Calculate the base of a triangle using multiple methods: area and height, three sides with Heron's formula, perimeter minus known sides, or SAS with the law of cosines.
| Method | Formula | Notes |
|---|---|---|
| Area & Height | b = 2A / h | Direct from A = ½bh |
| Three Sides (Heron's) | A = √[s(s−a)(s−b)(s−c)] | Then h = 2A/b, base = chosen side |
| Perimeter & 2 Sides | b = P − a − c | Subtract known sides from perimeter |
| SAS (Two Sides + Angle) | b² = a² + c² − 2ac·cos(B) | Law of Cosines for the base |
| Isosceles | b = 2√(a² − h²) | When two equal sides are known plus height |
| Right Triangle | b = √(c² − a²) | Pythagorean theorem |
The base of a triangle is the side upon which the perpendicular height is drawn. Although any side can serve as the base, it is typically the bottom side in diagrams and the one used in the classic area formula A = ½bh. Finding the base when you know other measurements is one of the most common tasks in geometry homework, construction, and engineering.
If you already know the area (A) and the height (h) to a particular side, the base is simply b = 2A/h — a direct rearrangement of the area formula. When three sides are given, Heron's formula yields the area, and from there the height and base relationship follow. If the perimeter and two sides are known, the third side (the base) is just P − a − c. In the SAS (side-angle-side) scenario, the law of cosines gives b² = a² + c² − 2ac·cos B, letting you solve for the base directly.
This calculator supports all four methods through a simple dropdown. Enter your knowns, and it computes the base plus supporting measurements: area, height, perimeter, all side lengths, and interior angles where applicable. Preset buttons let you load famous triangles (equilateral, 3-4-5, 5-12-13) to explore the results quickly. A reference table summarises every formula, and visual comparison bars show how the base relates to the other sides.
Use this page when the base is the missing side but the givens come in different forms. It keeps the direct area-height method, Heron-based route, perimeter relation, and SAS law-of-cosines method in one workflow so you can match the formula to the data you actually have.
From Area & Height: b = 2A / h
From Heron's: s = (a+b+c)/2, A = √[s(s−a)(s−b)(s−c)], h = 2A/b
From Perimeter: b = P − a − c
From SAS: b² = a² + c² − 2ac·cos(B)Result: Base = 6
For an equilateral triangle with sides 6, 6, and 6, the chosen base is 6. The surrounding outputs then show the matching area, perimeter, equal angles, and equal side lengths.
Calculate the base of a triangle using multiple methods: area and height, three sides with Heron Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
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Rearrange A = ½bh to get b = 2A/h. If the area is 30 and the height is 10, then the base is 2(30)/10 = 6.
Yes. The base is simply the side you choose to measure the perpendicular height from. Different choices give different heights but the same area.
Heron's formula calculates area from three sides: A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. With area known, h = 2A/base gives the height to any chosen base.
SAS stands for Side-Angle-Side. If you know two sides and the included angle, the law of cosines gives the third side (the base): b² = a² + c² − 2ac·cos B.
If a + b ≤ c (or any permutation), the three lengths cannot form a triangle. The calculator will show no result in that case.
If you know the area, h = 2A/b. Without the area, you need more information — either the other sides or an angle.
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