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Geometry

Practice MCQs

Geometry is an important quantitative aptitude topic that deals with shapes, sizes, angles, lines, triangles, circles, quadrilaterals, area, perimeter, surface area, and volume.

Quantitative Aptitude Geometry Competitive Exams

Geometry is an important quantitative aptitude topic that deals with shapes, sizes, angles, lines, triangles, circles, quadrilaterals, area, perimeter, surface area, and volume. It is commonly asked in competitive exams, aptitude tests, school-level mathematics, and entrance exams.

What is Geometry?

Geometry is the branch of mathematics that studies figures and spaces. It includes two-dimensional shapes such as triangles, rectangles, squares, and circles, and three-dimensional solids such as cubes, cuboids, cylinders, cones, and spheres.

In competitive exams, geometry questions usually test your understanding of formulas, properties of shapes, angle relations, and diagram-based reasoning.

Quick idea: First identify the shape, then write the known values, then apply the correct formula.
Concept Meaning Example
Perimeter Total boundary length of a 2D shape Fence around a field
Area Space covered by a 2D shape Floor area of a room
Surface Area Total outer area of a 3D solid Painting a box
Volume Space occupied by a 3D solid Water inside a tank

“Geometry becomes easy when the shape is identified correctly and the formula is applied carefully.”

Aptitude Tip
Key points
  • Identify whether the figure is 2D or 3D.
  • Use perimeter for boundary length.
  • Use area for flat surface measurement.
  • Use volume for space inside a solid.
  • Angle properties are very important in triangles.
  • Always keep units consistent.
area perimeter angles volume

Basic Geometry Diagrams

The following diagrams show commonly used shapes in geometry questions.

Triangle
Base = b h A B C
\[ A = \frac{1}{2} \times b \times h \]

Area of a triangle depends on base and height.

Circle
r Radius
\[ A = \pi r^2 \]

Area of a circle depends on its radius.

Rectangle
Length = l b Rectangle
\[ A = l \times b \]

Area of a rectangle is length multiplied by breadth.

Cube
Side = a
\[ V = a^3 \]

Volume of a cube depends on the cube of its side.

Important Geometry Formulas

Square
\[ A = a^2 \]
\[ P = 4a \]

Here, \(a\) is the side of the square.

Rectangle
\[ A = l \times b \]
\[ P = 2(l+b) \]

Here, \(l\) is length and \(b\) is breadth.

Triangle
\[ A = \frac{1}{2}bh \]
\[ \angle A + \angle B + \angle C = 180^\circ \]

The sum of angles in a triangle is \(180^\circ\).

Circle
\[ A = \pi r^2 \]
\[ C = 2\pi r \]

Here, \(r\) is the radius.

Cube
\[ V = a^3 \]
\[ TSA = 6a^2 \]

All edges of a cube are equal.

Cuboid
\[ V = lbh \]
\[ TSA = 2(lb+bh+lh) \]

Here, \(l\), \(b\), and \(h\) are length, breadth, and height.

Cylinder
\[ V = \pi r^2 h \]
\[ CSA = 2\pi rh \]

Cylinder has circular base and height.

Sphere
\[ V = \frac{4}{3}\pi r^3 \]
\[ SA = 4\pi r^2 \]

Here, \(r\) is the radius of the sphere.

Rule: Area is measured in square units, while volume is measured in cubic units.

Common Types of Geometry Questions

Area-Based Questions

Find the area of 2D figures.

  • Square
  • Rectangle
  • Triangle
  • Circle
Perimeter-Based Questions

Find total boundary length.

  • Square perimeter
  • Rectangle perimeter
  • Triangle perimeter
  • Circle circumference
Angle-Based Questions

Use angle properties.

  • Triangle angles
  • Straight angle
  • Complementary angles
  • Supplementary angles
Volume Questions

Find space occupied by solids.

  • Cube
  • Cuboid
  • Cylinder
  • Sphere
Exam approach: Identify whether the problem is based on area, perimeter, angle, surface area, or volume before using a formula.
Geometry Method Bank
Rectangle Area
\[ A = l \times b \]

Multiply length and breadth.

