The formulas
From base and height
Area = ½ × base × height
From three sides — Heron's formula
First find the semi-perimeter: s = (a + b + c) ÷ 2
Then: Area = √( s(s−a)(s−b)(s−c) )
Perimeter
P = a + b + c
Right-angled triangles — Pythagoras
a² + b² = c², where c is the hypotenuse.
Worked example
A triangle with sides 6, 8 and 10:
| Step | Working |
|---|---|
| Perimeter | 6 + 8 + 10 = 24 |
| Semi-perimeter | 24 ÷ 2 = 12 |
| Heron's | √(12 × 6 × 4 × 2) = √576 |
| Area | 24 square units |
| Type | 6² + 8² = 100 = 10², so right-angled |
Types of triangle
| By sides | Definition |
|---|---|
| Equilateral | All three sides equal, all angles 60° |
| Isosceles | Two sides equal, two angles equal |
| Scalene | All sides different |
| By angles | Definition |
|---|---|
| Acute | All angles under 90° |
| Right | One angle exactly 90° |
| Obtuse | One angle over 90° |
The triangle inequality
Three lengths only form a triangle if each pair adds to more than the third. Sides of 2, 3 and 9 cannot form a triangle, because 2 + 3 is less than 9. If Heron's formula gives you an error or an imaginary result, this is usually why.
Where this gets used
- Land measurement — irregular plots are commonly divided into triangles and their areas summed.
- Construction — the 3-4-5 rule is the standard site method for checking a right angle without instruments.
- Roofing and staircases — rise, run and rafter length form a right triangle.
- School and competitive exams — Heron's formula appears regularly in Class 9 onward.
Common mistakes
- Using a slanted side as the height. Height must be perpendicular to the base.
- Forgetting the ½ in the base-height formula.
- Mixing units — convert everything before calculating.
- Entering three lengths that cannot form a triangle.
Frequently Asked Questions
How do I find the area of a triangle?
With base and height, use ½ × base × height. With three side lengths, use Heron's formula.
What is Heron's formula?
Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter, half the sum of the three sides. It lets you find the area from sides alone.
How do I know if a triangle is right-angled?
Check whether the squares of the two shorter sides add up to the square of the longest. For 6, 8 and 10, 36 + 64 = 100, so it is right-angled.
Can any three lengths form a triangle?
No. Each pair must add to more than the third side. Lengths of 2, 3 and 9 cannot form a triangle.
What is the 3-4-5 rule?
A site method for checking a right angle: measure 3 units along one edge and 4 along the other, and if the diagonal is exactly 5, the corner is square.
How do I find the height if I only know the sides?
Find the area using Heron's formula, then rearrange the base-height formula: height = 2 × area ÷ base.
Sides se angle nikalna
Agar teenon bhujaayein pata hain to law of cosines se koi bhi kona nikal sakta hai:
cos A = (b² + c² − a²) / 2bc
Jaise 5, 6, 7 wale triangle mein sabse badi bhuja (7) ke saamne wala kona:
cos A = (25 + 36 − 49) / (2 × 5 × 6) = 12/60 = 0.2, isliye A = 78.5°
Baaki do kone bhi isi tarah nikal sakte hain, ya ek nikal kar 180 mein se ghata dijiye.
Rozmarra ke istemaal
- Chhat ka dhalaan — 4 meter chaudi chhat 1 meter oonchi uthti hai to dhalaan ki lambai √(4² + 1²) = 4.12 meter. Sheet kitni chahiye, ye isi se tay hota hai.
- Tirchhi plot ka kshetrafal — teen kone naap kar Heron's formula laga dijiye. Aayat wale formula se galat aayega.
- Seedhi ki lambai — 3 meter oonchi deewar par 1 meter door rakhi seedhi ko 3.16 meter lambi hona chahiye.
- Kona sahi hai ya nahi — 3-4-5 rule se mistri deewar ka kona check karte hain.
More questions
Teen bhujaon se kona kaise nikalein?
Law of cosines se — cos A = (b² + c² − a²) / 2bc. Ek kona nikaal kar baaki 180 mein se ghata sakte hain.
Tirchhe plot ka kshetrafal kaise nikalein?
Use do triangles mein baant lijiye aur har ek par Heron's formula lagaiye. Lambai × chaudai wala formula sirf aayat ke liye sahi hai.
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