How to Identify an Obtuse Angled Triangle (with Diagrams & Examples)
The concept of obtuse angled triangle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding what makes a triangle obtuse, how to classify and solve problems involving obtuse angled triangles, is vital for mastering geometry and performing well in tests.
What Is an Obtuse Angled Triangle?
An obtuse angled triangle is a triangle where one of its three angles measures more than 90 degrees but less than 180 degrees. The other two angles in an obtuse angled triangle are always acute (less than 90 degrees), because the sum of all angles in a triangle is always 180 degrees. You’ll find this concept applied in areas such as triangle classification, calculation of area and altitude, and geometric reasoning for exams.
Key Formula for Obtuse Angled Triangle
Here’s the standard formula for the area of an obtuse angled triangle:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
Alternatively, if all three sides are known,
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s = \frac{a+b+c}{2} \) (Heron's Formula).
Properties of an Obtuse Angled Triangle
| Property | Value/Explanation |
|---|---|
| Number of obtuse angles | Only one |
| Sum of angles | 180° |
| Longest side | Opposite obtuse angle |
| Other two angles | Both acute (<90°) |
| Altitude from acute angle | Falls outside triangle |
| Circumcentre/Orthocentre location | Outside the triangle |
How to Identify an Obtuse Angled Triangle?
Follow these steps to check if a triangle is obtuse angled:
- Check angle measures: Does one angle > 90°?
- Sum of all angles = 180°? (Always true for triangles).
- Use sides: For sides a, b, c (where c is the longest), if \( a^2 + b^2 < c^2 \), the triangle is obtuse angled.
Step-by-Step Illustration
- Given triangle sides: 3 cm, 4 cm, 6 cm.
Arrange: Largest side is 6 cm (let c = 6). - Calculate squares: \(3^2 = 9\), \(4^2 = 16\), \(6^2 = 36\).
- Check: \(9 + 16 = 25\), which is less than \(36\).
So, \(a^2 + b^2 < c^2\), the triangle is obtuse angled.
Solved Example
Find the area of an obtuse angled triangle with base 10 cm and height 6 cm.
1. Use the formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)2. Substitute the values: \( \frac{1}{2} \times 10 \times 6 = 30 \)
3. Final answer: The area is 30 cm².
Speed Trick or Vedic Shortcut
Here’s a quick way to check if a triangle is obtuse, just by the side lengths:
- Identify the longest side (let's call it c).
- Square c, and square the other two sides (a and b).
- If \( a^2 + b^2 < c^2 \), it is obtuse angled.
Use this in exams when angle measures are not given. Vedantu's live sessions teach such helpful tricks for time management and clarity in geometry.
Try These Yourself
- Given triangle angles: 120°, 35°, 25°. Is it an obtuse angled triangle?
- Check if triangle with sides 7 cm, 24 cm, 25 cm is obtuse.
- Find the area of an obtuse angled triangle with base 14 cm and height 5 cm.
- Which side is the longest in an obtuse angled triangle, and why?
Frequent Errors and Misunderstandings
- Confusing obtuse and right angled triangles.
- Assuming more than one obtuse angle is possible in a triangle.
- Using wrong base or height in area formula.
- Forgetting that the altitude from acute angles in obtuse triangles may fall outside the triangle.
Relation to Other Concepts
The idea of obtuse angled triangle connects closely with concepts like acute angled triangle and properties of triangle. Mastering obtuse angled triangles gives you confidence to solve a variety of geometric problems involving area, perimeter, and classification of triangles. For a broad comparison of all triangle types, see types of triangles.
Obtuse Angled Triangle in Real Life
Obtuse angled triangles can be found in architecture (like roof trusses), tool design, and corners of plots or sports fields. Noticing an angle that “opens wide” (greater than a right angle) is a good clue. If you want more hands-on practice, try triangle worksheets for additional problems.
Classroom Tip
A simple way to remember: "In an obtuse angled triangle, one angle is always greater than a right angle and the longest side is opposite that angle." Vedantu’s teachers often use illustrations and cut-out models to demonstrate this visually during live classes.
We explored obtuse angled triangle—from definition, formula, stepwise problems, common mistakes, and its link to other maths concepts. Continue practicing with Vedantu and attend live classes to become confident with obtuse and other types of triangles.
FAQs on Obtuse Angled Triangle: Properties, Formulas, and Examples
1. What is an obtuse angled triangle in Maths?
An obtuse angled triangle is a triangle with one interior angle measuring greater than 90° but less than 180°. This angle is called the obtuse angle. The other two angles are necessarily acute angles (less than 90°).
2. How do you identify an obtuse angle triangle?
To identify an obtuse angled triangle, check the measures of its angles. If one angle is greater than 90° and less than 180°, and the sum of all three angles equals 180°, then it's an obtuse angled triangle. Alternatively, if a², b², and c² represent the squares of the sides, and a² + b² < c², where c is the longest side, the triangle is obtuse.
3. What is the formula for the area of an obtuse triangle?
The area of an obtuse triangle can be calculated using the standard formula: Area = ½ × base × height. However, remember that the height (altitude) from the obtuse angle will fall outside the triangle. Alternatively, Heron's formula can be applied: Area = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter [(a+b+c)/2], and a, b, and c are the side lengths.
4. Can a triangle have two obtuse angles?
No, a triangle cannot have two obtuse angles. The sum of the angles in any triangle is always 180°. If two angles were greater than 90°, their sum alone would exceed 180°, making the formation of a triangle impossible.
5. What are real-life examples of obtuse angled triangles?
Obtuse angled triangles are found in many everyday objects. Examples include certain roof structures, some types of ramps, and the shape of certain tools or wedges. Think of a slice of pizza cut at an angle greater than 90°.
6. How does the altitude in an obtuse triangle differ from other triangles?
In an obtuse triangle, the altitude drawn from the obtuse angle lies outside the triangle. In acute and right-angled triangles, the altitudes always fall within the triangle itself.
7. What happens to the orthocenter in an obtuse angled triangle?
The orthocenter, the point where the altitudes intersect, lies outside the obtuse triangle. In acute triangles, it's inside; in right triangles, it's at the right angle.
8. How do side lengths help determine if a triangle is obtuse?
If the square of the longest side (c²) is greater than the sum of the squares of the other two sides (a² + b²), then the triangle is obtuse. This is a direct application of the Pythagorean inequality.
9. Why can't two angles in a triangle be obtuse?
Because the sum of angles in a triangle must equal 180°. Two obtuse angles (each > 90°) would already sum to more than 180°, violating this fundamental property of triangles.
10. How is the area calculated if side lengths and the obtuse angle are known?
You can use the formula: Area = ½ab sin(C), where a and b are two sides, and C is the included obtuse angle. Remember that the sine of an obtuse angle is positive.
11. What is the difference between an obtuse and an acute triangle?
An obtuse triangle has one angle greater than 90°, while an acute triangle has all three angles less than 90°.
12. Can an obtuse triangle be isosceles?
Yes, an obtuse triangle can be isosceles. An isosceles triangle has at least two equal sides. The obtuse angle could be between two equal sides or opposite one of the equal sides.
