Isosceles Triangle - Sides, Angles, Area, Heights

An isosceles triangle calculator is a geometry tool that turns a single pair of measurements on an isosceles triangle into every other useful value: the equal sides, the base, the perimeter, the area, the two heights, the base angles, the apex angle, the inradius, and the circumradius. You type either the base and the perpendicular height, or the two equal sides and the base, and the page reports the rest of the triangle so you do not have to chain the Pythagorean theorem, the area formula, and the angle formulas by hand.

• • Classroom and homework checks: Verify area, perimeter, and angle steps for an isosceles triangle problem, especially when the answer should be a clean Pythagorean triple such as 5-5-6 or 13-13-10.
• • Roof, gable, and awning layout: Convert base width and slant rafter length into a cut list: area of the gable face, height of the peak, and base angle for the bird's-mouth cut.
• • Incircle and circumcircle design: Recover the inradius and circumradius of an isosceles triangle so the largest inscribed circle and the smallest enclosing circle can be drawn or specified.

The two input methods cover the way an isosceles triangle is most often described. Both descriptions encode the same triangle, and the calculator treats them as interchangeable by using the half-base right triangle to switch between them. Because the page exposes intermediate values, you can see how the area, the angle, and the inradius are built from the inputs.

When the triangle is not isosceles, switch to the Triangle Calculator to handle sides, base and height, or two sides with an included angle, or to the Triangle Area Calculator for a more general area tool.

The page implements the standard Euclidean geometry formulas for an isosceles triangle. It takes the two known measurements, builds the half-base right triangle, then uses the Pythagorean theorem, the area formula, and the tangent of the half-apex angle to recover every other measurement.

• a: Length of one of the two equal sides (the leg).
• b: Length of the third, unequal side (the base).
• h_b: Perpendicular altitude from the base up to the apex.
• h_a: Perpendicular altitude from one equal leg to the opposite base corner.
• alpha: Base angle, equal at both base corners.
• beta: Apex (vertex) angle between the two equal sides.
• r: Inradius.
• R: Circumradius.

The same code path serves both input methods: the page accepts either the base and the perpendicular height, or the two equal sides and the base, and the half-base right triangle recovers every other length, angle, and radius on the triangle.

According to Wolfram MathWorld, the altitude of an isosceles triangle with equal sides s and base b is h = sqrt(s^2 - (b/2)^2), the area equals one-half of the base times the altitude, and the inradius equals (2 * area) / (perimeter).

The altitude splits an isosceles triangle into two congruent right triangles, and the Right Triangle Calculator solves that same half-base shape when you only have two of the three known sides or angles.

Four ideas explain why the formulas are short, why the two base angles are equal, and why a single input pair is enough to recover the whole triangle.

A golden triangle is the most famous special isosceles triangle: the leg is in the golden ratio to the base, which forces the apex angle to 36 degrees and the two base angles to 72 degrees.

The dedicated Golden Ratio Calculator explains the underlying phi, and the Equilateral Triangle Area Calculator handles the special case where a equals b and all three angles become 60 degrees.

Six short steps cover every common case, from a clean textbook example to a measured object with only the slant sides and the base.

1. 1 Pick the calculation method: Choose Base and Height for a textbook problem with a known altitude. Choose Two Equal Sides and Base for a measured object with known slant sides.
2. 2 Enter the base length: Type the length of the third, unequal side. The default is 6 units.
3. 3 Enter the second measurement: Type the height (4 by default) for Base and Height, or the equal side (5 by default) for Two Equal Sides and Base.
4. 4 Read the area first: Use the Area output as the main answer for material takeoffs, paint coverage, or area-based comparisons.
5. 5 Read the supporting values: Look at perimeter, equal side, both heights, the apex angle, and the base angle to confirm the arithmetic and feed other calculations.
6. 6 Read the inradius and circumradius: Use the inradius to size the largest circle that fits inside the triangle, and the circumradius to size the smallest circle that encloses all three vertices.

A woodworker is cutting a gable sign with a base of 10 inches and a slant side of 13 inches. The Two Equal Sides and Base method returns area 60.00 square inches, height to the apex 12.00 inches, height to the equal side 9.23 inches, perimeter 36.00 inches, apex angle 45.24 degrees, base angle 67.38 degrees, inradius 3.33 inches, and circumradius 7.04 inches.

The woodworker in this example would otherwise have to chain a half-base right triangle through three or four formulas by hand, but the Triangle Area Calculator accepts the same kind of input pair and returns the same area plus the same supporting values for a general triangle.

These benefits matter most when you are working a problem by hand and need a quick, trustworthy check on the arithmetic.

• • Two interchangeable input methods: Use the direct base-and-height path or the side-and-base path depending on what is already measured on the triangle.
• • All key outputs in one place: See area, perimeter, equal side, both heights, apex angle, base angle, inradius, and circumradius without re-entering values into another calculator.
• • Easy homework audit trail: The intermediate height, side, and angle values match the steps many geometry teachers expect to see in worked solutions.
• • Decimal friendly: Decimal base, height, and side values work for measured sketches, scaled drawings, and design dimensions.
• • Special cases handled: The page recognizes an equilateral triangle when a = b and rejects input pairs where the equal side is shorter than half the base.

If the same project includes a general scalene triangle next to the isosceles one, the Scalene Triangle Area Calculator covers the non-isosceles case with the same style of inputs and a Heron-style area formula.

The formulas are the same in every case, but a few measurement and labeling decisions affect whether the answer matches the real shape.

• • This calculator does not solve for the equal side from an area and a base alone, or for the base from an altitude and an equal side, without the user back-solving on paper first.
• • The results are geometric estimates only. Real material takeoffs may need allowances for seams, overlap, cutting waste, edge thickness, or coating that spreads beyond the face of the triangle.

According to Wolfram MathWorld, a golden triangle is an isosceles triangle in which the ratio of the longer leg to the base is the golden ratio phi, which makes the apex angle 36 degrees and the base angles 72 degrees.

The 5-12-13 and 3-4-5 right triangles that show up in the worked examples are both Pythagorean triples, and the Pythagorean Triples Calculator generates the full list so you can spot the next clean example before you start measuring.