2
out height | Height | cm | a * cosd(v / 2) | 2
out area | Area | cm² | 0.5 * base * height | 2