Calculate the area, height, base, legs, angles, and perimeter of an isosceles triangle from any two known values.
Isosceles Triangle Formula
An isosceles triangle has two equal sides called legs (a) and one unequal side called the base (b). A line dropped from the apex to the base is the height (h), and it splits the triangle into two equal right triangles. The calculator solves the whole triangle once you give it any two values.
Height from the leg and base:
h = sqrt(a^2 - (b/2)^2)
Area:
A = (1/2) * b * h = (1/2) * a^2 * sin(beta)
Perimeter:
P = 2a + b
Angles (the two base angles are equal):
alpha = (180 - beta) / 2 beta = 180 - 2*alpha
- a = leg, the length of each of the two equal sides
- b = base, the length of the unequal side
- h = height measured from the apex perpendicular to the base
- alpha = base angle, the angle each leg makes with the base (both are equal)
- beta = apex or vertex angle, the angle between the two legs
- A = area
- P = perimeter
The height formula comes from the Pythagorean theorem applied to one of the half triangles, where the half base (b/2) and the height are the two legs and the slant side a is the hypotenuse. The area uses the standard one half base times height, and the alternate form (1/2)*a^2*sin(beta) is useful when you know the two legs and the apex angle instead of the base. Because the angles of any triangle sum to 180 degrees and the two base angles are equal, knowing one angle fixes the other two.
Isosceles Triangle Quantities and When to Use Them
Use this table to pick the right formula based on which two values you already have.
| Quantity | Formula | Known inputs |
|---|---|---|
| Height | h = sqrt(a^2 - (b/2)^2) | Leg and base |
| Base | b = 2*a*sin(beta/2) | Leg and apex angle |
| Leg | a = (b/2) / cos(alpha) | Base and base angle |
| Area | A = (1/2)*b*h | Base and height |
| Base angle | alpha = (180 - beta)/2 | Apex angle |
| Apex angle | beta = 180 - 2*alpha | Base angle |
Several well known shapes are special cases of the isosceles triangle.
| Triangle | Apex angle | Base angles |
|---|---|---|
| Right isosceles (45-45-90) | 90° | 45° each |
| Equilateral (special case) | 60° | 60° each |
| Tall and narrow | 20° | 80° each |
| Short and wide | 120° | 30° each |
Example Problems
Example 1: leg and base. You have legs a = 5 and base b = 6. The height is h = sqrt(5^2 - (6/2)^2) = sqrt(25 - 9) = sqrt(16) = 4. The area is A = (1/2)*6*4 = 12. The perimeter is P = 2*5 + 6 = 16. The base angle is alpha = arccos((b/2)/a) = arccos(3/5) = 53.13 degrees, so the apex angle is beta = 180 - 2*53.13 = 73.74 degrees.
Example 2: leg and apex angle. You have legs a = 10 and an apex angle beta = 40 degrees. The base is b = 2*10*sin(40/2) = 20*sin(20) = 6.84. The height is h = 10*cos(20) = 9.40. The area is A = (1/2)*10^2*sin(40) = 50*0.6428 = 32.14. Each base angle is alpha = (180 - 40)/2 = 70 degrees.
FAQ
How do you find the height of an isosceles triangle?
Drop a line from the apex straight down to the base. It hits the midpoint of the base, creating a right triangle whose hypotenuse is a leg (a) and whose bottom side is half the base (b/2). Apply the Pythagorean theorem: h = sqrt(a^2 - (b/2)^2).
What are the angles of an isosceles triangle?
An isosceles triangle has two equal base angles and one apex angle. All three add up to 180 degrees. If you know the apex angle beta, each base angle is (180 - beta)/2. If you know one base angle alpha, the apex angle is 180 - 2*alpha.
Can an isosceles triangle be a right triangle?
Yes. A right isosceles triangle has a 90 degree apex angle and two 45 degree base angles. Its two legs are equal, and the base is the hypotenuse. This is the common 45-45-90 triangle.
