Area of a Triangle

Proof for Right Triangles

First, consider the particular case of a right triangle. It is always possible to reflect a right triangle across its hypotenuse to form a rectangle.

Note that the area of the rectangle formed is twice the area of the original right triangle. Because of this, the formula for the area of the rectangle,

A_{r} = ℓ w,

can be used to find the area of the right triangle.

A_{r} = 2 A_{t} \Rightarrow ℓ w = 2 A_{t}

Furthermore, the height and base of the right triangle have the same measures as the width and length of the rectangle formed by reflecting the triangle. Based on this observation,

b

and

h

can be substituted for

ℓ

and

w,

respectively, to solve for the area of the original right triangle in terms of its base and height.

ℓ w = 2 A_{t}

SubstituteII

$ℓ = b$ , $w = h$

b h = 2 A_{t}

▼

Solve for

A_{t}

Div

Divide by $2$

\frac{b h}{2} = A_{t}

MoveNumRight

$\frac{a}{b} = \frac{1}{b} \cdot a$

\frac{1}{2} b h = A_{t}

RearrangeEqn

Rearrange equation

A_{t} = \frac{1}{2} b h

This shows that the area of a right triangle can be calculated by using the formula

A = \frac{1}{2} b h .

Proof for Non-Right Triangles

To generalize the previous result, it is useful to note that any non-right triangle can be split into two right triangles by drawing one of its heights.

Note that the area of the non-right triangle

A

is equal to the sum of the individual areas of the smaller right triangles

A_{1}

and

A_{2} .

Therefore, it is possible to calculate the area of the non-right triangle by using the previous result for the areas of the smaller right triangles.

A = \frac{b _{1} h}{2} + \frac{b _{2} h}{2}

▼

Simplify right-hand side

AddFrac

Add fractions

A = \frac{b _{1} h + b _{2} h}{2}

FactorOut

Factor out $h$

A = \frac{( b _{1} + b _{2} ) h}{2}

MoveNumRight

$\frac{a}{b} = \frac{1}{b} \cdot a$

A = \frac{1}{2} (b_{1} + b_{2}) h

Substitute

$b_{1} + b_{2} = b$

A = \frac{1}{2} b h

It has been found that the area of the non-right triangle is half the product of its base

b

and its height

h .

This is the same result as the area for a right triangle. Therefore, the area of any triangle is half the product of its base

b

and its height

h .

$A = \frac{1}{2} b h$

Proof

Proof for Right Triangles

Proof for Non-Right Triangles

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