What is an isosceles triangle?

A triangle with exactly two equal sides, called the legs (a=c), and two equal angles opposite them (the base angles, A=C). The third side is the base (b), and the third angle is the apex angle.

What is the formula for the area of an isosceles triangle?

Given leg a and base b: K = b/4{4a^2-b^2}. For a=5, b=6: K = 6/4{100-36} = 6/4(8) = 12.

What is the formula for the height (altitude to the base)?

The altitude from the apex to the base is h_b = 1/2{4a^2-b^2}. It comes from the Pythagorean theorem: the height splits the triangle into two right triangles with legs b/2 and h_b, hypotenuse a, so h_b = {a^2-(b/2)^2}.

What about the altitude to one of the legs?

The altitude to a leg is h_a = h_c = b/2a{4a^2-b^2} — a different (usually shorter) length than h_b, since it drops from a base vertex onto the opposite leg rather than from the apex onto the base.

How do I find the base and apex angles?

The base angles are equal: A = C = \!(b/2/a). The apex angle is whatever remains of 180°: B = 180° - 2A. For a=5, b=6: A ≈ 53.13° and B ≈ 73.74°.

Why must the base be shorter than twice the leg?

This is the triangle inequality applied to an isosceles triangle: the base must be shorter than the sum of the two legs, b a+a = 2a. If b 2a, the two legs can't reach far enough to meet above the base at all, so no triangle closes.

Is an equilateral triangle a special case of an isosceles triangle?

Yes — when the base equals the leg (b=a), all three sides are equal and the triangle becomes equilateral. Every equilateral-triangle formula is what these isosceles formulas reduce to at b=a.

Isosceles Triangle Calculator: Area, Height, Angles

Find an isosceles triangle's area, height, perimeter, and both base/apex angles from the two equal legs and the base — with exact simplified-radical results.

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Isosceles triangle formulas

An isosceles triangle has two equal legs $a$ (also labeled $c$ ) and a base $b$ . The two base angles ( $A=C$ , opposite the equal legs) are equal, and the apex angle $B$ sits between the two legs.

Given the leg $a$ and base $b$ , every other measure follows from a single discriminant, $4a^2-b^2$ :

Quantity	Formula
Perimeter	$P = 2a+b$
Semi-perimeter	$s = a+\dfrac{b}{2}$
Area	$K = \dfrac{b}{4}\sqrt{4a^2-b^2}$
Altitude to the base	$h_b = \dfrac{1}{2}\sqrt{4a^2-b^2}$
Altitude to a leg	$h_a = h_c = \dfrac{b}{2a}\sqrt{4a^2-b^2}$
Base angles	$A = C = \arccos\!\left(\dfrac{b/2}{a}\right)$
Apex angle	$B = 180° - 2A$

Tip: The altitude to the base splits the triangle into two congruent right triangles, each with legs

b/2

and

h_b

and hypotenuse

a

. The Pythagorean theorem gives

h_b = \sqrt{a^2-(b/2)^2} = \frac{1}{2}\sqrt{4a^2-b^2}

, and the area follows from

K = \frac{1}{2}\cdot b \cdot h_b

Worked example: leg a = 5, base b = 6

$4a^2-b^2 = 4(25)-36 = 64 \qquad h_b = \frac{1}{2}\sqrt{64} = 4 \qquad K = \frac{6}{4}\sqrt{64} = \frac{6\times8}{4} = 12$

$P = 2(5)+6 = 16 \qquad s = 5+3 = 8 \qquad A = C = \arccos(0.6) \approx 53.13° \qquad B \approx 73.74°$

This is two 3-4-5 right triangles mirrored back-to-back across the height, which is why every value here comes out clean. Not every leg/base pair simplifies this neatly — leg $a=5$ , base $b=4$ gives a discriminant of $84 = 4\times21$ , so $h_b = \sqrt{21} \approx 4.5826$ : still an exact closed form, just not a whole number.

