Geometry basics

Geometry Basics – Study Notes

Program: NABARD Development Assistant
Subject: Quantitative Aptitude
Topic: Geometry Basics

1. Simple Explanation

Geometry studies the properties and relations of points, lines, angles, surfaces, and solids. In quantitative aptitude, we mainly use:

• Basic Euclidean figures (points, lines, triangles, quadrilaterals, circles).
• Perimeter (total boundary length).
• Area (space covered by a figure).
• Right‑angle relationships (Pythagoras theorem, trigonometric ratios).

Knowing a few standard formulas and being able to apply them quickly is key to solving most aptitude questions.

2. Key Points

• Area (base × height ÷ 2)
• Right‑angle: (c^2 = a^2 + b^2) | Most common figure in aptitude questions. | | Quadrilateral | Four sides. | - Rectangle: (P = 2(l+w)), (A = l \times w)
• Square: (P = 4a), (A = a^2) | Quick calculation of perimeter/area. | | Circle | All points equidistant from a center. | - Circumference = (2\pi r) or (\pi d)
• Area = (\pi r^2) | Used for circular objects or arcs. | | Pythagoras Theorem | In a right‑angle triangle, (a^2 + b^2 = c^2). | – | Find missing side/height. | | Trigonometric Ratios (Right‑angle) | (\sin = \frac{opposite}{hypotenuse}) etc. | – | Find missing angles or lengths. | | Similar Figures | Same shape, different sizes. | Ratios of corresponding sides are equal; areas scale as square of ratio. | Convert problems across sizes. |

3. Worked Example

Problem:
A rectangle has a length of 12 cm and a width that is 3 cm less than half its length. Find its perimeter and area.

Step‑by‑Step Solution

1. Find the width
   [ \text{Width} = \frac{\text{Length}}{2} - 3 = \frac{12}{2} - 3 = 6 - 3 = 3 \text{ cm} ]

2. Perimeter (P) – sum of all four sides
   [ P = 2(\text{Length} + \text{Width}) = 2(12 + 3) = 2 \times 15 = 30 \text{ cm} ]

3. Area (A) – length × width
   [ A = \text{Length} \times \text{Width} = 12 \times 3 = 36 \text{ cm}^2 ]

Answer:

• Perimeter = 30 cm
• Area = 36 cm²

4. Quick Revision Summary

• Perimeter:

  • Triangle: sum of 3 sides.
  • Rectangle: (2(l+w)).
  • Square: (4a).
  • Circle (circumference): (2\pi r).

• Area:

  • Triangle: ( \frac{1}{2} \times \text{base} \times \text{height}).
  • Rectangle: (l \times w).
  • Square: (a^2).
  • Circle: (\pi r^2).

• Pythagoras: (c^2 = a^2 + b^2) (right‑angle).

• Trigonometry (Right‑angle):
  [ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}},; \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}},; \tan\theta = \frac{\text{opposite}}{\text{adjacent}} ]

• Similar Figures:

  • Side ratios equal → area ratio = (side ratio)².

Quick Practice Questions (Try without looking at the solution!)

1. A circle has a diameter of 10 cm. Find its circumference and area.
2. A right‑angle triangle has legs of 9 cm and 12 cm. Find the hypotenuse.
3. A square’s perimeter is 48 cm. What is its area?
4. Two triangles are similar; one has sides 4 cm, 5 cm, 6 cm. The other has sides 8 cm, 10 cm, 12 cm. Confirm similarity by side ratios.

Tip: Always identify what the question asks for (perimeter, area, side, angle). Write down the relevant formula first, then substitute the given numbers. Practice a few problems each day to build speed and accuracy.