What Is an Isosceles Obtuse Triangle?
An isosceles obtuse triangle is a triangle that is both isosceles and obtuse at once. Isosceles means two sides are equal (and so the two angles opposite them are equal). Obtuse means one angle is greater than 90° but less than 180°. Put together: the triangle has one obtuse angle and two equal acute angles, with the two equal sides meeting at the obtuse corner.
Because a triangle's three angles add to 180°, only one angle can be obtuse, more than one would already overshoot 180°. That obtuse angle has to be the apex angle, the one between the two equal sides. The remaining two are the base angles, equal to each other and necessarily acute. A common example is angles of 120°, 30°, 30°; another is 100°, 40°, 40°.
You meet this triangle while classifying triangles by both their sides and their angles, which sits in NCERT Class 6, Chapter 5 (Understanding Elementary Shapes) and under CCSS-M 4.G.A.2, where triangles are sorted by angle and by side length.
Why an Isosceles Triangle Can Be Obtuse
This is the question that trips most people up, so it is worth settling directly: can an isosceles triangle be obtuse, and where does the obtuse angle go?
Yes, it can, but only at the apex. Here is the reasoning. The two base angles of an isosceles triangle are equal. Suppose one of them were obtuse, say 100°. Then the other base angle would also be 100° (they are equal), and 100° + 100° = 200° already exceeds 180° before the third angle is even counted. Impossible. So the obtuse angle cannot be a base angle. It must be the single odd angle out, the apex between the two equal sides.
Work the numbers from the other direction. If the apex angle is some obtuse value θ (with 90° < θ < 180°), each base angle is:
$$\text{base angle} = \frac{180^{\circ} - \theta}{2}.$$
When θ is just over 90°, each base angle is just under 45°; when θ approaches 180°, the base angles shrink toward 0°. So in any isosceles obtuse triangle, each base angle is less than 45°. That single fact is a fast way to recognise one.
Properties of the Isosceles Obtuse Triangle
Everything about this triangle flows from "one obtuse apex, two equal sides." The properties worth holding:
One obtuse angle, two equal acute angles. The obtuse apex angle lies between 90° and 180°; the two base angles are equal and each less than 45°.
Two equal sides meet at the obtuse angle. These are the legs; the side opposite the obtuse angle is the base, and it is the longest side of the triangle (the longest side always faces the largest angle).
Exactly one line of symmetry. It runs from the obtuse apex straight down to the midpoint of the base, and it is the perpendicular bisector of the base as well as the bisector of the apex angle.
Angles still sum to 180°. Like every triangle, no exception for being obtuse or isosceles.
Area and Perimeter of the Isosceles Obtuse Triangle
The formulas are the standard triangle formulas; what matters (per the "derive, don't just list" habit) is knowing what each symbol stands for and why the formula holds.
Perimeter. The perimeter is just the total distance around. With the two equal sides each of length a and the base b:
$$P = a + a + b = 2a + b.$$
Here a is the length of each equal side and b is the base. Units are simple length units (cm, m).
Area from base and height. Every triangle's area is half its base times its height, because a triangle is exactly half of the parallelogram (or rectangle) you get by copying and flipping it:
$$A = \frac{1}{2} \times b \times h,$$
where b is the base and h is the perpendicular height from the obtuse apex down to the base (along the line of symmetry). A caution specific to obtuse triangles: if you instead use one of the equal sides as the base, the corresponding height falls outside the triangle, so the base-and-the-perpendicular-height-to-it pairing must match.
Area from three sides (Heron's formula). When you know all three sides but no height, use Heron's formula. With equal sides a, base b, and the semi-perimeter $s = \frac{2a + b}{2}$:
$$A = \sqrt{s(s - a)(s - a)(s - b)} = (s - a)\sqrt{s(s - b)}.$$
The second form just uses the two equal (s − a) factors. Heron's formula works for any triangle, which is exactly why it is handy here, no need to find the height first.
Examples of the Isosceles Obtuse Triangle
With the definition, the why, and the formulas in place, here is the triangle in worked problems, moving from angle identification up to a Heron's-formula area.
Example 1 - The apex angle of an isosceles obtuse triangle is 110°. Find the two base angles.
The base angles are equal and share what is left of 180°:
$$\text{base angle} = \frac{180^{\circ} - 110^{\circ}}{2} = \frac{70^{\circ}}{2} = 35^{\circ}.$$
Final answer: each base angle is 35°.
Example 2 - A student is told a triangle is isosceles with one angle of 100° and is asked to find the other two angles. They write the two equal angles as 100° each.
A first instinct is to assume the 100° is one of the matching pair, so both equal angles are 100°. Check: 100° + 100° = 200°, which is already more than the 180° a triangle is allowed. That cannot work, so the 100° angle cannot be one of the equal base angles.
The correct reading: the obtuse 100° must be the apex (the single odd angle), and the two equal base angles share the rest:
$$\text{base angle} = \frac{180^{\circ} - 100^{\circ}}{2} = 40^{\circ}.$$
Final answer: the angles are 100°, 40°, 40°.
Example 3 - An isosceles obtuse triangle has equal sides of 8 cm each and a base of 15 cm. Find its perimeter.
$$P = 2a + b = 2(8) + 15 = 16 + 15 = 31 \text{ cm}.$$
Final answer: the perimeter is 31 cm.
