Geometry

“Geometry deals with the properties and relations of points, lines, surfaces, and solids.”

Key Concepts

Point: An exact location in space with no size or dimension.
Line: A straight one-dimensional figure having no thickness and extending infinitely in both directions.
Plane: A flat, two-dimensional surface that extends infinitely in all directions.
Angle: Formed by two rays (sides of the angle) sharing a common endpoint (vertex).
Triangle: A polygon with three edges and three vertices.
Circle: A set of all points in a plane that are at a given distance from a given point, the center.

Important Formulas

Area of Triangle: (1/2) × base × height
Area of Circle: πr²
Circumference of Circle: 2πr
Area of Rectangle: length × breadth
Area of Square: side²
Area of Parallelogram: base × height
Area of Trapezium: (1/2) × (sum of parallel sides) × height
Pythagoras Theorem: In a right-angled triangle, (Hypotenuse)² = (Base)² + (Height)²

Previous Year Questions with Explanations

SSC CGL 2020: The area of a triangle is 60 cm² and its base is 12 cm. Find its height.
Height = (2 × Area) / base = (2 × 60) / 12 = 10 cm
RRB JE 2019: Find the area of a circle with radius 7 cm.
Area = πr² = π × 7² = 154 cm²
SSC CHSL 2018: The perimeter of a square is 40 cm. Find its area.
Side = Perimeter / 4 = 10 cm; Area = side² = 100 cm²
HSSC 2021: The base and height of a parallelogram are 8 cm and 5 cm respectively. Find its area.
Area = base × height = 8 × 5 = 40 cm²
SSC GD 2023: The lengths of two sides of a right-angled triangle are 6 cm and 8 cm. Find the length of the hypotenuse.
Hypotenuse = √(6² + 8²) = √(36 + 64) = √100 = 10 cm
SSC CGL 2019: The diagonal of a rectangle is 13 cm and one side is 5 cm. Find the other side.
Other side = √(13² - 5²) = √(169 - 25) = √144 = 12 cm
RRB JE 2020: Find the circumference of a circle with diameter 14 cm.
Circumference = π × diameter = 3.14 × 14 = 43.96 cm
SSC CHSL 2021: The area of a trapezium is 72 cm², and the lengths of the parallel sides are 8 cm and 4 cm. Find its height.
Height = (2 × Area) / (sum of parallel sides) = (2 × 72) / (8 + 4) = 144 / 12 = 12 cm
HSSC JE 2022: A triangle has sides 7 cm, 24 cm, and 25 cm. Is it right angled?
Check: 7² + 24² = 49 + 576 = 625; 25² = 625 → Yes, it is a right-angled triangle.
SSC GD 2020: Find the area of a square whose diagonal is 10 cm.
Side = diagonal / √2 = 10 / 1.414 = 7.07 cm; Area = 7.07² = 50 cm² (approx.)
RRB ALP 2019: Find the area of a parallelogram with base 15 cm and height 6 cm.
Area = base × height = 15 × 6 = 90 cm²
SSC CGL 2018: The length and breadth of a rectangle are in the ratio 5:3 and its perimeter is 64 cm. Find its area.
Let sides be 5x and 3x; Perimeter = 2(5x + 3x) = 16x = 64 → x=4. Sides: 20 cm and 12 cm. Area = 20 × 12 = 240 cm²
HSSC 2020: Find the radius of a circle whose circumference is 31.4 cm.
Radius = Circumference / (2π) = 31.4 / (2 × 3.14) = 5 cm
SSC MTS 2021: A triangle has sides 3 cm, 4 cm, and 5 cm. Find its area.
Area = (1/2) × base × height = (1/2) × 3 × 4 = 6 cm²
RRB NTPC 2022: Find the area of a circle with circumference 44 cm.
Radius = Circumference / (2π) = 44 / (2 × 3.14) = 7 cm; Area = πr² = 3.14 × 7² = 153.86 cm²
SSC CHSL 2022: Find the length of the diagonal of a square whose area is 98 cm².
Side = √98 = 9.9 cm; Diagonal = side × √2 = 9.9 × 1.414 = 14 cm approx.
HSSC JE 2019: The base of a triangle is 10 cm, and the area is 50 cm². Find its height.
Height = (2 × Area) / base = (2 × 50) / 10 = 10 cm
SSC GD 2019: The sum of all angles of a polygon is 1260°. Find the number of sides.
Sum of interior angles = (n-2) × 180 = 1260 → n-2 = 7 → n = 9 sides
RRB JE 2021: Find the length of the hypotenuse of a right triangle if the other two sides are 9 cm and 12 cm.
Hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15 cm
SSC CGL 2021: The lengths of the parallel sides of a trapezium are 9 cm and 5 cm, and its height is 4 cm. Find the area.
Area = (1/2) × (sum of parallel sides) × height = (1/2) × (9 + 5) × 4 = 28 cm²

Note: Understanding the properties and formulas of different geometric shapes is crucial for solving problems efficiently.

Conclusion

Geometry is a fundamental branch of mathematics that deals with shapes, sizes, and the properties of space. Mastery of geometric principles is essential for success in various competitive exams.