A triangular sail with a 12-foot foot and an 8-foot hoist catches 48 square feet of wind, because any triangle's area is half its base times its height: A = 1/2 x b x h. Two copies of the triangle always assemble into a parallelogram, so the triangle owns exactly half that area. The one rule that cannot be bent is that the height must be perpendicular to the base. For the sail, the height is the vertical hoist, not the slanted leech edge, and measuring along the slant is the mistake that inflates nearly every first attempt. Gardeners laying out triangular flower beds, roofers sheathing gable ends, and contractors ordering sod for corner lots all use the identical arithmetic, and they all check the right angle first.

Start with the sail and the garden, the two anchors this calculator is built around. The sail, base 12 feet and height 8 feet, gives 1/2 x 12 x 8 = 48 square feet of canvas, before seams and waste are added. A triangular garden bed 10 feet by 6 feet gives 1/2 x 10 x 6 = 30 square feet of planting area, the number the mulch bags are counted from: at 2 square feet of coverage per bag, that bed needs 15 bags for a single layer. The calculator's Hypotenuse estimate output earns its keep on the garden job, because the bed's third side needs edging: the square root of (10 squared + 6 squared) is the square root of 136, about 11.66 feet of edging for the diagonal side, assuming the bed is a right triangle. If the bed is not right-angled, that hypotenuse output is only an estimate, and the real third side must be measured, not computed from base and height.

A gable end shows the same formula at house scale. A triangular gable with a 24-foot base and a 9-foot height covers 1/2 x 24 x 9 = 108 square feet of sheathing. Plywood sheets cover 32 square feet each, so 108 / 32 = 3.375 sheets, which means buying 4 sheets and cutting the fourth, because lumberyards do not sell three-eighths of a sheet and the offcut covers the inevitable miscut. Add 10 percent waste for cuts and the order becomes 108 x 1.10 = 118.8 square feet, still 4 sheets, but with a much smaller margin for error. Contractors who skip the waste factor on gables always seem to make a second trip, and the second trip costs more than the fourth sheet did.

Any side of a triangle can serve as the base, as long as the height is measured perpendicular to that side, and proving it once removes a lot of doubt. Take the classic 3-4-5 right triangle. With base 4 and height 3, the area is 1/2 x 4 x 3 = 6 square units. Flip it and use base 3 with height 4, and the area is 1/2 x 3 x 4 = 6 again. Use the hypotenuse 5 as the base, and the height to the right-angle corner is 2.4, because the area must still be 6, so height = 2 x 6 / 5 = 2.4. All three choices agree, as they must, because the triangle itself has not changed, only the viewpoint. The practical lesson is to pick whichever side is easiest to measure on the ground and find its perpendicular height, rather than forcing the triangle into one orientation.

When you know all three sides but no height, Heron's formula takes over, and it needs no angles at all. First find the semi-perimeter, s = (a + b + c) / 2, then the area is the square root of s x (s - a) x (s - b) x (s - c). For sides 5, 6, and 7, the semi-perimeter is (5 + 6 + 7) / 2 = 9, and the area is the square root of (9 x 4 x 3 x 2), the square root of 216, about 14.70 square units. Check it against the base-height method on the 3-4-5 triangle: s = (3 + 4 + 5) / 2 = 6, and the area is the square root of (6 x 3 x 2 x 1), the square root of 36, exactly 6, matching the base-height answer. Two different routes to the same number is the best evidence your measurements are right, and surveyors use exactly this cross-check on irregular plots.

A land-survey example shows Heron's formula doing work no tape measure can do alone. A triangular plot with sides of 20, 30, and 40 meters has a semi-perimeter of (20 + 30 + 40) / 2 = 45 meters, and an area of the square root of (45 x 25 x 15 x 5), the square root of 84,375, about 290.47 square meters. Before trusting any three-side measurement, run the triangle inequality test: the two shorter sides must add up to more than the longest side. Here 20 + 30 = 50, which beats 40, so the triangle is real. A claimed triangle with sides 3, 4, and 9 fails, because 3 + 4 = 7 is less than 9, and no such triangle exists; Heron's formula would try to take the square root of a negative number, which is the math telling you the measurements are wrong. Always run the inequality test before the area formula.

