Triangle Area Calculator - Base, Height, or Heron's

The triangle area calculator finds the area of any triangle from the measurements you already have. It accepts base and perpendicular height, three side lengths through Heron's formula, or two sides and the included angle, then returns the area. Only the three-sides method also reports the perimeter and semi-perimeter.

• • Geometry homework: Pick the method that matches the data, then copy the area into the next step. Use Three Sides for the perimeter too.
• • Roof and floor layouts: Translate a measured base and rise, or three corner-to-corner distances, into a usable area for sheathing, tile, or flooring.
• • Land and survey plots: Convert three boundary distances into the plot area when a base and height are not available.
• • Sailing and navigation: Use two side distances and a bearing or angle to estimate the swept area of a triangular region.

A triangle is the simplest polygon, but the available measurements change between problems. The calculator switches between the three common input sets so the right formula is one click away.

The three-sides method is the only mode whose primary inputs are the side lengths, so it is the only one that returns perimeter and semi-perimeter. The base-height and SAS methods focus on area. The third side of a SAS triangle is determined by the law of cosines, but this calculator is scoped to what the user entered.

Whatever the method, the result is an area in square units. Keep all length inputs in the same unit, then read the area in matching square units.

When the same job involves a rectangle, circle, or another standard shape, the Area Calculator keeps the formulas for those shapes in one place.

The calculator uses the input method you choose and applies the matching triangle area formula. The perimeter and semi-perimeter come back as numbers only for the three-sides method, where the side lengths are the inputs. The SAS method fixes the area from two sides and the included angle; the third side is determined by the law of cosines, but this calculator reports area only for that mode.

• base and height: one side of the triangle paired with the perpendicular distance to the opposite vertex
• sides a, b, c and semi-perimeter s: three side lengths; s is half the perimeter and feeds Heron's formula
• sides a, b and included angle C: two sides and the angle between them, used in the SAS method

The base-height formula is the simplest. The height must be perpendicular to the base; a slanted side is not the height. Switch to Heron's or SAS when the height is unknown.

Heron's formula builds the area from the three side lengths using the semi-perimeter s. Each factor inside the square root is s minus one side, so the result is non-negative when the sides satisfy the triangle inequality.

The SAS method uses the sine of the included angle, which is the angle between the two given sides. The angle is entered in degrees, then converted to radians. The third side is determined by the law of cosines, but the calculator reports only the area here.

According to Wolfram MathWorld, the area of a triangle is one-half the base times the height, and when only three sides are known the area is given by Heron's formula sqrt(s * (s - a) * (s - b) * (s - c)).

For triangle properties beyond area, such as the angles or the third side from two sides and an angle, the Triangle Calculator extends the same inputs to the full triangle solution.

These four ideas decide which input set matches the measurements and how to read the result.

The base-height method is exact only when the height is perpendicular to the base. If you know a slanted side and an angle, switch to SAS instead of forcing a non-perpendicular height into the base-height formula.

A right triangle is a special case. Either leg can be the base and the other is the height, so base-height gives the area as half the product of the two legs.

Right triangles have a built-in perpendicular relationship, so the Right Triangle Calculator is a useful neighbor when the triangle is clearly a right triangle.

Pick the input method that matches the data on hand, then read the result with the matching square unit.

1. 1 Choose an input method: Select Base and Height, Three Sides (Heron's formula), or Two Sides and Included Angle from the dropdown. Three Sides is the only one that returns a numeric perimeter.
2. 2 Enter the matching measurements: Use the same length unit for every input, and measure the included angle in degrees for the SAS method.
3. 3 Read the area: The primary result is the area in square units. Use this number for material, paint, or coverage estimates.
4. 4 Read the perimeter and semi-perimeter when shown: A dash means the chosen method does not take the side lengths as inputs. Switch to Three Sides to get both as numbers.
5. 5 Fix invalid input errors: If the result is an error, check for a non-perpendicular height, a triangle-inequality violation, or an included angle outside 0 to 180 degrees.

A roof gable has a base of 24 feet, a rafter length of 16 feet, and a peak angle of 50 degrees. Switch to the SAS method, enter side a = 24, side b = 16, and angle C = 50. The calculator returns 0.5 * 24 * 16 * sin(50) = 147.08 square feet.

If the shape is a polygon that needs to be split into triangles before adding the parts, the Polygon Area Calculator is the right place to start.

Switching input methods in one triangle area calculator keeps the math straight.

• • Three methods in one place: Base-height, Heron's, and SAS sit side by side so the right formula is one click away.
• • Honest support values: The perimeter and semi-perimeter are reported only for the three-sides method, where they keep the Heron's formula steps auditable.
• • Input validation: Negative lengths, triangle-inequality violations, and out-of-range angles are caught and explained instead of a misleading zero.
• • Unit-agnostic output: The same calculator handles inches, feet, centimeters, and meters because the square-unit label follows the input unit.
• • Real-time feedback: Every input change updates the result, so it is easy to try a different measurement.

The base-height and SAS methods focus on area, so perimeter and semi-perimeter stay as a dash. The three-sides method is the only one that returns perimeter and semi-perimeter as numbers.

Picking the right method also matters. A perpendicular height makes base-height the fastest. Three measured side lengths with no clear right angle make Heron's the safer pick.

For rectangular pieces that sit alongside the triangle in a layout, the Length Width Area Rectangle Calculator covers the same workflow in a single pass.

The math is stable, but the input method and the measurement precision decide how trustworthy the result is.

• • The base-height method assumes the height is perpendicular to the base. Slanted heights overstate the area.
• • Heron's formula and SAS do not detect ambiguous units. Mixing feet and a metric angle will mix the two systems.
• • The calculator returns a geometric area only. Real-world takeoffs may need to add waste, overlap, seam allowances, or subtract openings.

If the base-height method seems wrong, the most common cause is that the height was measured along a slanted side rather than perpendicular to the base. Switch to SAS, or remeasure with a right-angle check.

Heron's formula is exact only when the three sides close into a real triangle. If the calculator rejects a set of sides, re-measure the longest side first because small errors there are most likely to break the inequality.

According to Khan Academy, Heron's formula computes the area of a triangle from three side lengths using the semi-perimeter s = (a + b + c) / 2.

According to OpenStax, the area of an oblique triangle is one-half the product of two sides times the sine of the included angle.

After the area is in square feet, meters, or inches, the Area Converter can move the result to a different square unit without re-entering the original sides.