Area of Scalene Triangle

Area of Scalene Triangle Definition:

A Scalene triangle has three random(Unequal) sides/lengths and three random (unequal) angles.

A simple definition of a scalene triangle is “A Scalene triangle is a triangle with three different sides and angles.”

Example: The sail on a sailboat is also likely to be raised in the shape of a scalene triangle, with no side of the sail being the same length as any other side.

Area of Scalene Triangle

As there are different types of triangles, how is the Scalene Triangle different from other triangles? (Read the Difference table below).

Differences

Scalene

Isosceles

Equilateral

Sides

3 sides with random lengths

2 Equal Sides

3 Equal Sides

Angles

3 random angles

One Right Angle(90)

3 Equal Angles

For finding out the area of Scalene Triangle, you need the following measurements.

a) The length of one side and the perpendicular distance of that side to the opposite angle.

The Area of Scalene Triangle with any side as base ‘b’ and height ‘h’ (an altitude on that base) is

A =\(\frac{1}{2}\times b\times h\)

Area of Scalene triangle Example:

Q1) Find the area of the triangle with base 20 cm and height 12 cm?

Solution: Base (b) = 20 cm

Height (h) = 12cm

A = \(\frac{1}{2}\times b\times h\)

= \(\frac{1}{2}\times 20\times 12\)

=120cm2

OR

b) The lengths of all three sides.

Area of Scalene triangle formula:

Step 1: If you know the length of the three sides of the triangle (a,b,c).

Step 2: Find the semi-perimeter, S.

The formula for finding the Semi-perimeter of a triangle is

S = \(\frac{a+b+c}{2}\)

Step 3: Then apply the Heron’s formula to find the area of Scalene triangle

A=\(\sqrt{S(S-a)(S-b)(S-c)}\)sq.units

Area of a scalene triangle with sides:

Q1) Find the area of a triangle whose sides are 122 cm, 120 cm, and 22 cm.

Solution: S = \(\frac{a+b+c}{2}\)

= \(\frac{122+120+22}{2}\)cm = \(\frac{264}{2}\)cm = 132 cm

S = 132 cm

A = \(\sqrt{S(S-a)(S-b)(S-c)}\)

= \(\sqrt{132(132-122)(132-120)(132-22)} cm^{2^{}}\)

=\(\sqrt{132(10)(12)(100)}cm^{2}\)

=\(\sqrt{12\times 11\times 10\times 12\times 11\times 10}cm^{2}\)

=\(\sqrt{12^{^{2}}\times 11^{2}\times 10^{2}}cm^{2}\)

=\(\sqrt{1320^{2}}cm^{2}\)

Area of this triangle is = \(1320cm^{2}\)

OR

c) The value of one of the angles (Suppose ∠C) as well as the lengths of the two sides (a and b) that form it.

Area = \(\frac{ab}{2}\times Sin C\)

Practice Example:

Q1) Find the area of the triangle with two sides as 28 cm and 35 cm and the angle between these sides as 60°?

Solution: Area = \(\frac{ab}{2}\times Sin C\)

Area = \(\frac{28\times 35}{2}\times sin 60^{\circ}\) \(cm^{2}\)

=\(\frac{28\times 35}{2}\times \sqrt{\frac{3}{2}}\)\(cm^{2}\)

=\(490\sqrt{2}\) \(cm^{2}\)

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Practise This Question

If you write each letter of these words on a piece of paper and pick a piece randomly, the probability of picking a vowel is highest for which one?