Solving with a different base

Example: leg a = 5, base b = 8

$4a^2-b^2 = 4(25)-64 = 36 \qquad h_b = \frac{1}{2}\sqrt{36} = 3 \qquad K = \frac{8}{4}\sqrt{36} = \frac{8\times6}{4} = 12$

Interestingly, this triangle has the same area (12) as the $a=5,b=6$ example above, even though its height ( $3$ ) is shorter — a wider base compensates for a smaller height. Its base angles are wider apart too: $A = C = \arccos(0.8) \approx 36.87°$ , so the apex angle $B \approx 106.26°$ is much larger — this is a wide, "flatter" isosceles triangle rather than a tall, "pointier" one.

Common mistakes

Using a leg as if it were the height. The altitude to the base, $h_b$ , is always shorter than the leg $a$ — mixing them up overstates the height and the area.
Forgetting the existence condition $b \lt 2a$ . The base must be shorter than twice the leg, or no triangle can close at all — this calculator flags it rather than showing a nonsensical negative-square-root result.
Mixing up $h_b$ and $h_a=h_c$ . The altitude to the base and the altitude to a leg are generally different lengths; only in special cases do they coincide.

Where this shows up

Roof trusses and gables: the classic gable-roof cross-section is an isosceles triangle, with the ridge as the apex and the span as the base.
Symmetric design: road signs, pennants, and architectural motifs often use an isosceles silhouette specifically for its single line of symmetry.
Surveying and construction: any layout with two known equal braces and a known span between their feet reduces to this same leg/base relationship.

An isosceles triangle generalizes the equilateral triangle (the special case $b=a$ ) and is itself a special case of the general triangle covered by triangle theorems and the law of cosines. Its own height calculation is a direct application of the right triangle relationships, since the altitude to the base splits it into two congruent right triangles.

Frequently asked questions

What is an isosceles triangle?: A triangle with exactly two equal sides, called the legs ( $a=c$ ), and two equal angles opposite them (the base angles, $A=C$ ). The third side is the base ( $b$ ), and the third angle is the apex angle.
What is the formula for the area of an isosceles triangle?: Given leg $a$ and base $b$ : $K = \frac{b}{4}\sqrt{4a^2-b^2}$ . For $a=5$ , $b=6$ : $K = \frac{6}{4}\sqrt{100-36} = \frac{6}{4}(8) = 12$ .
What is the formula for the height (altitude to the base)?: The altitude from the apex to the base is $h_b = \frac{1}{2}\sqrt{4a^2-b^2}$ . It comes from the Pythagorean theorem: the height splits the triangle into two right triangles with legs $b/2$ and $h_b$ , hypotenuse $a$ , so $h_b = \sqrt{a^2-(b/2)^2}$ .
What about the altitude to one of the legs?: The altitude to a leg is $h_a = h_c = \frac{b}{2a}\sqrt{4a^2-b^2}$ — a different (usually shorter) length than $h_b$ , since it drops from a base vertex onto the opposite leg rather than from the apex onto the base.
How do I find the base and apex angles?: The base angles are equal: $A = C = \arccos\!\left(\frac{b/2}{a}\right)$ . The apex angle is whatever remains of $180°$ : $B = 180° - 2A$ . For $a=5$ , $b=6$ : $A \approx 53.13°$ and $B \approx 73.74°$ .
Why must the base be shorter than twice the leg?: This is the triangle inequality applied to an isosceles triangle: the base must be shorter than the sum of the two legs, $b \lt a+a = 2a$ . If $b \geq 2a$ , the two legs can't reach far enough to meet above the base at all, so no triangle closes.
Is an equilateral triangle a special case of an isosceles triangle?: Yes — when the base equals the leg ( $b=a$ ), all three sides are equal and the triangle becomes equilateral. Every equilateral-triangle formula is what these isosceles formulas reduce to at $b=a$ .