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Agniveer Army CEE Triangles — Area & Properties

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This page covers Agniveer Army CEE Triangles — Area & Properties with complete concept notes, 8 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

Sum of all angles = 180°

Sum of any two sides > third side

Exterior angle = sum of two opposite interior angles

In right triangle: a² + b² = c² (Pythagoras theorem)

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

8graded MCQs · easy to hard · full solution & trap analysis

All MCQs →

Practice 1easy

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

Practice 2easy

In a right-angled triangle, the two legs are 5 cm and 12 cm. What is the length of the hypotenuse?

Practice 3easy

An equilateral triangle has a side length of 6 cm. What is the sum of all its interior angles?

Practice 4medium

An equilateral triangle has a side length of 10 cm. What is its area? (Use: Area = (√3/4) × side²)

Practice 5medium

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

Practice 6medium

In a right-angled triangle, the two perpendicular sides are 6 cm and 9 cm. Find the area of the triangle.

Practice 7medium

A triangle has sides of length 5 cm, 5 cm, and 6 cm. Using Heron's formula, find its area. (Note: s = 8 cm)

Practice 8hard

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

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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