Triangles on the Same Base and Between the Same Parallels

What Is the Theorem for Triangles on the Same Base and Between the Same Parallels?

The theorem states: triangles on the same base and between the same parallels are equal in area.

Two triangles qualify when they share one base and their opposite vertices (apexes) both lie on a single line parallel to that base. Because the two lines are parallel, the perpendicular distance between them, the triangles' common height, is identical. This is Proposition 37 in Book I of Euclid's Elements.

The area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. With both the base and the height shared, that product is the same for every such triangle, so their areas are equal.

How Do You Prove Triangles on the Same Base Are Equal in Area?

There are two clean routes. Both are worth seeing.

Route 1 - the direct altitude argument.

Let triangles $ABC$ and $ABD$ share base $AB$, with $C$ and $D$ on a line parallel to $AB$. Drop perpendiculars from $C$ and from $D$ to line $AB$; call both heights $h$. They are equal because the distance between two parallel lines is constant.

$$\text{ar}(ABC) = \frac{1}{2} \times AB \times h$$

$$\text{ar}(ABD) = \frac{1}{2} \times AB \times h$$

The right-hand sides are identical, so:

$$\text{ar}(ABC) = \text{ar}(ABD)$$

Route 2 - half of a parallelogram.

Complete each triangle into a parallelogram on the same base $AB$ and between the same parallels. A diagonal splits a parallelogram into two congruent triangles, so each triangle is exactly half its parallelogram's area. By the companion result, parallelograms on the same base and between the same parallels are equal in area. Halves of equal parallelograms are equal, so the two triangles are equal in area.

You can read Euclid's own version in Elements Book I, Proposition 37. Both routes belong to the wider umbrella theorem, same base and between the same parallels.

What Is the Half-a-Parallelogram Relationship?

A key link falls straight out of Route 2: a triangle and a parallelogram on the same base and between the same parallels satisfy $\text{ar(triangle)} = \frac{1}{2},\text{ar(parallelogram)}$.

This is the same $\frac{1}{2}$ that separates the triangle area formula from the parallelogram area formula, now given a geometric meaning: the triangle really is one of the two congruent halves a diagonal produces. It is the most useful single fact to carry out of this topic.

Why Does a Median Divide a Triangle into Equal Areas?

Here is the theorem's most quoted consequence. A median of a triangle joins a vertex to the midpoint of the opposite side. In triangle $ABC$, the median $AD$ meets $BC$ at its midpoint $D$, so $BD = DC$.

Triangles $ABD$ and $ACD$ then have equal bases ($BD = DC$) and share the same apex $A$, so they have the same height from $A$.

$$\text{ar}(ABD) = \frac{1}{2} \times BD \times h = \frac{1}{2} \times DC \times h = \text{ar}(ACD)$$

Every median splits a triangle into two equal-area halves. This is why all three medians meet at the centroid, the triangle's balance point, a fact used to find centres of mass. The idea threads into the midpoint theorem as well.

Examples of Triangles on the Same Base and Between the Same Parallels

The set moves from direct area to a median split to a converse.

Example 1

Two triangles share a base of 12 cm and lie between parallels 7 cm apart. Find each area.

$$\text{ar} = \frac{1}{2} \times 12 \times 7 = 42 \text{ cm}^2$$

Final answer: both are 42 cm².

Example 2

A student says a tall, skinny triangle on a base must have more area than a short, wide one on the same base between the same parallels. Check this.

The instinct is that a taller-looking triangle covers more, but "taller-looking" here just means the apex sits off to one side, not that the height is greater. Between the same parallels, the perpendicular height is fixed. Only the apex's horizontal position shifts, and horizontal position does not enter the area formula.

For base 10 cm and height 6 cm, every such triangle has area:

$$\frac{1}{2} \times 10 \times 6 = 30 \text{ cm}^2$$

Final answer: 30 cm² for all of them, skinny or wide alike.

Example 3

A parallelogram on a base has area 88 cm². Find the area of a triangle on the same base and between the same parallels.

The triangle is half the parallelogram.

$$\frac{1}{2} \times 88 = 44 \text{ cm}^2$$

Final answer: 44 cm².

Example 4

In triangle $ABC$, median $AD$ is drawn. If $\text{ar}(ABC) = 36$ cm², find $\text{ar}(ABD)$.

A median splits the triangle into two equal areas.

$$\text{ar}(ABD) = \frac{1}{2} \times 36 = 18 \text{ cm}^2$$

Final answer: 18 cm².

