Base of a Triangle Calculator
Enter any two known measurements to find the base of a triangle instantly. Choose your solving method - area and height, two sides with the included angle, known perimeter with the other two sides, or the Pythagorean theorem for right triangles. The calculator shows step-by-step working so you can follow the math.
What is the base of a triangle?
The base of a triangle is any side you choose to treat as the bottom reference side for a given calculation. Because a triangle has three sides, any of them can serve as the base, as long as you measure the perpendicular height to the opposite vertex from that same side. In practice, the base is often the side you want to find, and the choice of method depends on which other values you already know.
Four methods to find the base
The most common approach uses the area formula rearranged: if you know the area A and the height h, then base = 2A / h. When you know two sides and the angle between them, the law of cosines gives the third side directly: b = sqrt(a² + c² - 2ac·cos(B)). If the total perimeter and the other two sides are known, simple subtraction gives the base: b = P - a - c. For right triangles specifically, the Pythagorean theorem (a² + b² = c²) works in two directions: find a missing leg from the hypotenuse and the other leg, or find the hypotenuse (as the base) from both legs.
Step-by-step example using area and height
Suppose you have a triangle with area 24 m² and a perpendicular height of 6 m. Step 1: write out the area formula: A = (b × h) / 2. Step 2: rearrange for b: b = 2A / h. Step 3: substitute numbers: b = (2 × 24) / 6 = 48 / 6 = 8 m. The base is 8 m. You can verify this by multiplying back: (8 × 6) / 2 = 24 m².
Using the law of cosines (SAS method)
When two sides and the included angle are known, the law of cosines is the right tool. For sides a = 7 m and c = 7 m with included angle B = 60 degrees: b² = 7² + 7² - 2(7)(7)cos(60°) = 49 + 49 - 98 × 0.5 = 98 - 49 = 49, so b = 7 m. This is an equilateral triangle where all three sides are equal, which is a good sanity check. The same formula works for any angle between 0 and 180 degrees.
Which method should I use?
| Method | Known values | Formula |
|---|---|---|
| Area and height | Area (A) and height (h) | b = 2A / h |
| SAS (two sides + angle) | Sides a, c and included angle B | b = sqrt(a² + c² - 2ac·cos B) |
| Perimeter | Total perimeter (P), sides a and c | b = P - a - c |
| Right triangle (hyp + leg) | Hypotenuse and one leg | b = sqrt(hyp² - leg²) |
| Right triangle (two legs) | Both legs | b = sqrt(leg1² + leg2²) |
Choose the method that matches the values you already know.
Frequently asked questions
How do I find the base of a triangle if I only know the area and height?
Rearrange the standard area formula. Because Area = (base x height) / 2, you can solve for base: base = 2 x Area / height. For example, a triangle with area 30 m² and height 5 m has a base of 2 x 30 / 5 = 12 m.
Can any side of a triangle be the base?
Yes. The base is simply the side you select as the reference for a given height. When the base changes, the corresponding perpendicular height also changes, but the area calculated stays the same because area = (base x height) / 2 for each valid base-height pair.
What formula is used for the SAS (two sides and included angle) method?
The law of cosines: b² = a² + c² - 2ac·cos(B), where a and c are the known sides and B is the angle between them. Take the square root of both sides to get b. This formula reduces to the Pythagorean theorem when B = 90 degrees, because cos(90°) = 0.
How do I find the base of a right triangle?
Use the Pythagorean theorem (a² + b² = c²). If you know the hypotenuse c and one leg a, the missing leg (your base) is b = sqrt(c² - a²). If you know both legs and want the hypotenuse as the base, use b = sqrt(a² + leg2²). For example, a right triangle with hypotenuse 10 and one leg 6 has the other leg as sqrt(100 - 36) = sqrt(64) = 8.
What is the difference between the base and a side of a triangle?
Every base is a side, but not every side is currently acting as the base. "Base" is a positional label: it marks the side from which the perpendicular height is measured. Once you choose a side as the base, the opposite vertex determines the height, and together they define the area. Switching which side you call the base changes the height value, but the resulting area calculation stays consistent.
Can I find the base from just the perimeter?
Not with only the perimeter, because infinitely many triangles share the same perimeter. You also need the lengths of the other two sides. Once you have all three pieces of information, the base is simply: base = perimeter - side a - side c.