With this calculator you can calculate the angles, altitudes, area, perimeter, inradius and circumradius of an isosceles triangle. You can select to how many decimal places the calculated values should be rounded and further calculations are always performed with the rounded values.
On this page, formulas are used and lengths are named as they appear on German websites about isosceles triangles. For example, the length of the base of the triangle is called c instead of b. (German Wikipedia article about isosceles triangles)
What is an isosceles triangle?
An isosceles triangle is a triangle in which at least 2 sides have the same length. On this site, the lengths of the two sides of equal length are called a and b, where a = b. The length of the third side is called c on this site. The angle opposite from the side with length c is called γ and the other two angles α and β.
Altitudes
The two altitudes h_{a} and h_{b} have the same length in an isosceles triangle. If the side length c differs from a and b, then the altitude h_{c} also differs from the altitudes h_{a} and h_{b}.
The altitude h_{c} divides the isosceles triangle into 2 right triangles. The side lengths and the angles of both triangles are equal.
Calculation of the altitudes:
The altitude h_{a} can be calculated with the following formula:
The altitude h_{b} corresponds to h_{a}.
The following formula can be used to calculate h_{c}:
Example:
An isosceles triangle has the side lengths a = 5 cm, b = 5 cm and c = 4 cm. The altitudes are to be calculated.
h_{a} | = | ∙4∙a²−c² |
| = | ∙4∙(5 cm)²−(4 cm)² |
| = | ∙4∙25 cm²−16 cm² |
| = | 0.4∙100 cm²−16 cm² |
| = | 0.4∙84 cm² |
| ≈ | 3.66606 cm |
h_{b}=h_{a}≈3.66606 cm
Angles
The angle γ (gamma) is between the two sides of equal length (resp. opposite from the side with length c). If the isosceles triangle is not also an equilateral triangle, then γ is different from the two angles α and β. However, the two angles α and β are equal. The sum of the 3 interior angles is 180°. Since the two angles α and β are equal in size:
γ=180°−2∙α
If the angle γ is known and you want to calculate the angle α, then you can solve the equation for α.
Example:
It is known that the angle γ of an isosceles triangle is 40°. We are looking for α and β.
Since α and β are equal, β = 70° is also valid.
The angles α and β are each smaller than 90°. The angle γ can also be larger than 90°, but is smaller than 180°.
Calculate c with the help of γ:
If you draw the altitude h_{c} in the above isosceles triangle, then the isosceles triangle is divided into 2 right triangles, which have the same side lengths and the same angles. Since a = b and α = β, b and β have been replaced by a and α in the following figure.
h
_{c} runs exactly in the middle between the two sides of equal length. The angle between the altitude and a or b is therefore
.
The height h
_{c} hits the side with length c exactly in the middle. Thus the right-angled triangles have in each case a cathetus with the length
.
In a right triangle, the following applies for the acute angles:
sine of angle =opposite of angle |
hypotenuse |
One of the acute angles in the right triangles is
. The opposite side of this angle has length
and the hypotenuse has length a.
If you solve the formula for c you get:
This formula can be used if the length of the two sides of equal length and the angle γ are known and the side length c is to be calculated.
Calculate γ with the help of the side lengths:
If the side lengths are known and the angle γ is to be calculated, then you can solve the above formula for γ and obtain:
Calculate α with the help of a and h_{c}:
The opposite side of α is h_{c} and the hypotenuse is a. Thus:
If you solve this for α you get:
Area of an isosceles triangle
The area of any triangle can be calculated by choosing one side as the base side, multiplying it by the corresponding height, and dividing the result by 2. The following therefore applies:
Example:
A triangle has the side lengths a = 2 cm, b = 2 cm and c = 3 cm. The area is to be calculated.
First, one of the altitudes is calculated. In this example, the altitude h_{c} is chosen.
Then h_{c} is used in the formula for the calculation of the area:
Calculate the area with the help of γ:
If only the angle γ and the side length a are known, the area can still be calculated. The following formula can be used for this purpose:
Perimeter of an isosceles triangle
The perimeter of a triangle is calculated by adding the lengths of the 3 sides. a and b have the same length. Therefore applies:
P=2∙a+c
Example:
A triangle has the side lengths a = 2 cm, b = 2 cm and c = 3 cm. The perimeter is to be calculated.
P | = | 2∙a+c |
| = | 2∙2 cm+3 cm |
| = | 7 cm |
Inradius
An incircle of a triangle is a circle that lies inside the triangle and touches each side of the triangle at exactly one point. For an isosceles triangle with side lengths a, b and c, where a = b and a ≠ c, the incircle touches the side with length c exactly in the middle. The sides with lengths a and b, on the other hand, are touched at a point other than the center.
The inradius can be calculated with the following formula:
Circumradius
The circumcircle of a triangle is a circle whose circular line passes through each of the 3 corners. If one of the angles is known, then the radius of the circumcircle can be calculated using one of the following formulas:
However, the following formula can also be used to calculate the radius of the circumcircle: