Area and Perimeter of Triangles
Area and Perimeter of Triangles
Triangle mensuration in NTPC is formula recall plus one Pythagoras step: Heron's formula for three sides, √3/4 × a² for an equilateral triangle, and half base × height for right and isosceles triangles. The traps are using the wrong height and forgetting that a "5 : 12 : 13" triangle is right-angled.
1. Formula Box
| Triangle | Area | Other results |
|---|---|---|
| Any (base b, height h) | (1/2) × b × h | Height must be perpendicular to that base |
| Any (sides a, b, c) | √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2 | Heron's formula |
| Two sides and included angle | (1/2) ab sin C | 60° gives (√3/4) ab |
| Equilateral (side a) | (√3/4) a² | Height (√3/2) a; inradius a/(2√3); circumradius a/√3 |
| Right (legs a, b) | (1/2) ab | Hypotenuse² = a² + b²; circumradius = hypotenuse/2 |
| Isosceles (equal sides a, base b) | (b/4) √(4a² − b²) | Height to base = √(a² − b²/4) |
| Isosceles right (leg a) | a²/2 | Hypotenuse a√2 |
Pythagorean triplets to recognise instantly: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (20, 21, 29) and their multiples.
✗ Equilateral triangle of side 16 has area 16²/2 | ✓ Area = (√3/4) × 256 = 64√3 ≈ 110.85
✗ Isosceles triangle 13, 13, 10: area = (1/2) × 10 × 13 | ✓ Use the height, not the slant side: √(169 − 25) = 12, area = 60
हिंदी नोट: त्रिभुज का क्षेत्रफल = (1/2) × आधार × ऊँचाई, जहाँ ऊँचाई आधार पर लंब होनी चाहिए। तीन भुजाएँ दी हों तो हीरोन सूत्र, समबाहु त्रिभुज के लिए (√3/4)a²।
Exam Pointer: Verified NTPC patterns: equilateral triangle area from the perimeter (February 2021) and right triangle area from perimeter and hypotenuse (January 2021). Heron's formula and isosceles triangles had no verified NTPC shift in our check and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Heron's formula and triangles with sides in a ratio
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. Find the area of a triangle with sides 9 cm, 10 cm and 17 cm.
s = 18. Area = √(18 × 9 × 8 × 1) = √1,296 = 36 cm².
Answer: 36 cm²
EXAMATLAS LEVEL
Q. The sides of a triangle are in the ratio 5 : 12 : 13 and its perimeter is 120 cm. Find its area and its circumradius.
Sides 20, 48, 52 (multiply by 4). Since 5 : 12 : 13 is a Pythagorean triplet, the triangle is right-angled: area = (1/2) × 20 × 48 = 480 cm². Circumradius = hypotenuse/2 = 26 cm. Heron gives the same 480 but takes much longer.
Answer: 480 cm²; 26 cm
Pattern 2: Equilateral triangle
[PYQ: NTPC CBT-1 3-Feb-2021 Shift-1]
EXAM LEVEL
Q. The perimeter of an equilateral triangle is 48 cm. Find its area.
Side = 16 cm. Area = (√3/4) × 256 = 64√3 cm² (about 110.85 cm²).
Answer: 64√3 cm²
EXAMATLAS LEVEL
Q. The height of an equilateral triangle is 9√3 cm. Find its area and its inradius.
Height = (√3/2)a, so a = 18 cm. Area = (√3/4) × 324 = 81√3 cm². Inradius = height/3 = 3√3 cm.
Answer: 81√3 cm²; 3√3 cm
Pattern 3: Right triangle from perimeter, hypotenuse or legs
[PYQ: NTPC CBT-1 8-Jan-2021 Shift-2]
EXAM LEVEL
Q. The legs of a right triangle are 9 cm and 12 cm. Find its hypotenuse, area and perimeter.
Hypotenuse = 15 cm (3-4-5 triplet × 3). Area = 54 cm². Perimeter = 36 cm.
Answer: 15 cm, 54 cm², 36 cm
EXAMATLAS LEVEL
Q. A right triangle has perimeter 56 cm and hypotenuse 25 cm. Find its area.
a + b = 31 and a² + b² = 625. Then 2ab = 31² − 625 = 961 − 625 = 336, so ab = 168 and area = 84 cm². (The legs are 7 and 24.)
Answer: 84 cm²
Pattern 4: Isosceles and isosceles right triangles
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. An isosceles triangle has equal sides of 13 cm and base 10 cm. Find its area.
Height = √(13² − 5²) = 12 cm. Area = (1/2) × 10 × 12 = 60 cm².
Answer: 60 cm²
EXAMATLAS LEVEL
Q. An isosceles right triangle has hypotenuse 10√2 cm. Find its area and perimeter.
Each leg = 10 cm. Area = 100/2 = 50 cm². Perimeter = 20 + 10√2 cm (about 34.14 cm).
Answer: 50 cm²; 20 + 10√2 cm
60-Second Revision
- Area = (1/2) base × perpendicular height; Heron for three sides.
- Equilateral: (√3/4)a², height (√3/2)a, inradius = height/3.
- Right triangle: (a + b)² − c² = 2ab gives the area from perimeter and hypotenuse.
- Isosceles: height = √(a² − b²/4); learn the Pythagorean triplets.