How this instrument works
An isosceles triangle's two base angles are equal for a reason that has nothing to do with measurement and everything to do with symmetry: reflect the triangle across the line from its apex to the midpoint of the unequal side and the two legs swap places, leaving the shape unchanged. Since nothing moved, the angle that was on the left must equal the angle that's now on the left — Euclid proved exactly this in the Elements, Book I, Proposition 5, a result students nicknamed the Pons Asinorum for how many got stuck on it. This sheet turns that symmetry into a number by dropping a perpendicular from the apex, which lands at that same midpoint and splits the triangle into two congruent right triangles. Each has the leg as its hypotenuse and half that unequal side as the other leg, so the base angle is the angle whose cosine equals that ratio — hence the inverse cosine in the formula.
The apex angle costs nothing extra once a base angle is in hand: every triangle's three interior angles sum to 180°, and here two of them are already known to match, so the apex angle is simply whatever 180° has left over after subtracting twice the base angle. No second trigonometric call is needed, and skipping one avoids a place where independent rounding could quietly disagree with the first.
The two extremes are worth picturing. Stretch the legs far longer than the unequal side and that same ratio shrinks toward zero, so the base angles climb toward 90° — the triangle becomes a tall, narrow spire standing almost straight up on a short base. Shrink the legs toward exactly half that length instead and the ratio climbs toward one, so the base angles collapse toward 0° and the apex angle opens toward 180° — the triangle flattens toward a straight line, the shortest legs that can still close a triangle at all.
- Enter the triangle's unequal side into the Base field.
- Enter the length of either matching side into Equal leg length.
- Read Base angle (both equal) — one figure that applies to both ends of the unequal side.
- Read Apex angle for the angle between the two legs, and use its unit toggle to switch between degrees, radians, and turns.
Worked example — base 6, legs 5
Take a frame with a base of 6 units and two equal legs of 5 units, this sheet's own default figures. Halving that side gives 3, and 3 ÷ 5 = 0.6, so the base angle is cos⁻¹(0.6) = 0.9272952180016123 radians. Converted to degrees that's 53.130102°, which the sheet rounds for display to about 53.13°.
The right triangle hiding inside that split has sides 3, 4, and 5 — the altitude comes out to exactly 4, because 3-4-5 is a Pythagorean triple, so 53.13° is a value worth remembering on sight. The apex angle takes whatever is left of 180°: 180° − 2 × 53.130102° = 73.739795°, matching the sheet's apexAngle reading of 1.2870022175865685 radians, or about 73.74°.
Questions
Why are the two base angles of an isosceles triangle always equal?
Because the triangle is a mirror image of itself across the line from the apex to the midpoint of the unequal side — swapping the two equal legs swaps the two base angles but leaves the triangle unchanged, so the angles must match. Euclid proved this in the Elements, Book I, Proposition 5, a result nicknamed the Pons Asinorum, 'bridge of asses,' for the students it separated.
How is the base angle actually worked out from the base and leg lengths?
A perpendicular dropped from the apex lands exactly at the midpoint of that side by symmetry, splitting the triangle into two congruent right triangles. Each has the leg as its hypotenuse and half that length as the other side, so the base angle is cos⁻¹((base ÷ 2) ÷ leg) — the angle whose adjacent-over-hypotenuse ratio matches that fraction.
Why is the apex angle just 180° minus twice the base angle, instead of its own arccos formula?
Every triangle's interior angles sum to 180°, and two of these three are already known to be equal, so the apex angle is only ever whatever remains: 180° − 2 × base angle. Computing it that way, rather than with a second inverse-cosine call, also keeps the two answers from disagreeing by a rounding hair.
What happens to the angles as the legs get very long compared to the unequal side?
The base angles climb toward 90° and the apex angle shrinks toward 0°, because (base ÷ 2) ÷ leg approaches zero and cos⁻¹(0) is 90°. Picture that side staying fixed while the legs rise almost straight up: the triangle stretches into a tall, narrow spire, and the corners near it look increasingly square.
How small can the base angles get?
They shrink toward 0° as the legs shrink toward exactly half that length, the shortest a leg can be and still close the shape. There, (base ÷ 2) ÷ leg approaches 1 and cos⁻¹(1) is 0°, so the triangle flattens toward a line and the apex angle opens toward 180°. The sheet's own check stops legs any shorter than that, since no triangle would exist.
Does this sheet also report the triangle's height or area?
No — this one solves only for the base angle and the apex angle. A companion sheet on this site handles the height, using the same right-triangle split solved for the vertical side instead of an angle, and another handles the area; each starts from the same inputs but reports a single different quantity.