Isosceles Triangle Calculator
Solve isosceles triangle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Height = sqrt(a^2 - (b/2)^2), Area = (1/2) * b * h
Where a is the length of the equal sides (legs), b is the base length, and h is the height from the apex to the base. The height is derived from the Pythagorean theorem applied to the right triangle formed by the altitude, half the base, and the equal side.
Worked Examples
Example 1: Roof Truss Calculation
Problem:A roof truss has equal rafters of 10 feet and a span (base) of 16 feet. Find the height, area, and angles.
Solution:Height = sqrt(10^2 - 8^2) = sqrt(100 - 64) = sqrt(36) = 6 feet\nArea = (1/2)(16)(6) = 48 square feet\nApex angle = 2 * arcsin(8/10) = 2 * 53.13 = 106.26 degrees\nBase angles = (180 - 106.26) / 2 = 36.87 degrees each\nPerimeter = 2(10) + 16 = 36 feet
Result:Height = 6 ft, Area = 48 sq ft, Apex = 106.26 deg, Base angles = 36.87 deg
Example 2: Isosceles Right Triangle
Problem:Find all properties of an isosceles right triangle with equal sides of 5 units.
Solution:Base (hypotenuse) = 5 * sqrt(2) = 7.071\nHeight = sqrt(25 - 12.5) = sqrt(12.5) = 3.536\nArea = (1/2)(7.071)(3.536) = 12.5 square units\nAlternatively: Area = (1/2)(5)(5) = 12.5 (confirmed)\nApex angle = 90 degrees, Base angles = 45 degrees each\nInradius = (5 + 5 - 7.071) / 2 = 1.464
Result:Base = 7.071, Height = 3.536, Area = 12.5, Angles: 90-45-45
Frequently Asked Questions
What is an isosceles triangle?
An isosceles triangle is a triangle that has at least two sides of equal length. The two equal sides are called the legs, and the third side is called the base. The angles opposite the equal sides are also equal, known as the base angles, while the angle between the two equal sides is called the apex angle or vertex angle. The word isosceles comes from the Greek iso meaning equal and skelos meaning leg. Isosceles triangles appear frequently in architecture, engineering, and nature, from Gothic arches and roof trusses to the cross-sections of many natural crystal formations. Every equilateral triangle is also isosceles, but not every isosceles triangle is equilateral.
How do you calculate the height of an isosceles triangle?
The height (altitude) of an isosceles triangle drawn from the apex to the base can be calculated using the Pythagorean theorem. Since the altitude from the apex bisects the base into two equal halves, it forms a right triangle with the equal side as the hypotenuse. The height h = sqrt(a^2 - (b/2)^2), where a is the length of the equal side and b is the base length. For example, if the equal sides are 10 and the base is 8, the height is sqrt(100 - 16) = sqrt(84) = 9.165. This altitude is also the perpendicular bisector of the base, the median from the apex, and the angle bisector of the apex angle, all in one line segment, which is a unique property of isosceles triangles.
How do you find the area of an isosceles triangle?
The area of an isosceles triangle can be calculated using several methods. The most direct formula uses the base and height: Area = (1/2) times base times height, where height = sqrt(a^2 - (b/2)^2). Alternatively, you can use Heron's formula with s = (2a + b)/2: Area = sqrt(s(s-a)(s-a)(s-b)). A third method uses trigonometry: Area = (1/2) times a^2 times sin(apex angle), using only the equal side length and the apex angle. For a triangle with equal sides of 10 and base of 8, the area is (1/2)(8)(sqrt(100-16)) = 4 times sqrt(84) = 36.66. Each formula is useful depending on which measurements are known.
What are the special properties of an isosceles triangle?
Isosceles triangles have several remarkable properties that set them apart from scalene triangles. The altitude from the apex vertex is simultaneously the median, the perpendicular bisector of the base, and the angle bisector of the apex angle, creating an axis of symmetry. This line of symmetry means the triangle can be folded in half along this line and the two halves match perfectly. The circumcenter, incenter, centroid, and orthocenter all lie on this axis of symmetry. The two base angle bisectors are equal in length, and the two medians to the equal sides are also equal in length. These symmetry properties make isosceles triangles particularly useful in structural engineering where balanced force distribution is needed.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy