Study Material — 18 PYQs (2018–2019) · Concept Notes · Shortcuts
RRB NTPC Triangles — Area & Properties is a frequently tested subtopic — 18 previous year questions from 2018–2019 papers are included below with concept notes, key rules and shortcut tricks.
A triangle has a base of 12 cm and a height of 8 cm. What is its area?
Exam Q 42019Previous Year Pattern
The area of a triangle is 24 cm² and its base is 6 cm. What is the height of the triangle?
Exam Q 52019Previous Year Pattern
A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its area using Heron's formula?
Exam Q 62019Previous Year Pattern
In a triangle, the sum of any two sides must be greater than the third side. If two sides of a triangle are 7 cm and 10 cm, which of the following can be the length of the third side?
Exam Q 72018Previous Year Pattern
An equilateral triangle has a side length of 8 cm. What is the ratio of its area to the area of a square with the same perimeter?
Exam Q 82018Previous Year Pattern
A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area?
Exam Q 92018Previous Year Pattern
In triangle ABC, the altitude from vertex A to side BC is 12 cm. If the area of triangle ABC is 96 cm², what is the length of side BC?
Exam Q 102019Previous Year Pattern
A triangle has sides of length 13 cm, 14 cm, and 15 cm. Using Heron's formula, find its area.
Exam Q 112019Previous Year Pattern
An equilateral triangle has a side length of 8 cm. Find the ratio of its area to its perimeter (in numerical form, without units).
Exam Q 122019Previous Year Pattern
In triangle ABC, angle B = 90°, AB = 5 cm, and BC = 12 cm. A line parallel to BC intersects AB at point D such that AD = 2 cm. Find the length of DE, where E is on AC and DE is parallel to BC.
Exam Q 132019Previous Year Pattern
A triangle with vertices at coordinates A(0, 0), B(8, 0), and C(4, 6) is given. Find its area using the coordinate formula.
Exam Q 142018Previous Year Pattern
A triangle has an area of 60 cm² and a base of 15 cm. If the base is increased by 20% and the height is decreased by 20%, what is the new area?
Exam Q 152019Previous Year Pattern
In triangle ABC, the angle bisector from vertex A meets side BC at point D. If AB = 18 cm, AC = 24 cm, and BD = 9 cm, what is the length of DC (in cm)?
Exam Q 162019Previous Year Pattern
In triangle ABC, the incircle touches side BC at point P, side AC at point Q, and side AB at point R. If AB = 13 cm, BC = 14 cm, and CA = 15 cm, what is the length of AR (in cm)?
Exam Q 172019Previous Year Pattern
A triangle has sides of length 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. What is the length of this perpendicular (in cm)?
Exam Q 182019Previous Year Pattern
In triangle ABC, the median from vertex A to side BC has length 13 cm. If AB = 12 cm and AC = 14 cm, what is the length of BC (in cm)?
Concept Notes
Triangles — Area & Properties— Rules & Concept
💡
Core Concept
Read this first — the foundation of the topic
→Core Concept
Triangle area measures the space inside the triangle. Properties tell us relationships between sides and angles
💡Key Rules and Properties
Sum of all angles = 180°
2. Sum of any two sides > third side
3. Exterior angle = sum of two opposite interior angles
4. In right triangle: a² + b² = c² (Pythagoras theorem)
5.
Area depends on base and height OR three sides OR two sides with included angle
🔢
Formula Block
Memorise — at least one formula appears in every paper
Basic Area = (1/2) × base × height
Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
SAS Formula: Area = (1/2) × a × b × sin C
Equilateral triangle area = (√3/4) × side²
Isosceles triangle area = (b/4)√(4a² - b²) where a = equal sides, b = base
📊
Exam Patterns
What examiners ask — read before attempting PYQs
✏️Worked Example 1
1
Use Heron's formula
2
s = (13+14+15)/2 = 21
3
Area = √[21(21-13)(21-14)(21-15)]
4
Area = √[21 × 8 × 7 × 6]
5
Area = √[7056] = 84 sq units
Worked Example 2: Triangle with vertices A(0,0), B(4,0), C(0,3). Find area.
