An isosceles triangle is a triangle in which at least two sides are equal; sometimes the equal sides are called the legs. The other side is called the base. The angles opposite of the equal sides are also equal. This helps to simplify some calculations, like the area formula, which uses the height (that is, the perpendicular distance from the base to the opposite vertex) and the length of the base. The isosceles triangle, characterized by equal sides and angles, offers unique properties that make it easier to work within various mathematical and real-world applications.
The formula for the area of an isosceles triangle is given as,
Where,
Base: the length of one of the triangle sides taken as a base (it can be the base of the isosceles triangle).
Height: Perpendicular distance from the base to the opposite vertex.
Also,
The perimeter of the isosceles triangle: P = 2a+b
The altitude of the isosceles triangle:
Base and Height: A = ½ × b × h
All three sides: A = ½[√(a2 − b2 ⁄4) × b]
Length of 2 sides and an angle between them: A = ½ × b × a ×A sin(α)
Two angles and length between them: A = [a2×sin(β/2)×sin(α)]
Isosceles right triangle: A = ½ × a2
where,
b is the base of the isosceles triangle
a is the measure of equal sides of the isosceles triangle
α is the measure of equal angles of the isosceles triangle
Β is the measure of the angle opposite to the base
The area of an isosceles triangle can be derived easily using Heron's formula as explained below.
Heron's formula states that,
Area = √[s(s−a)(s−b)(s−c)]
Where, s = ½(a + b + c)
Now, for an isosceles triangle,
s = ½ (a + a + b)
Or,
s = ½ (2a + b)
Or, s = a + (b/2)
Now,
Area = √ [s(s−a)(s−b)(s−c)]
Or, Area =
⇒ Area = (s−a) × √ [s (s−b)]
Put in the value of "s"
⇒ Area = (a + b/2 − a) × √[(a + b/2) × ((a + b/2) − b)]
⇒ Area = b/2 × √[(a + b/2) × (a − b/2)]
Or, area of isosceles triangle =
Isosceles Right Triangle Area= ½ × a
Area = ½ ×base × height
area = ½ × a × a =
P = a(2+√2)
Derivation:
The perimeter of an isosceles right triangle is the sum of all the sides of an isosceles right triangle itself.
Let the two equal sides be a. Using Pythagoras theorem the unequal side can be found to be a√2.
Therefore, the perimeter of the isosceles right triangle = a+a+a√2
= 2a+a√2
= a(2+√2)
= a(2+√2)
Using the Length of two Sides and the Angle Between Them
A = ½ × b × c × sin(α)
Using 2 Angles and Length Between Them
Problem 1: Find the area of an isosceles triangle, if b = 12 cm and h = 17 cm.
Solution:
The base of the triangle (b) = 12 cm
Height of the triangle (h) = 17 cm
Area of Isosceles Triangle = (1/2) × b × h
= (1/2) × 12 × 17
= 6 × 17
= 102