Base of a Triangle

Q: How do you find the base of a triangle if you know the area and height?

Rearrange the area formula: $b = \frac{2A}{h}$. For example, if the area is 24 cm² and the height is 8 cm, the base is $\frac{2 \times 24}{8} = 6$ cm.

Base of a Triangle

The side of a triangle which is perpendicular to the altitude.

Triangle ABC with vertex C at top, altitude drawn perpendicular from C to base AB, with point A marking the foot.

See also

Area of a triangle

Key Formula

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

Where:

$A$ = Area of the triangle
$b$ = Length of the base (the chosen side)
$h$ = Height (altitude) drawn perpendicular to the base

Worked Example

Problem: A triangle has a base of 10 cm and a corresponding altitude of 6 cm. Find its area.

Step 1: Identify the base and the altitude perpendicular to it.

b = 10 \text{ cm}, \quad h = 6 \text{ cm}

Step 2: Write the area formula for a triangle.

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

Step 3: Substitute the known values into the formula.

A = \frac{1}{2} \times 10 \times 6

Step 4: Calculate the result.

A = \frac{60}{2} = 30 \text{ cm}^2

Answer: The area of the triangle is 30 cm².

Another Example

This example demonstrates that a triangle has three valid bases and shows that switching the base (with the correct altitude) always gives the same area. It also uses a right triangle where the legs double as altitudes.

Problem: A triangle has sides of length 8 cm, 15 cm, and 17 cm. The altitude drawn to the 15 cm side is 8 cm, and the altitude drawn to the 8 cm side is 15 cm. Verify that the area is the same regardless of which side you choose as the base.

Step 1: Note that this is a right triangle (since 8² + 15² = 17²). The two legs are perpendicular to each other, so each leg serves as the altitude to the other.

8^2 + 15^2 = 64 + 225 = 289 = 17^2

Step 2: Choose the 15 cm side as the base. The corresponding altitude is 8 cm.

A = \frac{1}{2} \times 15 \times 8 = 60 \text{ cm}^2

Step 3: Now choose the 8 cm side as the base instead. The corresponding altitude is 15 cm.

A = \frac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2

Step 4: Both choices yield the same area, confirming that any side can serve as the base as long as you use the correct corresponding altitude.

60 = 60 \checkmark

Answer: The area is 60 cm² regardless of which side is chosen as the base.

Frequently Asked Questions

Can any side of a triangle be the base?

Yes. Any of the three sides of a triangle can be chosen as the base. When you pick a different side as the base, the corresponding altitude changes so that it is always the perpendicular distance from that base to the opposite vertex. The calculated area remains the same no matter which side you designate as the base.

What is the difference between the base and the height of a triangle?

The base is one of the sides of the triangle — a segment connecting two vertices. The height (altitude) is the perpendicular distance from the base to the opposite vertex. The base lies along the triangle itself, while the altitude may fall inside or outside the triangle depending on the triangle's shape.

How do you find the base of a triangle if you know the area and height?

Rearrange the area formula:

b = \frac{2A}{h}

. For example, if the area is 24 cm² and the height is 8 cm, the base is

\frac{2 \times 24}{8} = 6

cm.

Base of a Triangle vs. Altitude of a Triangle

	Base of a Triangle	Altitude of a Triangle
Definition	A chosen side of the triangle	The perpendicular segment from the base to the opposite vertex
Orientation	Lies along a side of the triangle	Perpendicular to the base
Location	Always part of the triangle's boundary	May fall inside or outside the triangle (e.g., in obtuse triangles)
Role in area formula	Multiplied by height, then halved: A = ½bh	Multiplied by base, then halved: A = ½bh
How many per triangle	3 possible bases (one per side)	3 altitudes (one per base)

Why It Matters

The base of a triangle is central to computing area, which appears throughout geometry, trigonometry, and real-world applications like land surveying and architecture. You will use the base-height relationship every time you calculate the area of a triangle, whether in a basic geometry course or in more advanced work involving coordinate geometry and vectors. Understanding that any side can serve as the base also helps when solving problems where only certain measurements are given.

Common Mistakes

Mistake: Using a side length that is not perpendicular to the chosen altitude.

Correction: The base and altitude must form a right angle. If you pick a side as the base, the altitude must be the perpendicular distance from that side (or its extension) to the opposite vertex. Always check that the base-altitude pair corresponds correctly.

Mistake: Thinking the base must be the bottom or horizontal side of the triangle.

Correction: Any side can be the base, regardless of the triangle's orientation. A triangle drawn with a vertical side can use that vertical side as its base. Choose whichever side makes the calculation easiest.

Related Terms

Altitude of a Triangle — The height drawn perpendicular to the base
Area of a Triangle — Calculated using base times height divided by two
Triangle — The polygon whose side is the base
Side of a Polygon — General term for a segment forming a polygon's boundary
Perpendicular — Describes the right-angle relationship between base and altitude
Vertex — The opposite vertex determines where the altitude meets the base