A third route covers the case where you know two sides and the angle between them: area = 1/2 x a x b x sin(C), where C is the included angle. For sides of 10 and 14 units with a 35-degree angle between them, the area is 1/2 x 10 x 14 x sin(35 degrees), which is 70 x 0.5736, about 40.15 square units. This route is the workhorse of navigation and land work, where angles are often easier to measure than perpendicular heights, and it also explains why the base-height formula works: the height is simply the second side times the sine of the included angle. One warning travels with this route: the calculator must be in degree mode, not radian mode, or sin(35) will come out as a different number entirely and the area will be nonsense.

Special triangles have shortcuts worth memorizing. An equilateral triangle with all sides equal to 10 has a height of 5 times the square root of 3, about 8.66, and an area of 1/2 x 10 x 8.66 = 43.30 square units, which also equals the square root of 3 divided by 4 times the side squared. An isosceles triangle with two equal sides lets you split it down the middle into two mirror-image right triangles, and the height is then a straightforward Pythagorean calculation. A right triangle is the simplest of all: its two legs are already a base and a perpendicular height, so the area is half their product with no extra work. Recognizing which special case you hold saves a Heron computation, and on a job site, saved computations are saved mistakes.

A small ladder of base-height pairs builds number sense fast, and it is worth running through once. Base 5 with height 4 gives 10 square units. Base 9 with height 7 gives 31.5. Base 14 with height 11 gives 77. Base 18 with height 13 gives 117. Notice the pattern that matters for estimating: doubling both the base and the height quadruples the area, because the formula multiplies two lengths, so a bed twice as wide and twice as deep needs four times the mulch, not twice. Halving one dimension while doubling the other leaves the area unchanged, which is why a long skinny triangle and a short wide one can hold the same garden. Area scales with the square of length, and every material order that ignores that scaling comes up short.

Units deserve one careful paragraph, because a triangle measured in mixed units has no meaningful area. If the base is in feet and the height in inches, convert first: a base of 10 feet with a height of 72 inches is a height of 6 feet, giving 30 square feet, not the 360 that the unconverted numbers suggest. To convert square feet to acres, divide by 43,560: a triangular lot with base 200 feet and height 150 feet covers 1/2 x 200 x 150 = 15,000 square feet, which is 15,000 / 43,560, about 0.34 acre. To convert square meters to square feet, multiply by 10.7639, so the 290.47-square-meter survey plot from earlier is about 3,126 square feet. Always convert the lengths before multiplying, not the area after, because a unit error inside the multiplication does not scale back out cleanly.

Five mistakes account for most wrong triangle areas. First, using a slanted side as the height instead of the perpendicular distance. Second, forgetting the one-half, which doubles every answer and every material order. Third, mixing units between base and height. Fourth, running Heron's formula without the semi-perimeter step, or forgetting to subtract each side from it. Fifth, trusting a three-side measurement that fails the triangle inequality, which produces an impossible answer instead of a warning. One habit catches all five: after computing an area, ask whether it fits inside the rectangle with the same base and height. A triangle area larger than base times height is impossible, and a triangle area equal to it means the one-half was forgotten.

To use this calculator, type the base in the Base box and the perpendicular height in the Height box, then read the Area output, and use the Hypotenuse estimate output when the layout needs the third side. Sanity-check against the anchors on this page: 12 by 8 should give 48, 10 by 6 should give 30 with a hypotenuse estimate of about 11.66, and a 24 by 9 gable should give 108. If your answer is exactly double what you expected, the one-half went missing. And if you know three sides instead of a height, run Heron's formula by hand with the steps above, or check the companion calculators on this site for right triangles, trapezoids, and rhombus layouts, which share the same base-height logic with different shapes.

With the pre-filled defaults — Base (b) 6, Height (h) 4 — the Triangle Area Calculator returns Area 12.0000, Hypotenuse estimate 7.2111. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.