Question

Derive the formula to calculate area of an isosceles triangle.

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Answer 1

Topic

Triangles

To Derive

The formula to calculate area of an isosceles triangle.

Derivation

Area using base and height of triangle

Area of a parallelogram = Base × Height

A diagonal of a parallelogram divides it into two triangles of equal areas.

Area of parallelogram = 2 × Area of Triangle

Base × Height = 2 × Area of Triangle

Area of Triangle = ( Base × Height ) / 2

Area using all three sides of Triangle

Let length of side opposite to vertex be 'b' and length of equal sides be 'a'.

Draw a perpendicular from vertex which extends to two equal sides.

Now, apply Pythagoras Theorem

(Hypotenuse)² = (Perpendicular)² + (Base)²

a² = (Perpendicular)² + (b/2)²

a² - b²/4 = (Perpendicular)²

Perpendicular = √ ( a² - b²/4 )

Now, apply Area of Triangle

( Base × Height ) / 2

[ ( b × √( a² - b²/4 ) ] / 2

Area using angle between same side and different side

Let a and be different sides of a parallelogram.

Area of Parallelogram = a × b × sin∅

∅ is angle between different sides in Parallelogram.

Area of Parallelogram = 2 × Area of Triangle

a × b × sin∅ = 2 × Area of Triangle

Area of Triangle = ( a × b × sin∅ ) / 2

Note :- Check attachment for reference.

Answer 2

Area of Isosceles Triangle Using Only Sides = ½[√(a2 − b2 /4) × b]

1. b = base of the isosceles triangle.
2. h = height of the isosceles triangle.
3. a = length of the two equal sides

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