1
This forms right triangle with base on x-axis
2
Base = 4 units, Height = 3 units
3
Area = (1/2) × 4 × 3 = 6 sq units
Alternative: Use coordinate formula = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
Shortcut Trick #3: For coordinate geometry triangles, if vertices have zeros, use simple base × height method instead of coordinate formula.
Common Mistake #1: Students forget to take square root in Heron's formula. They calculate s(s-a)(s-b)(s-c) and stop there. Always remember the square root symbol! This single mistake costs many students easy marks
⚠️Additional Common Mistakes
Confusing perimeter with semi-perimeter in Heron's formula. Using wrong angle in SAS formula. Not checking if given sides can form a triangle before calculating area.
Key Points to Remember
Basic area formula: (1/2) × base × height works for all triangles
Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = semi-perimeter
Equilateral triangle area = (√3/4) × side² - memorize this shortcut
Sum of angles in any triangle = 180° always
Pythagoras theorem: a² + b² = c² for right triangles only
Triangle inequality: sum of any two sides > third side
SAS area formula: (1/2) × a × b × sin C for two sides and included angle
Coordinate triangle area = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
Right triangle sides often in ratios 3:4:5, 5:12:13, 8:15:17
Always check if three given sides can form triangle before solving
Exam-Specific Tips
Heron of Alexandria discovered Heron's formula in 60 AD
In equilateral triangle, all angles = 60° each
Right triangle with sides 3:4:5 has area = 6 square units
Isosceles triangle has two equal sides and two equal angles
Triangle with sides 5, 12, 13 is right-angled triangle
Sum of exterior angles of any triangle = 360°
Median divides triangle into two equal areas
Altitude from vertex to opposite side creates two right triangles
Practice MCQs
Triangles — Area & Properties — Practice Questions
14graded MCQs · easy to hard · full solution & trap analysis
A triangle has a base of 12 cm and a height of 8 cm. What is its area?
Practice 2easy
A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its area using Heron's formula?
Practice 3easy
An equilateral triangle has a side length of 10 cm. What is its perimeter?
Practice 4easy
A right-angled triangle has legs of length 6 cm and 8 cm. What is the length of its hypotenuse?
Practice 5easy
A triangle has a base of 16 cm and a height of 12 cm. What is its area?
Practice 6easy
A triangle has an area of 24 cm² and a base of 8 cm. What is its height?
Practice 7medium
An equilateral triangle has a side length of 8 cm. What is the ratio of its area to its perimeter (in cm)?
Practice 8medium
A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area in cm²?
Practice 9medium
An equilateral triangle has a side length of 8 cm. What is the length of its altitude in cm?
Practice 10medium
A triangle has sides of length 13 cm, 14 cm, and 15 cm. Find its area using Heron's formula.
Practice 11medium
A right-angled triangle has legs of 9 cm and 12 cm. A circle is inscribed in this triangle. What is the radius of the inscribed circle in cm?
Practice 12hard
A right-angled triangle has legs of 20 cm and 21 cm. A circle is inscribed in this triangle. What is the radius of the inscribed circle (in cm)?
Practice 13hard
A triangle has sides of length 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. What is the length of this perpendicular (in cm)?
Practice 14hard
A triangle has sides of length 13 cm, 14 cm, and 15 cm. If a perpendicular is drawn from the vertex opposite the 14 cm side to that side, what is the length of this perpendicular (in cm)?
60-Second Revision — Triangles — Area & Properties
Remember: Always take square root in Heron's formula final step
Formula: Equilateral area = (√3/4) × side² - fastest method
Trap: Check triangle inequality before calculating area
Shortcut: Recognize 3:4:5 ratio triangles for instant right triangle identification
Formula: Basic area = (1/2) × base × height works universally
Remember: Semi-perimeter s = (a+b+c)/2 in Heron's formula
Quick check: For coordinates with zeros, use base × height method