The Area Method

In one sentence

The area of a figure is a number, and it does not change however you compute it. The area method relies on this to do two things:

Compute it twice. Take the same figure, compute its area again with a different base and height, and set the two results equal. That gives an equation between lengths.
Turn ratios of lengths into ratios of areas. For two triangles with the same height, the ratio of the areas equals the ratio of the bases. Ratios of segments are hard to handle, but areas can be added, subtracted and computed in different ways.

\frac{1}{2} \times base_{1} \times height_{1} = S = \frac{1}{2} \times base_{2} \times height_{2}

Predict first

A right triangle has legs $3$ and $4$ and hypotenuse $5$ . How long is the altitude to the hypotenuse?

Choose an answer first; then you can see the result.

Stop 1 · Compute it twice: the altitude to the hypotenuse (Vol. 4 · Right triangles)

A right triangle has legs $8$ and $6$ . How long is the altitude to the hypotenuse?

Example: legs 8 and 6

Step 1 / 4

First way
The two legs are perpendicular: take one as the base, and the other is the height.

$S = \frac{1}{2} \times 8 \times 6 = 24$
Second way
Set them equal
The general fact

Stop 2 · Compute it twice: the sum of three distances (Vol. 4 · Triangle area)

$P$ is any point inside an equilateral triangle of side $6$ . What is the sum of the distances from $P$ to the three sides?

Predict first

Drag $P$ from the center of the triangle to somewhere near a vertex. What happens to the sum of the three distances?

It gets biggerIt gets smallerIt stays the same

Choose an answer first; then you can see the result.

Example: distances from a point inside an equilateral triangle

Step 1 / 4

Three perpendiculars
Drop perpendiculars from $P$ to the three sides and call their lengths $d_{1}$ , $d_{2}$ , $d_{3}$ . They point in three different directions, so you cannot add them directly.

$d_{1} + d_{2} + d_{3} = ?$
Cut it into three
Compute the whole area again
Cancel the common base

“Equilateral” is used in only one place: the three bases are equal, so they can be cancelled.

Stop 3 · Turn a ratio of segments into a ratio of areas (Vol. 4 · Proportional segments)

In $△ A B C$ , $A B = 6$ , $A C = 4$ and $B C = 8$ . $A D$ bisects $∠ A$ and meets $B C$ at $D$ . Find $B D$ .

Example: how an angle bisector divides the opposite side

Step 1 / 4

First, any point D
The bases $B D$ and $D C$ of $△ A B D$ and $△ A C D$ lie on the same line, and the two triangles share one height. So the ratio of their areas equals the ratio of the bases. This is true for every point $D$ on $B C$ .

$\frac{S _{△ A B D}}{S _{△ A C D}} = \frac{B D}{D C}$
Put D on the angle bisector
New base and height, compute again
The two results are equal

In the figure above, tap “△ABP : △ACP” and drag the point $P$ on $A D$ : this ratio also always equals $B D : D C$ . The two triangles share only the side $A P$ , yet the ratio of their areas is decided by the point $D$ on the opposite side. This is the common-side theorem, the most used tool of the area method.

Stop 4 · See a product as an area (Vol. 1 · Using the formulas)

Why does $a^{2} - b^{2}$ equal $(a + b) (a - b)$ ? See every term as a piece of area.

Example: 5² − 2² = 7 × 3

Step 1 / 4

A big square minus a small one
From a square of side $a$ , cut a small square of side $b$ out of the top-right corner. The area left is $a^{2} - b^{2}$ .

$a^{2} - b^{2} = 25 - 4 = 21$
Make one cut
Move it and fit it on
The area has not changed

Practice

Exercise 1Basic

A right triangle has legs $5$ and $12$ . How long is the altitude to the hypotenuse? (A fraction is fine.)

Exercise 2Advanced

In $△ A B C$ , $A B = 9$ , $A C = 6$ and $B C = 10$ . $A D$ bisects $∠ B A C$ and meets $B C$ at $D$ . Find $B D$ .

Exercise 3Challenge

In $△ A B C$ , $D$ is on $B C$ with $B D : D C = 2 : 3$ . $P$ is a point on the segment $A D$ , and $△ A B P$ has area $8$ . What is the area of $△ A C P$ ?

In one sentence

Stop 1 · Compute it twice: the altitude to the hypotenuse (Vol. 4 · Right triangles)

Example: legs 8 and 6

Stop 2 · Compute it twice: the sum of three distances (Vol. 4 · Triangle area)

Example: distances from a point inside an equilateral triangle

Stop 3 · Turn a ratio of segments into a ratio of areas (Vol. 4 · Proportional segments)

Example: how an angle bisector divides the opposite side

Stop 4 · See a product as an area (Vol. 1 · Using the formulas)

Example: 5² − 2² = 7 × 3

Practice

Mistake Museum · Exhibits from this stop

Base and height that do not belong together

Writing a ratio of areas as a ratio of sides without thinking

Stops on this line