How to Find the Height of an Isosceles Triangle

Geometry Lessons

The short answer

h = √(a² – (b/2)²)

Here a is one of the two equal sides and b is the base. Halve the base first, square it, subtract that from the square of the equal side, then take the root.

An isosceles triangle has two equal sides and a third that differs. That symmetry is exactly what makes the height findable from the side lengths alone, because the altitude drawn to the base lands on its midpoint and hands you a right triangle to work with.

Why the base is bisected but the other sides are not

The altitude to the base splits the triangle into two congruent halves, so the base divides evenly. This holds only for the altitude drawn to the base, meaning the unequal side. Drop a perpendicular onto one of the two equal sides instead and nothing is bisected, because there is no symmetry along that direction to exploit.

The distinction matters more than it looks. The rule gets remembered as altitudes bisect things, which is false in general and false even elsewhere in this same triangle. It is one specific altitude that behaves this way.

1

Halve the base

Divide the unequal side by two. That result is one leg of your right triangle.

2

Square and subtract

Square one equal side, square the half base, then subtract the second from the first.

3

Take the root

What comes out is the perpendicular height measured to the base.

A worked example

Take equal sides of 13 and a base of 10. Half the base is 5. The height squared is 169 minus 25, which is 144, so the height is exactly 12. The area follows as half of 10 times 12, or 60 square units.

Cross check it with Heron on sides of 13, 13 and 10. The semiperimeter is 18, and 18 times 5 times 5 times 8 is 3600, whose square root is 60. Same area by a completely different route, which is the strongest confirmation available.

Two panels of the same isosceles triangle with equal sides of 13 and a base of 10. The left panel halves the base to 5 and reaches a height of 12, while the right panel uses the full base of 10 and reaches an incorrect 8.31.
Both panels show the same triangle, and the base really does split into 5 and 5 in each. The only difference is which number went into Pythagoras, and it costs almost four units of height.

Halve the base before squaring, not after

Skipping the halving is the error that catches almost everyone, and it is dangerous precisely because the answer it returns looks reasonable. There is no right triangle anywhere in this shape with legs of 13 and 10, so the number it produces describes nothing at all.

half base 5givesh² = 13² – 5² = 14412

the altitude meets the base at its midpoint

full base 10givesh² = 13² – 10² = 698.31

no right triangle in this shape has legs of 13 and 10

A second example with less friendly numbers

Equal sides of 25 and a base of 14. Half the base is 7, so the height squared is 625 minus 49, which is 576, giving a height of exactly 24. The area is half of 14 times 24, or 168 square units.

The other two heights

Every triangle has three heights, and an isosceles triangle has two that equal each other and one that differs. The formula above gives the one measured to the base. To get the height onto one of the equal sides instead, work from the area: with an area of 168 and a side of 25, that height is 2 times 168 divided by 25, which is 13.44. A different number for the same triangle, and neither is more correct than the other.

When the equal sides are too short

If the equal side is shorter than half the base, the subtraction goes negative and there is no real square root. That is not an arithmetic slip. It means the two equal sides cannot reach far enough to meet above the base, so no such triangle exists. Equal sides of 6 with a base of 20 is an example.

Mistakes that catch people out

  • Forgetting to halve the base. The single most common error, and it returns a plausible looking number rather than an obvious one.
  • Using the base as the hypotenuse. The hypotenuse is one of the equal sides, always the longest side of the right triangle you built.
  • Assuming every altitude bisects. Only the one drawn to the base does.
  • Pairing the base height with an equal side in the area formula. Each height belongs to its own base.
  • Treating a negative result as a mistake. It means the triangle cannot exist as described.

Practice questions

  1. Equal sides of 5 with a base of 6. Find the height.
  2. Equal sides of 10 with a base of 12. Find the height.
  3. Equal sides of 17 with a base of 16. Find the height.
  4. Equal sides of 13 with a base of 10. Find the area.
  5. Equal sides of 6 with a base of 20. Find the height.
Show the answers
  1. half base 3, then h² = 25 – 9 = 16, so h = 4
  2. half base 6, then h² = 100 – 36 = 64, so h = 8
  3. half base 8, then h² = 289 – 64 = 225, so h = 15
  4. h = 12, then ½ x 10 x 12 = 60 square units
  5. half base 10, and 36 – 100 is negative, so no such triangle exists

Common questions

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

The height to the base is the square root of the equal side squared minus half the base squared. Halve the base before squaring it.

Why is the base halved?

Because the altitude to the base of an isosceles triangle lands on its midpoint, splitting the shape into two congruent right triangles.

Does the altitude always bisect the side it meets?

Only the altitude to the base. Perpendiculars drawn onto either of the two equal sides do not bisect anything.

What if I get a negative number under the square root?

The two equal sides are too short to meet above that base, so the triangle cannot exist with the lengths given. Check the measurements.

How do I find the height onto one of the equal sides?

Work from the area. Double the area and divide by that side length, since every base has its own matching height.

Is an equilateral triangle also isosceles?

Yes. An equilateral triangle satisfies the definition, and this formula works on it, though the root three over two shortcut is quicker.

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