Example 4 - An isosceles obtuse triangle has a base of 24 cm and a perpendicular height of 8 cm to that base. Find its area.
$$A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 24 \times 8 = 96 \text{ cm}^2.$$
Final answer: the area is 96 cm².
Example 5 - The apex angle of an isosceles obtuse triangle is 4 times a base angle. Find all three angles, and confirm it is obtuse.
Let each base angle be x; the apex is 4x. The three sum to 180°:
$$x + x + 4x = 180^{\circ} ;\Rightarrow; 6x = 180^{\circ} ;\Rightarrow; x = 30^{\circ}.$$
So the base angles are 30° each and the apex is 4(30°) = 120°.
Final answer: 120°, 30°, 30°. Since 120° > 90°, the triangle is obtuse, and the two equal base angles confirm it is isosceles.
Example 6 - An isosceles obtuse triangle has equal sides of 5 cm and a base of 8 cm. Find its area using Heron's formula.
First the semi-perimeter, with a = 5, b = 8:
$$s = \frac{2a + b}{2} = \frac{10 + 8}{2} = 9 \text{ cm}.$$
Now Heron's formula:
$$A = \sqrt{s(s - a)(s - a)(s - b)} = \sqrt{9 \times 4 \times 4 \times 1} = \sqrt{144} = 12 \text{ cm}^2.$$
Final answer: the area is 12 cm². (Quick check that it is obtuse: the base 8 is longer than each leg 5, and 8² = 64 > 5² + 5² = 50, so the apex angle exceeds 90°.)
Why the Isosceles Obtuse Triangle Matters
A triangle that is both obtuse and isosceles is more than a classification puzzle, the shape and its wide, symmetric span turn up wherever a stable, spreading form is needed.
Roof and bracket design. A low, wide-spanning gable or a shallow support bracket is often an isosceles obtuse triangle: the obtuse apex lets the structure cover a broad base while staying symmetric, and the equal sides distribute load evenly to both supports.
Reading shape from angles. This triangle is the cleanest illustration of a rule that runs through all of geometry, the longest side faces the largest angle. Because the obtuse angle is the biggest, the base opposite it is always the longest side, a fact students later use to reason about any triangle's sides without measuring.
Where heights go outside. Obtuse triangles are the case that forces students to confront that an altitude can land outside the triangle. Meeting it in the symmetric isosceles version, where one altitude is clean (down the line of symmetry) and the others are external, builds the intuition needed for coordinate geometry and trigonometry later.
Tessellation and design. Pairs of isosceles obtuse triangles tile into rhombi and broad parallelograms, which is why the shape appears in patterned tiling, truss webs, and decorative lattices.
For a Class 6 or Class 7 student, this triangle is where "classify by sides" and "classify by angles" stop being two separate exercises and become one combined idea, the moment a triangle earns two labels at once.
Where Do Students Trip Up on the Isosceles Obtuse Triangle?
Mistake 1: Putting the obtuse angle at a base
Where it slips in: A student is told a triangle is isosceles and obtuse and sets both equal base angles to the obtuse value.
Don't do this: Treat the obtuse angle as one of the equal pair, giving something like 100°, 100°, and a leftover.
The correct way: Two equal obtuse base angles would already exceed 180°. The obtuse angle must be the single apex angle between the equal sides; the two equal base angles are acute and each less than 45°.
Mistake 2: Pairing the wrong height with the base
Where it slips in: Finding the area, a student multiplies an equal side by a height meant for a different base.
Don't do this: Use one side as the base while plugging in a height measured to a different side.
The correct way: The height must be the perpendicular distance to the same base you are using. In an obtuse triangle, two of the three altitudes fall outside the triangle, so the clean pairing is the base opposite the obtuse apex with the height down the line of symmetry, or use Heron's formula and skip the height entirely.
Mistake 3: Thinking a triangle can be obtuse and equilateral
Where it slips in: A student blurs "isosceles" and "equilateral" and looks for an obtuse equilateral triangle.
Don't do this: Expect equal sides to force equal angles and allow an obtuse one.
The correct way: An equilateral triangle has all angles 60°, never obtuse. Only the isosceles (exactly-two-equal-sides) version can be obtuse, because its two equal angles are the small base angles, not the apex.
Key Takeaways
An isosceles obtuse triangle has one obtuse apex angle and two equal acute base angles, each under 45°.
The obtuse angle can only sit at the apex between the equal sides; two equal obtuse base angles would break the 180° sum.
The base, opposite the obtuse angle, is the longest side, and the triangle has exactly one line of symmetry.
Perimeter is 2a + b; area is ½ × base × height, or Heron's formula when only the three sides are known.
An equilateral triangle can never be obtuse, only the isosceles version can.
Practice These Problems to Solidify Your Understanding
The apex angle of an isosceles obtuse triangle is 130°. Find the two base angles.
An isosceles obtuse triangle has equal sides of 9 cm and a base of 16 cm. Find its perimeter.
An isosceles obtuse triangle has a base of 20 cm and a height of 6 cm to that base. Find its area.
Answer to Question 1: 25° each. Answer to Question 2: 34 cm. Answer to Question 3: 60 cm². If Question 1 gave you anything 90° or larger, recheck that the obtuse angle is the apex, not a base angle (see Mistake 1).
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