Isosceles Triangle

The isosceles triangle is a polygon of three sides with two equal sides. The other side unequal is called the base of the triangle.

Therefore, the angles will also be two equal (α) and the other different (β), this being the angle formed by the two equal sides (a).

Two special cases of isosceles triangles are the equilateral triangle and the isosceles right triangle.

Altitude of an Isosceles Triangle

The altitude (h) of the isosceles triangle (or height) can be calculated from Pythagorean theorem. The sides a, b/2 and h form a right triangle. The sides b/2 and h are the legs and a the hypotenuse.

By Pythagorean theorem:

And it is obtained that the height h is:

In a isosceles triangle, the height corresponding to the base (b) is also the angle bisector, perpendicular bisector and median.

Area of an Isosceles Triangle

The area of an isosceles triangle is calculated from the base b (the non-repeated side) and the altitude (h) of triangle corresponding to the base. The area is the product of the base and the altitude divided by two, being its formula the following one:

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Perimeter of an Isosceles Triangle

The perimeter of an isosceles triangle is obtained as the addition of the three sides of the triangle. Having two equal sides, the perimeter is twice the repeated side (a) plus the different side (b).

If the repeating side (a) and the angle of the two equal sides are known, the other side (b) should be found by law of cosines.

Resolved Exercises

Exercise of the Isosceles Triangle Area

Determine the area of a isosceles triangle knowing its two equal sides (a=3 cm) and the unequal one, whose length is 2 cm (b=2 cm).

What is its area?

Calculate the area using the above formula by multiplying the base by the height:

The isosceles triangle’s area is 2.83 cm².

Exercise of the Isosceles Triangle Perimeter

Being an isosceles triangle with two equal sides, a=3 cm and a different side of b=2 cm.

What is its perimeter?

To calculate this perimeter we add the repeated side multiplied by two plus the unequal side, i.e.:

It is obtained that the isosceles triangle’s perimeter is 8 cm.

Exercise of the Altitude of an Isosceles Triangle

Find the sides and perimeter of an isosceles triangle whose height referred to the uneven side measures h = 6 cm and the opposite angle, also uneven, 40°.

Found by trigonometric relationships from one of the right triangle into which divides the isosceles triangle by height h.

The opposite leg to the angle β/2, which is the segment b/2, we found it through the tangent:

The side b measures 4.36 cm.

The right triangle‘s hypotenuse formed, i.e. the a side is found by cosine:

The side a measures 6.38 cm.

Finally, the triangle’s perimeter will measure:

It is obtained that the perimeter of this isosceles triangle will measure 17.12 cm.