terican

Last verified · v1.0

Calculator · math

Obtuse Triangle Area Calculator

Compute obtuse triangle area using base & height, two sides with angle (SAS), or all three sides via Heron's Formula — instantly and accurately.

FreeInstantNo signupOpen source

Inputs

Triangle Area

—

Explain my result

0/3 free

Get a plain-English breakdown of your result with practical next steps.

Triangle Area—sq units

The formula

How the
result is
computed.

How to Calculate the Area of an Obtuse Triangle

An obtuse triangle contains exactly one interior angle greater than 90°. This characteristic affects how altitude is measured and which formula is most practical. Three established methods cover every measurement scenario when using an area obtuse triangle calculator — choose the one that matches the known values.

Method 1: Base and Perpendicular Height (½ × b × h)

The fundamental triangle area formula applies directly to obtuse triangles:

A = ½ × b × h

The critical difference for obtuse triangles is the location of the altitude. When the obtuse angle sits at a base vertex, the perpendicular height h falls outside the triangle boundary. To measure it correctly, extend the base line beyond the obtuse vertex and drop a perpendicular from the opposite vertex down to that extended line. That perpendicular length is the true height — never substitute the slant side length.

Worked Example: A triangular roof panel has a base of 12 m and a perpendicular altitude of 5 m. Area = ½ × 12 × 5 = 30 m². If the altitude were incorrectly taken as a slant edge of 7 m, the result would be wrong by over 40%.

Method 2: Two Sides and the Included Angle — SAS Formula

When two side lengths and the angle between them are known, the trigonometric SAS formula provides the most direct solution:

A = ½ × a × b × sin(C)

Angle C is the obtuse angle (90° < C < 180°) formed between sides a and b. Because the sine function is positive for all angles between 0° and 180°, sin(C) yields a valid positive value even when C is obtuse — the formula never breaks down. The Texas A&M University Open Math 150 textbook on the Law of Cosines identifies this SAS area formula as a standard tool for all non-right triangles, derived directly from trigonometric identities.

Worked Example: A surveyed land parcel has sides a = 9 m, b = 6 m, with an obtuse included angle C = 135°. Since sin(135°) = sin(45°) ≈ 0.7071, the area = ½ × 9 × 6 × 0.7071 ≈ 19.09 m².

Method 3: Heron's Formula — All Three Sides Known (SSS)

When all three side lengths are available but no angle is directly measured, Heron's Formula computes the area without any trigonometry:

Step 1 — Semi-perimeter: s = (a + b + c) / 2

Step 2 — Area: A = √[ s(s − a)(s − b)(s − c) ]

In an obtuse triangle, side c — directly opposite the obtuse angle — is always the longest side, satisfying the inequality c² > a² + b². The University of Akron geometry curriculum on Heron's Formula confirms that this formula applies universally to acute, right, and obtuse triangles without modification. The Grand Valley State University open textbook on Triangles and Vectors further documents the geometric derivations linking side-length relationships to area across all triangle types.

Worked Example: A structural truss forms an obtuse triangle with sides a = 5 m, b = 6 m, c = 9 m. Verify: 9² = 81 > 5² + 6² = 61 — confirmed obtuse. Semi-perimeter: s = (5 + 6 + 9) / 2 = 10. Area = √[10 × (10−5) × (10−6) × (10−9)] = √[10 × 5 × 4 × 1] = √200 ≈ 14.14 m².

Recognizing a Valid Obtuse Triangle

Before entering measurements, confirm the triangle is genuinely obtuse using any of these tests:

  • One interior angle directly measures more than 90°
  • For sides a ≤ b ≤ c: the inequality c² > a² + b² holds
  • The cosine of the largest angle (computed via the Law of Cosines) is negative

The interior angles of any triangle always sum to exactly 180°, so an obtuse triangle has precisely one obtuse angle and two acute angles.

Variable Reference

  • b — Any side chosen as the base
  • h — Perpendicular altitude to the base; may require extending the base line for obtuse triangles
  • a, b — The two sides enclosing the known included angle C (SAS method)
  • C — The obtuse included angle in degrees; must satisfy 90° < C < 180°
  • c — The side opposite the obtuse angle; always the longest side
  • s — Semi-perimeter = (a + b + c) / 2, used in Heron's method

Reference

Frequently asked questions

What is the formula for the area of an obtuse triangle?
Three formulas apply depending on available measurements. Use A = ½ × base × height when the perpendicular altitude is known, being careful that the altitude may fall outside the triangle. Use A = ½ × a × b × sin(C) when two sides and the obtuse included angle are known. Use Heron's Formula A = √[s(s−a)(s−b)(s−c)] when all three side lengths are given. All three methods produce identical areas for the same triangle.
Why does the height of an obtuse triangle fall outside the triangle?
The perpendicular altitude must connect a vertex to the line containing the opposite side at a 90° angle. When the obtuse angle sits at a base vertex, that 90° foot point lands beyond the triangle's edge on an extension of the base line — a direct geometric consequence of the angle exceeding 90°. Extend the base line, drop the perpendicular from the opposite vertex, and measure that external perpendicular distance as the true height for the area formula.
Can the SAS formula A = ½ab·sin(C) be used when angle C is obtuse?
Yes, absolutely. The sine function returns a positive value for every angle between 0° and 180°, meaning sin(C) remains positive even when C exceeds 90°. For instance, sin(120°) = sin(60°) ≈ 0.8660 and sin(150°) = sin(30°) = 0.5. The formula A = ½ × a × b × sin(C) therefore yields a correct, positive area for any obtuse angle C, making it one of the most reliable methods for obtuse triangle problems.
How do I identify which side is c in Heron's formula for an obtuse triangle?
Side c is always the longest side of the triangle and sits directly opposite the obtuse angle. To confirm a side qualifies as c, verify that c² > a² + b² (the Pythagorean inequality that defines an obtuse triangle). Label the two shorter sides a and b in any order — Heron's formula is fully symmetric with respect to a and b, so their assignment does not affect the result. Correctly identifying the longest side also serves as a built-in validity check before computing the semi-perimeter s.
What is the difference between calculating area for an obtuse vs. an acute triangle?
The mathematical formulas are identical for both triangle types, but obtuse triangles require extra care with altitude measurement. For acute triangles, all three altitudes fall inside the triangle and can be measured directly. For obtuse triangles, two of the three altitudes fall outside the boundary, requiring a base-line extension before dropping the perpendicular. The SAS and Heron's methods are unaffected by this distinction since they rely on side lengths and angles, not external altitudes.
Can an obtuse triangle have a larger area than an acute triangle with the same base?
Yes. Area equals ½ × base × height, so it depends entirely on perpendicular height, not on angle classification. An obtuse triangle with a tall altitude easily exceeds the area of a flat acute triangle sharing the same base but a shorter altitude. For example, an obtuse triangle with base 10 m and height 20 m covers 100 m², far exceeding a low acute triangle of base 10 m and height 3 m at only 15 m². The obtuse or acute label describes angle geometry, not area magnitude.