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Area of Isosceles Right Triangle

The area of an isosceles right triangle (derived from 0.5bh) is equal to the square of its base divided by 4, where b is the base and h is the height of the isosceles right triangle.

Recall that base angles of an isosceles triangle are equal. Since an isosceles right triangle is a right-angled isosceles triangle, the base angles of an isosceles right triangle are equal.

Question 1:
Where is the base of an isosceles right triangle?

Answer:
The side between the base angles is the base.

Question 2:
Where is the height of an isosceles right triangle?

Answer:
The height is a straight line drawn from the right angle to bisect the base.

An isosceles right triangle has two 45° angles and one 90° angle. The two 45° angles are the base angles and the 90° angle is the right angle. The base is the straight line between the two 45° angles and the height is the straight line drawn from the right angle to the mid-point between the two 45° angles.

Consider an isosceles right triangle ABC, where A and B are the base angles and C is the right angle. The three sides of the triangle are AB, BC and AC. The height is a vertical line drawn from C to the mid-point of AB, which means that the height and the base of an isosceles right triangle are two perpendicular lines.

Let the mid-point of the base be M. Then, the height is equal to CM.

In an isosceles right triangle, CM = AM = MB. Hence, the height is equal to half It's base.

h = b/2

Recall that the general formula of the area A of a triangle is 0.5bh.

Substituting 0.5b for h,
A (Area) = 0.5b × 0.5b
                = 0.25 (b × b)

Example:
The base of an isosceles right triangle is 4cm. What is the area of the isosceles right triangle?

Solution:
b (base)     = 4cm
h (height) = 0.5b
                   = 0.5 × 4cm
                   = 2cm

Area (A) = 0.5bh
                = 0.5 × 4cm × 2cm
                = 4cm (squared)

Alternatively,
A = 0.25 (b × b)

b = 4cm

Substituting 4 for b,
A = 0.25 (4cm × 4cm)
    = 0.25 × 16cm (squared)
    = 4cm (squared) 

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