Circle Area
\[ A = \pi r^2 \]

Use radius, not diameter.

Triangle Angle Sum
\[ A+B+C = 180^\circ \]

Useful in angle problems.

Cube Volume
\[ V = a^3 \]

Side multiplied three times.

Tip: In circle questions, if diameter is given, first find radius using \(r = \frac{d}{2}\).

Cylinder Diagram
r h Cylinder
A cylinder has a circular base of radius \(r\) and height \(h\).
\[ V = \pi r^2h \]
\[ TSA = 2\pi r(r+h) \]

Step-by-Step Solving Method

Step Action Example
Step 1 Identify the shape. Rectangle, circle, triangle, cube, or cylinder.
Step 2 Write the given values. Length = 12 cm, breadth = 5 cm.
Step 3 Select the correct formula. \(A = l \times b\)
Step 4 Substitute values carefully. \(A = 12 \times 5\)
Step 5 Write answer with correct unit. \(A = 60 \text{ cm}^2\)
Important: Area answers use square units like \(\text{cm}^2\), while volume answers use cubic units like \(\text{cm}^3\).

Solved Examples

Question Method Answer
Find the area of a rectangle with length 12 cm and breadth 5 cm.
\[ A = l \times b \]
\[ A = 12 \times 5 = 60 \]
60 cm²
Find the perimeter of a square with side 8 cm.
\[ P = 4a \]
\[ P = 4 \times 8 = 32 \]
32 cm
Find the area of a triangle with base 10 cm and height 6 cm.
\[ A = \frac{1}{2}bh \]
\[ A = \frac{1}{2} \times 10 \times 6 = 30 \]
30 cm²
Find the area of a circle with radius 7 cm. Use \(\pi = \frac{22}{7}\).
\[ A = \pi r^2 \]
\[ A = \frac{22}{7} \times 7 \times 7 = 154 \]
154 cm²
Find the volume of a cube with side 5 cm.
\[ V = a^3 \]
\[ V = 5^3 = 125 \]
125 cm³
Find the volume of a cuboid with length 8 cm, breadth 4 cm, and height 3 cm.
\[ V = lbh \]
\[ V = 8 \times 4 \times 3 = 96 \]
96 cm³
Find the missing angle of a triangle if two angles are \(50^\circ\) and \(70^\circ\).
\[ A+B+C = 180^\circ \]
\[ C = 180^\circ - 50^\circ - 70^\circ = 60^\circ \]
60°
Find the volume of a cylinder with radius 7 cm and height 10 cm. Use \(\pi = \frac{22}{7}\).
\[ V = \pi r^2h \]
\[ V = \frac{22}{7} \times 7 \times 7 \times 10 = 1540 \]
1540 cm³

Note: Always check whether the question asks for perimeter, area, surface area, or volume.

Common Traps and Shortcuts

Common Traps
  • Using diameter instead of radius in circle formulas.
  • Writing area units instead of volume units.
  • Confusing perimeter with area.
  • Forgetting the factor \(\frac{1}{2}\) in triangle area.
  • Using wrong formula for curved surface area and total surface area.
  • Ignoring unit conversion before calculation.
Useful Shortcuts
  • If diameter is given, use \(r = \frac{d}{2}\).
  • Area uses square units.
  • Volume uses cubic units.
  • For rectangle, area is \(l \times b\).
  • For cube, volume is side cubed.
  • Triangle angle sum is always \(180^\circ\).
Exam approach: Identify the figure first, then decide whether the question asks for perimeter, area, surface area, volume, or angle.

