Triangle Area Calculator
Calculates the area of any triangle from its base and perpendicular height using A = ½·base·height. Perfect for geometry homework, land surveying, construction layout, and architectural design when the perpendicular height is known.
Last updated: May 2026
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About this calculator
The area of a triangle is exactly half the area of a rectangle (or parallelogram) with the same base and height. This gives the formula A = ½ × base × height, where the height must be the perpendicular (vertical) distance from the chosen base to the opposite vertex — not the length of a slanted side. Variables: base (any side of the triangle you choose as the reference line, typically the bottom side), height (the perpendicular distance from that base to the opposite vertex). The formula works for all triangle types — scalene, isosceles, equilateral, right-angled, acute, and obtuse — as long as you use the true perpendicular height. The factor of ½ comes from a geometric proof: any triangle can be doubled to form a parallelogram with area base × height, so the triangle is half of that. Edge cases: for obtuse triangles, the perpendicular height may fall outside the triangle when extended from the base — the formula still works, but you must extend the base line conceptually. If only the three side lengths are known, use Heron's formula A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. If two sides and the included angle are known, use A = ½ × a × b × sin(C). For coordinates of three vertices, use the shoelace formula. All produce identical results — choose whichever method matches your known inputs.
How to use
Example 1: A triangular plot of land with base 14 m and perpendicular height 9 m. Step 1: multiply base × height = 14 × 9 = 126. Step 2: divide by 2 = 63 m². Verify: a rectangle of the same base and height would have 126 m² area; the triangle is half of that, confirming 63 m². If seeding at 50 g/m², you need 63 × 50 = 3,150 g = 3.15 kg of seed. Example 2: A roof gable triangle with base 8 ft and ridge height 5 ft above the base. Step 1: 8 × 5 = 40. Step 2: 40 / 2 = 20 sq ft. Verify: this is the side-elevation area of the gable end of the roof. If sheathing costs $4/sq ft, the gable triangle costs 20 × $4 = $80 to sheath.
Frequently asked questions
How do I find the height of a triangle if it is not given directly?
If the perpendicular height is not stated, you can calculate it from other measurements. For a right triangle, the height is simply one of the two legs (the leg perpendicular to the chosen base). For other triangles, you can drop a perpendicular from a vertex to the opposite base and use trigonometry: height = side × sin(angle), where 'side' is the slanted side and 'angle' is the angle that side makes with the base. If all three sides are known, use Heron's formula A = √(s(s−a)(s−b)(s−c)) where s is the semi-perimeter — this skips the height entirely. For coordinate geometry, the shoelace formula gives area directly from three (x, y) vertices.
Why does the triangle area formula use one half times base times height?
A triangle can always be paired with an identical copy of itself (rotated 180°) to form a parallelogram or rectangle. The parallelogram has area = base × height, so each triangle is exactly half of that, giving A = ½ × base × height. This geometric proof holds for all triangle types: acute, right, and obtuse. The key insight is that the height must be perpendicular to the chosen base, not a slanted side length. The same formula also follows from integral calculus by integrating the width of the triangle from base to apex, which is a linear function whose definite integral is the average width times the height — half the base times the height.
What is the difference between the base and the height in a triangle area calculation?
The base is any side of the triangle that you choose as the reference line — usually but not necessarily the bottom side. The height (also called altitude) is the perpendicular distance from that base to the opposite vertex, measured at a strict 90° angle. For an acute triangle, the height falls inside the triangle; for an obtuse triangle, the height may fall outside the triangle and require extending the base line conceptually. For a right triangle, if you choose one of the two legs as the base, the height is simply the other leg. Mixing up slanted side lengths with the true perpendicular height is the single most common mistake in triangle area calculations.
What are common mistakes when calculating triangle area?
Using a slanted side length as 'height' instead of the perpendicular distance gives an overestimate, often by 10–30% depending on the angle. Forgetting the ½ factor doubles the area — a frequent homework error. Mixing units between base and height (cm and m) produces nonsense without conversion. For obtuse triangles, choosing a base where the perpendicular foot lies outside the triangle requires conceptually extending the base — many students get confused at this step. Confusing the semi-perimeter formula (Heron's) with the base-height formula and mixing variables is another pitfall. For triangles defined by coordinates, the shoelace formula's sign convention (counterclockwise positive) trips up beginners — always take the absolute value.
When should I NOT use this base-times-height formula?
When you don't know the perpendicular height directly, use a more appropriate formula: Heron's formula for three sides (SSS), the SAS formula A = ½·a·b·sin(C) for two sides and included angle, or the shoelace formula for vertex coordinates. For curved-edge 'triangles' like spherical triangles on a sphere's surface, use spherical excess formulas (area = R² × (A + B + C − π) where A, B, C are angles in radians). Triangles in non-Euclidean (hyperbolic) geometry follow yet different rules. For 3D triangles in space, use the cross-product formula A = ½ × |AB × AC| where AB and AC are edge vectors. For very thin or degenerate triangles (where three points are nearly collinear), floating-point precision can produce noisy results — verify with a sanity check against a known method.