Surface area of Triangles

At that place 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 information technology is easy.

Information technology is simply half of b times h

Expanse = 1 2 bh

(The Triangles page explains more)

The virtually important thing is that the base of operations and top are at right angles. Have a play here:

../geometry/images/triangle.js?way=area

Example: What is the area of this triangle?

Triangle
(Note: 12 is the height, non the length of the left-manus side)

Height = h = 12

Base = b = xx

Area = ½ bh = ½ × xx × 12 = 120

627,723, 3132, 3133

Knowing Three Sides

SSS Triangle

There's also a formula to notice the area of whatever triangle when we know the lengths of all iii of its sides.

This tin can be constitute on the Heron's Formula folio.

Knowing Two Sides and the Included Angle

SSS Triangle

When nosotros know two sides and the included angle (SAS), there is some other formula (in fact three equivalent formulas) we can apply.

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

Area = 1 2 ab sin C

Area = 1 2 bc sin A

Area = one two ca sin B

They are actually the aforementioned formula, just with the sides and angle changed.

Example: Observe the surface area of this triangle:

trig area example

First of all we must decide what we know.

We know angle C = 25º, and sides a = vii 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 figurer work: 35 × 0.4226...

Surface area = 14.8 to one decimal place

How to Think

Just call back "abc": Surface area = ½ a b sin C

Information technology is also good to recollect that the angle is always between the two known sides, called the "included bending".

How Does it Work?

We kickoff with this formula:

Expanse = ½ × base × height

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

trig triangle b sinA
the peak is b × sin A

So nosotros go:

Expanse = ½ × (c) × (b × sin A)

Which can be simplified to:

Area = 1 2 bc sin A

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

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

1 more example:

Example: Find How Much Land

trig area example

Farmer Rigby owns a triangular piece of country.

The length of the contend AB is 150 g. 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 make up one's mind which lengths and angles we know:

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

So we employ:

Area = one 2 ca sin B

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

Do some computer work: 17,325 × 0.838... thou2

 Area = 14,530 m2

Farmer Rigby has fourteen,530 one thousandii of land

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