Practice

A) Multiple Choice Questions
  1. Find the area of a square with side 9 cm.
    72 cm² 81 cm² 90 cm² 99 cm²
  2. Find the perimeter of a rectangle with length 10 cm and breadth 6 cm.
    16 cm 26 cm 32 cm 60 cm
  3. Find the area of a triangle with base 12 cm and height 8 cm.
    36 cm² 48 cm² 60 cm² 96 cm²
  4. Find the volume of a cube with side 4 cm.
    16 cm³ 48 cm³ 64 cm³ 80 cm³
  5. Two angles of a triangle are \(65^\circ\) and \(45^\circ\). Find the third angle.
    60° 70° 80° 90°
B) Solve the Higher-Order Problems
  1. Find the area of a circle with radius 14 cm. Use \(\pi = \frac{22}{7}\). (Hint: \(A = \pi r^2\).)
  2. Find the volume of a cuboid with length 12 cm, breadth 5 cm, and height 4 cm. (Hint: \(V = lbh\).)
  3. A rectangle has area 96 cm² and length 12 cm. Find its breadth. (Hint: \(A = l \times b\).)
  4. Find the curved surface area of a cylinder with radius 7 cm and height 20 cm. Use \(\pi = \frac{22}{7}\). (Hint: \(CSA = 2\pi rh\).)
  5. The perimeter of a square is 48 cm. Find its area. (Hint: First find side using \(P = 4a\).)
C) Match the Shape with the Correct Formula
Shape / Concept Correct Formula
Square Area \(A = a^2\)
Rectangle Area \(A = l \times b\)
Triangle Area \(A = \frac{1}{2}bh\)
Circle Area \(A = \pi r^2\)
Cube Volume \(V = a^3\)
Cylinder Volume \(V = \pi r^2h\)
Geometry Reminder

Geometry questions become simple when the figure is identified correctly. Always decide whether the question asks for perimeter, area, surface area, volume, or angle. Use the correct formula and write the answer with proper units.

Task: Create five geometry questions using square, rectangle, triangle, circle, and cylinder formulas.

Show Suggested Answers
Multiple Choice
  1. 81 cm²
    \[ A = a^2 = 9^2 = 81 \]
  2. 32 cm
    \[ P = 2(l+b) = 2(10+6) = 32 \]
  3. 48 cm²
    \[ A = \frac{1}{2} \times 12 \times 8 = 48 \]
  4. 64 cm³
    \[ V = a^3 = 4^3 = 64 \]
  5. 70°
    \[ \text{Third Angle} = 180^\circ - 65^\circ - 45^\circ = 70^\circ \]
Higher-Order Problems
  1. Circle radius \(r = 14\) cm:
    \[ A = \frac{22}{7} \times 14 \times 14 = 616 \]
    Answer = 616 cm².
  2. Cuboid dimensions: \(l = 12\), \(b = 5\), \(h = 4\):
    \[ V = lbh = 12 \times 5 \times 4 = 240 \]
    Answer = 240 cm³.
  3. Rectangle area \(A = 96\), length \(l = 12\):
    \[ b = \frac{A}{l} = \frac{96}{12} = 8 \]
    Answer = 8 cm.
  4. Cylinder radius \(r = 7\), height \(h = 20\):
    \[ CSA = 2 \times \frac{22}{7} \times 7 \times 20 = 880 \]
    Answer = 880 cm².
  5. Perimeter of square \(P = 48\):
    \[ a = \frac{48}{4} = 12 \]
    \[ A = a^2 = 12^2 = 144 \]
    Answer = 144 cm².
Concept Matching
  1. Square Area → \(A = a^2\)
  2. Rectangle Area → \(A = l \times b\)
  3. Triangle Area → \(A = \frac{1}{2}bh\)
  4. Circle Area → \(A = \pi r^2\)
  5. Cube Volume → \(V = a^3\)
  6. Cylinder Volume → \(V = \pi r^2h\)
Clue Explanation

Geometry requires formula selection based on the shape and the quantity asked. Perimeter is boundary, area is surface covered, and volume is space occupied.

Exam tips
  • Draw a rough diagram if needed.
  • Check whether radius or diameter is given.
  • Use square units for area.
  • Use cubic units for volume.
  • Remember triangle angle sum is \(180^\circ\).
  • Do not confuse CSA and TSA.
Practice MCQs