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SSC MTS Triangles — Area & Properties

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This page covers SSC MTS Triangles — Area & Properties with complete concept notes, 14 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

14graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

An equilateral triangle has a side length of 6 cm. What is its perimeter?

Practice 2easy

A right-angled triangle has legs of length 5 cm and 12 cm. What is the length of its hypotenuse?

Practice 3easy

A triangle has a base of 12 cm and a height of 8 cm. What is its area?

Practice 4easy

An isosceles triangle has two equal sides of 10 cm each and a base of 12 cm. What is its perimeter?

Practice 5easy

The area of a triangle is 60 cm² and its base is 15 cm. What is the height of the triangle?

Practice 6medium

A triangle has sides 5 cm, 5 cm, and 6 cm. What is the length of the altitude drawn to the side of length 6 cm?

Practice 7medium

In triangle ABC, the altitude from vertex A to side BC is 12 cm. If the area of the triangle is 96 cm², find the length of BC.

Practice 8medium

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?

Practice 9medium

An equilateral triangle has a side length of 8 cm. What is the ratio of its area to its perimeter (in cm²:cm)?

Practice 10medium

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

Practice 11hard

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 12hard

In triangle ABC, the angle bisector from vertex A meets side BC at point D. If AB = 18 cm, AC = 24 cm, and BC = 21 cm, what is the ratio BD:DC?

Practice 13hard

In triangle ABC, the medians AD, BE, and CF intersect at centroid G. If the area of triangle ABC is 72 cm², what is the area of triangle BGC (in cm²)?

Practice 14hard

A triangle has sides 25 cm, 39 cm, and 40 cm. What is the area of the triangle (in cm²)?

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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