Area of Triangles Without Right Angles

triangle b h

Knowing Base and Height

It is easy to find the area of a right-angled triangle, or any triangle where we are given the base and the height.

It is simply half of b times h

Area = 1 bh
2

(The Triangles page tells you more about this).

Example: What is the area of this triangle?

Triangle

Height = h = 12

Base = b = 20

Area = ½ bh = ½ × 20 × 12 = 120

Knowing Three Sides

SSS Triangle

There's also a formula to find the area of any triangle when we know the lengths of all three of its sides.

This can be found on the Heron's Formula page.

Knowing Two Sides and the Included Angle

SSS Triangle

When we know two sides and the included angle (SAS), there is another formula (in fact three equivalent formulas) we can use.

Depending on which sides and angles we know, the formula can be written in three ways:

Either: Area = 1 ab sin C
2
Or: Area = 1 bc sin A
2
Or: Area = 1 ca sin B
2

They are really the same formula, just with the sides and angle changed.

Example: Find the area of this triangle:

First of all we must decide what we know.

We know angle C = 25º, and sides a = 7 and b = 10.

So let's get going:

Start with:   Area = (½)ab sin C
     
Put in the values we know:   Area = ½ × 7 × 10 × sin(25º)
     
Do some calculator work:   Area = 35 × 0.4226...
    Area = 14.8 to one decimal place

How to Remember

Just think "abc": Area = ½ a b sin C

How Does it Work?

Well, we know that we can find an area when we know a base and height:

Area = ½ × base × height

In this triangle:

  • the base is: c
  • the height is: b × sin A

Putting that together gets us:

Area = ½ × (c) × (b × sin A)

Which is (more simply):

Area = 1 bc sin A
2

By changing the labels on the triangle we can also get:

  • Area = ½ ab sin C
  • Area = ½ ca sin B

One more example:

Example: Find How Much Land

Farmer Jones owns a triangular piece of land.

The length of the fence AB is 150 m. The length of the fence BC is 231 m.

The angle between fence AB and fence BC is 123º.

How much land does Farmer Jones own?

 

First of all we must decide which lengths and angles we know:

  • AB = c = 150 m,
  • BC = a = 231 m,
  • and angle B = 123º

So we use:

Area = 1 ca sin B
2

 

Start with:   Area = ½ ca sinB
     
Put in the values we know:   Area = ½ × 150 × 231 × sin(123º) m2
     
Do some calculator work:   Area = 17,325 × 0.838... m2
    Area = 14,530 m2

 

Farmer Jones has 14,530 m2 of land