Area Obtuse Triangle Calculator - SAS Area From Two Sides

The area obtuse triangle calculator finds the area of any triangle that has one interior angle greater than 90 degrees. Enter the two sides that meet at that obtuse angle and the obtuse angle in degrees, and the tool returns the area, perpendicular height, third side, and perimeter in a single step.

• • Roof or rafter layout: Estimate the footprint of a roof face where two rafters meet at an obtuse ridge or eave angle.
• • Irregular survey plots: Compute the area of a lot corner that flares out beyond a right angle.
• • Classroom SAS problems: Verify homework answers when the SAS area method is taught on an obtuse triangle.

An obtuse triangle has exactly one interior angle greater than 90 degrees, and the side opposite that obtuse angle is always the longest side. The 0.5 * base * height shortcut still works on an obtuse triangle: drop the perpendicular from the obtuse vertex to the opposite side, and the foot lands on that side between the two acute vertices. The SAS area formula is the more direct path because it skips the height step, returning 0.5 * a * b * sin(C) as the same area in a single multiplication.

When you need the same SAS area workflow for any non-right triangle rather than only the obtuse case, the Area Oblique Triangle Calculator handles acute and obtuse inputs in a single page.

The calculator multiplies the two known sides, multiplies by the sine of the included obtuse angle, and divides by two. Heron's formula cross-checks the result when the third side is known.

• a: length of one side that meets the obtuse vertex.
• b: length of the other side that meets side a at the obtuse included angle.
• C: obtuse angle between side a and side b, in degrees, strictly between 90 and 180.
• sin(C): sine of the obtuse angle. The SAS area formula 0.5 * a * b * sin(C) equals 0.5 * base * effective height, where b * sin(C) is the altitude from the acute vertex onto side a. The calculator reports the altitude from the obtuse vertex onto the opposite (longest) side, which lands inside the triangle.
• third side c: longest side of the obtuse triangle, from the law of cosines or used as the Heron cross-check input.

The SAS area formula is the easiest path because only one trigonometric function is needed. The calculator converts degrees to radians, computes sine and cosine, and returns area, perpendicular height, third side, and perimeter.

When side c is also entered and the three sides satisfy the triangle inequality, the calculator adds a Heron's formula row that should agree with the SAS area within a few hundredths of a square unit.

According to Wolfram MathWorld, the area of a triangle is one half of the base multiplied by the perpendicular height, which gives 0.5 * a * b * sin(C) for any triangle whose two sides meet at angle C, including obtuse triangles where C is greater than 90 degrees.

Setting the included angle to 90 degrees turns the SAS formula into the right-triangle shortcut 0.5 * a * b, and the Right Triangle Calculator can be used to cross-check that special case with full Pythagorean details.

These four ideas decide whether the SAS area formula gives the answer you actually need for an obtuse triangle.

The perpendicular height needed for 0.5 * base * height can be recovered from the third side and the area: h = 2 * area / c. Once that step is visible, the SAS area formula is a direct application of the standard triangle area definition.

When the input set is mixed, such as all three sides or a base and a height, the general Triangle Calculator picks the right method automatically without forcing you back to the SAS form.

Use the area obtuse triangle calculator with matching length units and the included angle measured in degrees, then read the area result in square units.

1. 1 Enter side a: Type the length of one side that meets at the obtuse vertex. Use the same length unit on both sides.
2. 2 Enter side b: Type the length of the other side that meets side a at the obtuse angle. Do not enter the third side here.
3. 3 Enter the obtuse angle in degrees: Type the angle between side a and side b. Values below 90 are rejected because the triangle would be right or acute. Setting the angle to 90 returns a right triangle result.
4. 4 Optionally enter side c: Add the third side if you have it. The calculator shows a Heron's formula cross-check when the three sides satisfy the triangle inequality.
5. 5 Read the area: Use the SAS area as the primary result in square units. The height is the perpendicular from the obtuse vertex to the opposite (longest) side, which lands inside the triangle.

A roof face has two measured edges of 5 feet and 8 feet meeting at a 120 degree ridge angle. Enter a = 5, b = 8, obtuse included angle = 120, and leave side c at 0. The calculator returns 17.32 square feet of area, 3.05 feet of height, an 11.36 foot third side, and a 24.36 foot perimeter.

When the same worksheet also needs a flat area for a rectangle, circle, or sector, the general Area Calculator covers those shapes alongside this obtuse triangle page.

The SAS area formula plus an optional Heron cross-check gives you a single tool for the most common obtuse triangle scenarios.

• • Two inputs do the work: Two side lengths and the included obtuse angle are enough for the primary area result. There is no need to derive the third side first.
• • Height, third side, and perimeter included: The perpendicular height, the law-of-cosines third side, and the perimeter come back with the area, so a second calculator is not needed.
• • Heron's formula cross-check: When the third side is known, the calculator shows a Heron's formula row that agrees within a few hundredths of a square unit, which is a fast way to catch a typing error.
• • Right triangle as a special case: Setting the included angle to 90 degrees recovers the right-triangle formula 0.5 * a * b, which makes this calculator a useful cross-check for the dedicated right-triangle page.

Because the calculator returns the area, height, third side, and perimeter together, the user can pick the value that belongs in the next step. A takeoff sheet may only need area. A diagram may only need height. A class worksheet may need every intermediate value.

After reading the area in square inches, square feet, or square meters, the Area Converter can convert the finished result into another square unit without redoing the SAS calculation.

The SAS area formula is compact, but a few measurement choices decide whether the calculator returns the right value for an obtuse triangle.

• • The calculator only handles the SAS workflow. If you only know the three side lengths, use the law of cosines to recover the included angle first.
• • Rounded output can differ by a few hundredths of a square unit from a hand calculation that rounds after each intermediate step. Keep full precision inside the formula.

Heron's formula works for any triangle as long as the three sides form a valid triangle, which is why the calculator can use it as an independent cross-check against the SAS area.

According to Wolfram MathWorld, the law of cosines gives the third side of any triangle from two sides and the included angle, and that third side is always the longest side of an obtuse triangle when the included angle is greater than 90 degrees.

According to Wolfram MathWorld, an obtuse triangle has exactly one interior angle greater than 90 degrees, and the side opposite that angle is the longest side of the triangle.

If the figure is more than three sides and the obtuse triangle is only one panel inside a larger shape, the Polygon Area Calculator handles regular and irregular polygon outlines with the same square-unit outputs.