Example 5

Triangle $PQR$ has area 24 cm² on base $PQ$. Triangle $PQS$ on the same base also has area 24 cm². What does this tell you about $R$ and $S$?

Equal area on the same base forces equal height, so $R$ and $S$ are the same perpendicular distance from line $PQ$. That means $R$ and $S$ lie on one line parallel to $PQ$.

Final answer: $RS$ is parallel to $PQ$; the two apexes share a parallel line.

Example 6

In triangle $ABC$, $D$ is the midpoint of $BC$ and $E$ is the midpoint of $AD$. Find $\text{ar}(BED)$ as a fraction of $\text{ar}(ABC)$.

Median $AD$ gives $\text{ar}(ABD) = \frac{1}{2}\text{ar}(ABC)$.

In triangle $ABD$, $BE$ is a median (since $E$ is the midpoint of $AD$), so it halves that area:

$$\text{ar}(BED) = \frac{1}{2} \times \frac{1}{2}\text{ar}(ABC) = \frac{1}{4}\text{ar}(ABC)$$

Final answer: $\text{ar}(BED) = \frac{1}{4},\text{ar}(ABC)$.

Students handle the direct area problems well but often freeze on Example 5, because it runs the theorem backwards, from equal area to a parallel line. Reading it height-first, "equal area on the same base means equal height means same parallel," is what unlocks the converse.

Why Does This Theorem Matter?

"Move the apex along the parallel, keep the area." That single freedom is quietly powerful.

• Medians and the centroid. Because every median halves the area (Example 4), the three medians balance a triangular plate at the centroid, which is how engineers locate a triangle's centre of mass.

• Area-preserving redrawing. A slanted triangle can be replaced by an upright one of equal area on the same base, the trick behind converting awkward regions into computable ones when finding the area of polygons.

• Proving lines parallel. The converse (Example 5) turns an equal-area statement into a parallel-line conclusion, a standard step in coordinate and Olympiad geometry.

The insight that a triangle's area depends only on base and height, never on where the apex slides, is the two-dimensional seed of Cavalieri's principle in three dimensions. Euclid set it out in Elements Book I, Proposition 37, already linked in the proof above.

What Are the Most Common Mistakes With This Theorem?

Mistake 1: Treating a slanted apex as extra height

Where it slips in: When one triangle looks taller because its apex leans far to one side.

Don't do this: Assuming a lopsided triangle has more area than an upright one on the same base.

The correct way: Height is the perpendicular distance to the base, not the slant to the apex. Between the same parallels, that height is fixed, so the areas match.

Mistake 2: Forgetting the factor of ½ against a parallelogram

Where it slips in: When a triangle and a parallelogram share a base and parallels.

Don't do this: Setting the triangle's area equal to the parallelogram's.

The correct way: The triangle is half the parallelogram. The check that fixes this is asking "triangle or parallelogram?" before writing the area, since one carries the $\frac{1}{2}$ and the other does not.

Mistake 3: Assuming equal area means congruent

Where it slips in: When two triangles come out with the same area.

Don't do this: Concluding the triangles are identical.

The correct way: Equal area is weaker than congruence. On the same base between the same parallels, the triangles share area but usually differ in shape.

Conclusion

• Triangles on the same base and between the same parallels are equal in area.

• The reason is a shared base and a shared perpendicular height.

• Each such triangle is half of a parallelogram on the same base between the same parallels.

• Every median splits a triangle into two equal-area halves, which is why the centroid balances it.

• Equal area does not mean congruent; the triangles usually differ in shape.

To take this further with a teacher, explore Bhanzu's geometry tutor or a high school math tutor, with wider support through math classes online.

Where Should You Practice This Theorem Next?

Work through these three.

1. Two triangles share a base of 18 cm between parallels 6 cm apart. Find each area.

2. A parallelogram on a base has area 130 cm². Find the triangle on the same base between the same parallels.

3. Median $CF$ of triangle $ABC$ has area of $\triangle ABC$ equal to 50 cm². Find $\text{ar}(BFC)$.

Write the base and the perpendicular height before computing. To have a live trainer connect the triangle case to the parallelogram case for you, you can book a free demo class with Bhanzu.

Read More

• Properties of a triangle

• Congruence in triangles

• Area of a triangle in coordinate geometry

• The triangle inequality theorem

• Triangle sum theorem