Working out the Areas of Triangles

 

It is well known that to work out the area of a rectangle you multiply the length by the width.

Almost as well known is the formula for the area of a triangle.

That is, the area of a triangle is half of the area that the rectangle that the triangle fits into.

But this formula requires the length of the base and the height of the triangle. There is another formula, which is used when the lengths of the triangle's three sides are known.

This formula was discovered by the Greek Mathematician, Hero (or Heron), round about 75 AD. But like many mathematical formulae, it was discovered and lost both before and since the time of the person whose name it now carries.

The lengths of the three sides of the triangle are labelled a,b and c.

Add these together to get the perimeter and divide by 2 to get the semi-perimeter

Now put a,b,c and s into Heron’s formula to work out the area of the triangle

An example

Now try it out!

  1. Use the formula and Heron’s formula to work out the area of this triangle and check that they both give the same answer.
  • Use Heron’s formula to work out the
  • area of this regular hexagon.
    •

    The third formula for working out the area of a triangle uses the lengths of two sides and the angle between them.

    I have used T for the area of the Triangle, in order not to get it mixed up with angle A on the diagram.
    So for this triangle:-

    Of course, the way that the sides and angles are labelled will not affect the formula, as long as two sides and the angle between them is used.

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