Session 6: Area of Triangle – Formula, Definition, Interactives and Examples

Area of Triangle

For finding the area of a triangle you need to identify the base and the height. The base of a triangle is generally considered the bottom of a shape; however, the shape might not always be oriented such that the base is on the bottom. The height of a triangle is the distance from the base to the opposite vertex at a right angle. The height of a triangle is also known as the altitude.

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Formula for Area of Triangle

You can think of any triangle as half a parallelogram. If you rotate a triangle 180∘about the midpoint of one of its sides, the original triangle and the new triangle will be a parallelogram.

Therefore, the area of a triangle is base times height divided by two.

Remember that any of the three sides can be the base. Also, remember that the height must be perpendicular to the base and extend to the highest point of the triangle.

Area_triangle=\(\frac{bh}{2}\)=\(\frac{1}{2}\)bℎ
Perimeter_triangle=Sum of three sides

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Formula for Area of Right Angled Triangle

Perimeter=a+b+c
Area=\(\frac{1}{2}\)ab

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Formula for Area of Equilateral Triangle

Perimeter=3a
Altitude=
Area=

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Formula for Area of Isosceles Triangle

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Area of Triangle – Examples

Example 1

Find the area of a right-angled triangle with a base of 6 cm and a height of 4 cm.

We know,

Area of right-angled triangle=\(\frac{1}{2}\)× base× height

=\(\frac{1}{2}\)×6×4
=12 cm2

Example 2

Find the area of an equilateral triangle with sides of 8 cm.

We know,

Area of an equilateral triangle with sides 

Area of an equilateral triangle with sides                                                                             

Example 3

Find the area of an isosceles triangle with equal sides of length of 17 cm and unequal sides of length of 30 cm.

We know,

For an isosceles triangle with equal sides of length a and unequal side of length b,

Area      

Example 4

Find the length of the side of an equilateral triangle having an area 576\(\sqrt{3}\) cm².

We know,

For an equilateral triangle with equal sides of length a,

     Area=

Example 5

Find the length of the equal sides of an isosceles triangle whose area is 420 cm² and the unequal side (base) measures 40 cm.

We know,

For an isosceles triangle with equal sides of length a and unequal side of length b,

       Area=

Squaring both sides, we get

Example 6

Find the base of a triangle whose height is 30 cm and area is 750 cm².

We know,

Area of triangle =\(\frac{1}{2}\)× base×height

750=\(\frac{1}{2}\)×base×(30)
base=\(\frac{750×2}{30}\)
base=25×2
base=50 cm

Example 7

Find the area of the △PQR, whose base PR is 7 cm and altitude QS is 6 cm. If the length of the base PQ is 14 cm, then find the length of the altitude RT.

Area of                            

Also, area of the                                  

Example 8

The base of a triangular field is one-third its height. If the cost of cultivating the field at ₹845 per hectare is ₹11407.50, find its base and height.

The total cost of cultivating the field₹=₹11407.50.

Rate of cultivation₹=₹845/ hectare.

∴ Area of the field=\(\frac{cost}{rate}\)=\(\frac{11407.50}{845}\)

 =13.5 hectare
=13.5×10000 (∵1 hectare=10000 m²)
=135000 m²

Let the base of the field be x meters.

Then, its height=3x meters.

∴ Area of the field=\(\frac{1}{2}\)×bℎ                  

∴ Base =x=300 m and height=3x=3(300)=900 m.