How do you find the area of an isosceles triangle?

Find the height by dropping a perpendicular from the apex to the midpoint of the base. Height = sqrt(leg squared minus (base/2) squared). Area = (1/2) x base x height. For legs 5 and base 6: height = sqrt(25-9) = 4, area = (1/2) x 6 x 4 = 12 square units.

What is the apex angle of an isosceles triangle?

The apex angle is at the vertex between the two equal legs. It equals 2 x arcsin((base/2) / leg). For legs 5 and base 6: apex = 2 x arcsin(3/5) = 2 x 36.87 = 73.74 degrees.

What makes a triangle isosceles?

An isosceles triangle has at least two sides of equal length, called the legs, and a third side called the base. The two base angles (opposite the equal legs) are always equal to each other. An equilateral triangle is a special case where all three sides are equal.

Isosceles Triangle Calculator

An isosceles triangle has two sides of equal length, called legs, and a third side called the base. The two angles at the ends of the base are always equal to each other. You see isosceles triangles everywhere: the gable of a roof, a yield sign, a coat hanger, and many structural frames are isosceles shapes chosen for their symmetry and balance. This calculator takes the leg length and the base length and returns the height, area, perimeter, apex angle, and base angles. The key to the isosceles triangle is its line of symmetry. A perpendicular dropped from the apex hits the midpoint of the base, splitting the shape into two identical right triangles. The height is found from Pythagoras: it is the square root of the leg squared minus half the base squared. From there, the area is half the base times the height, and the perimeter is two legs plus the base. The apex angle is found using the inverse sine of half the base divided by the leg, then doubled. The two base angles each equal half of what remains when the apex angle is subtracted from 180. The calculator checks that your inputs form a valid triangle: the two legs must be long enough to span the base, meaning twice the leg length must exceed the base. Enter any valid leg length and base and all five results appear instantly and update as you type. Use this tool for geometry homework, construction layout of gable ends and roof peaks, and any practical application involving a symmetric triangle.

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Equal leg length

units

Base length

units

area (square units)

Height4

Perimeter16

Apex angle73.74°

Base angles53.13°

How it works

With equal legs a and base b: height h = √(a² − (b/2)²). Area = (1/2) × b × h. Perimeter = 2a + b. Apex angle = 2 × arcsin((b/2) / a). Each base angle = (180 − apex angle) / 2. The three angles always sum to 180 degrees.

Worked example

With legs = 5 and base = 6: half base = 3. Height = √(25 − 9) = √16 = 4 units. Area = (1/2) × 6 × 4 = 12 square units. Perimeter = 10 + 6 = 16 units. Apex angle = 2 × arcsin(3/5) = 2 × 36.87 = 73.74 degrees. Base angles = (180 − 73.74) / 2 = 53.13 degrees each. These match the pre-filled defaults above.

Related calculators

Equilateral Triangle Calculator: area, height and radii when all three sides are equal.
Square Calculator: area, perimeter and diagonal of a square.
SOHCAHTOA Calculator: solve right triangles from an angle and a side.
Law of Sines and Cosines: solve any triangle from sides and angles.

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