Area of Triangles

In that location are several ways to find the area of a triangle.

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Knowing Base and Tiptop

triangle b h

When nosotros know the base and height it is like shooting fish in a barrel.

It is simply half of b times h

Surface area = 1 2 bh

(The Triangles folio explains more)

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

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Example: What is the surface area of this triangle?

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

Height = h = 12

Base = b = 20

Area = ½ bh = ½ × twenty × 12 = 120

627,723, 3132, 3133

Knowing Three Sides

SSS Triangle

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

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

Knowing 2 Sides and the Included Angle

SSS Triangle

When nosotros know ii 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 means:

Expanse = i two ab sin C

Area = 1 2 bc sin A

Expanse = i 2 ca 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 nosotros must decide what we know.

Nosotros know angle C = 25º, and sides a = vii and b = 10.

So let'south get going:

Expanse = (½)ab sin C

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

Practice some calculator work: 35 × 0.4226...

Surface area = 14.8 to i decimal identify

How to Retrieve

Only think "abc": Area = ½ a b sin C

It is besides skilful to remember that the angle is always between the 2 known sides, called the "included angle".

How Does it Work?

We kickoff with this formula:

Surface area = ½ × base × pinnacle

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

trig triangle b sinA
the tiptop is b × sin A

So nosotros become:

Area = ½ × (c) × (b × sin A)

Which tin can be simplified to:

Surface area = 1 ii bc sin A

Past changing the labels on the triangle nosotros can as well get:

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

Ane more example:

Example: Find How Much Land

trig area example

Farmer Rigby owns a triangular slice of state.

The length of the fence AB is 150 thou. The length of the fence BC is 231 grand.

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

How much country does Farmer Rigby own?

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

  • AB = c = 150 m,
  • BC = a = 231 1000,
  • and bending B = 123º

So we use:

Area = 1 2 ca sin B

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

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

 Area = fourteen,530 mtwo

Farmer Rigby has xiv,530 g2 of country

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