Area of Triangles

There are several ways to find the area of a triangle.

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Knowing Base and Height

triangle b h

When we know the base and height it is easy.

It is simply half of b times h

Area = 12bh

(The Triangles page explains more)

The most important thing is that the base and height are at right angles. Have a play here:

Example: What is the area of this triangle?

Triangle
(Note: 12 is the height, not the length of the left-hand side)

Height = h = 12

Base = b = 20

Area = ½ bh = ½ × 20 × 12 = 120

627,723, 3132, 3133

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:

Area = 12ab sin C

Area = 12bc sin A

Area = 12ca sin B

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

Example: Find the area of this triangle:

trig area example

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:

Area =(½)ab sin C

Put in the values we know:½ × 7 × 10 × sin(25º)

Do some calculator work:35 × 0.4226...

Area = 14.8 to one decimal place

How to Remember

Just think "abc": Area = ½ a b sin C

It is also good to remember that the angle is always between the two known sides, called the "included angle".

How Does it Work?

We start with this formula:

Area = ½ × base × height

We know the base is c, and can work out the height:

trig triangle b sinA
the height is b × sin A

So we get:

Area = ½ × (c) × (b × sin A)

Which can be simplified to:

Area = 12bc sin A

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

trig area example

Farmer Rigby 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 Rigby 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 = 12ca sin B

Put in the values we know:½ × 150 × 231 × sin(123º) m²

Do some calculator work:17,325 × 0.838... m²

Area =14,530 m²

Farmer Rigby has 14,530 m² of land

259, 1520, 1521, 1522,260, 1523, 2344, 2345, 3940, 3941