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SSC CHSL Triangles — Area & Properties

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This page covers SSC CHSL Triangles — Area & Properties with complete concept notes, 15 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Triangles — Area & Properties— Rules & Concept

Core ConceptRead this first — the foundation of the topic

Triangles are three-sided closed figures. Finding their area and understanding their properties is crucial for SSC CGL success. This topic appears in 2-3 questions per paper. Core Concept: Triangle area measures the space inside the triangle. Properties tell us relationships between sides and angles.

Key RulesCore rules you must know cold
1

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 BlockMemorise — 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 PatternsWhat examiners ask — read before attempting PYQs

SSC loves asking area with given coordinates, Heron's formula problems, and finding missing sides when area is given. Questions often combine area with similarity or congruence.

ShortcutsUse these to save 30–60 seconds per question

#1: For right triangles, if sides are in ratio 3:4:5 or 5:12:13 or 8:15:17, instantly recognize them. Area = (1/2) × product of perpendicular sides. Shortcut Trick #2: When three sides are given, check if a² + b² = c². If yes, it's right triangle.

Use simple area formula instead of Heron's.

Worked ExampleSolve this step-by-step before moving on
1
Step 1

Use Heron's formula

2
Step 2

s = (13+14+15)/2 = 21

3
Step 3

Area = √[21(21-13)(21-14)(21-15)]

4
Step 4

Area = √[21 × 8 × 7 × 6]

5
Step 5

Area = √[7056] = 84 sq units Worked Example 2: Triangle with vertices A(0,0), B(4,0), C(0,3). Find area.

1
Step 1

This forms right triangle with base on x-axis

2
Step 2

Base = 4 units, Height = 3 units

3
Step 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.

Exam TrapsCommon mistakes students make — avoid these

#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

15graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

An equilateral triangle has a side length of 10 cm. What is its perimeter?

Practice 2easy

An isosceles triangle has two equal sides of 7 cm each and a base of 10 cm. What is its perimeter?

Practice 3easy

The area of a triangle is 24 cm² and its base is 6 cm. What is the height of the triangle?

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 12 cm and a height of 8 cm. What is its area?

Practice 6easy

A triangle has sides of length 5 cm, 12 cm, and 13 cm. What is its semi-perimeter?

Practice 7medium

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?

Practice 8medium

A triangle has sides of length 13 cm, 14 cm, and 15 cm. What is its area in cm²?

Practice 9medium

A triangle has vertices at A(0, 0), B(8, 0), and C(4, 6). Find the area of the triangle using the coordinate formula.

Practice 10medium

A triangle has sides of length 13 cm, 14 cm, and 15 cm. Using Heron's formula, find its area.

Practice 11hard

In an isosceles triangle, the base is 16 cm and each of the equal sides is 10 cm. A line parallel to the base divides the triangle into two parts of equal area. Find the distance of this parallel line from the base (in cm).

Practice 12hard

In triangle PQR, the sides are PQ = 25 cm, QR = 29 cm, and PR = 36 cm. What is the area of the triangle (in cm²)?

Practice 13hard

Two triangles have the same height. The base of the first triangle is 12 cm and its area is 60 cm². The area of the second triangle is 80 cm². Find the base of the second triangle (in cm).

Practice 14hard

A triangle has sides 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 15hard

A triangle has sides 13 cm, 14 cm, and 15 cm. A perpendicular is drawn from the vertex opposite the 14 cm side to that side. Find 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
Circles — Area & Circumference→
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