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SSC CPO Triangles — Area & Properties

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This page covers SSC CPO Triangles — Area & Properties with complete concept notes, 13 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

13graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

A triangle has a base of 15 cm. If its area is 60 cm², what is the perpendicular height from the opposite vertex to the base?

Practice 2easy

An equilateral triangle has a side length of 10 cm. What is its area? (Use √3 ≈ 1.73)

Practice 3easy

A right-angled triangle has legs of 6 cm and 8 cm. What is the length of its hypotenuse?

Practice 4easy

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

Practice 5easy

A triangle has sides of 5 cm, 12 cm, and 13 cm. What is its area using Heron's formula?

Practice 6easy

The area of a triangle is 24 cm² and its base is 8 cm. What is the height of the triangle?

Practice 7medium

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

Practice 8medium

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

Practice 9hard

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 10hard

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, find the length of DC (in cm).

Practice 11hard

In triangle PQR, the medians from P and Q intersect at the centroid G. If the median from P has length 18 cm, what is the distance PG (in cm)?

Practice 12hard

A triangle has vertices at A(0, 0), B(12, 0), and C(6, 8). A line parallel to AB passes through the centroid of the triangle and intersects AC at point P and BC at point Q. What is the length of PQ (in cm)?

Practice 13hard

In triangle ABC, the sides are in the ratio 3:4:5. If the area of the triangle is 96 cm², what is the length of the longest side (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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