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Geometry, Mensuration and Area Short Notes for GATE CS

Geometry, Mensuration and Area short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

geometry mensuration and area short notes

Quick Revision: Similarity and Area

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Quick Revision: Area Computation

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Quick-Recall: Solid Geometry Formula Sheet

Solid Volume Surface Cuboid l bh LSA = 2 h (l + b), TSA = 2 (l b + bh + h l) Cube a^{3} TSA = 6 a^{2}, diagonal = a 3 Cylinder π r^{2} h CSA = 2 π r h, TSA = 2 π r (r + h) Cone \frac{1}{3} π r^{2} h CSA = π r l, TSA = π r (r + l) Sphere \frac{4}{3} π r^{3} SA = 4 π r^{2} Hemisphere \frac{2}{3} π r^{3} CSA = 2 π r^{2}, TSA = 3 π r^{2} Cube diagonal angle. cos θ = \frac{1}{3} with any edge. Sheet \to cylinder. Joined edges \to height; other edge \to circumference. Sheet \to closed cube. 6 a^{2} = sheet area .

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Question 1

Level 1: Warm-up

Assertion (A): If a triangle is half the area of a square and also half the area of a rectangle, then the square and rectangle have equal areas. Reason (R): The area of a triangle is always measured in cubic units. Choose the correct option:

Question 2

Level 1: Warm-up

Assertion (A): If a triangle and a parallelogram share the same base and lie between the same parallel lines, the area of the triangle is half the area of the parallelogram. Reason (R): The area of a triangle is measured in cubic units when the base and height are in linear units. Choose the correct option:

Question 3

Level 1: Warm-up

Assertion (A): The angle θ between the body diagonal of a cube and any of its edges satisfies cos θ = \frac{1}{3} . Reason (R): The body diagonal of a cube with edge a has a length of a 3, which is measured in cubic units. Choose the correct option:

Question 4

Level 1: Warm-up

Consider the statements about the area under a step function: f (x) = 4 for 0 \leq x < 1 f (x) = 2 for 1 \leq x \leq 3 Statement P: The width of the second rectangular region is 3 units. Statement Q: The total area under f (x) from x = 0 to x = 3 is 8 square units. Which of the following is correct?

Question 5

Level 1: Warm-up

Consider a step function g (x) defined as g (x) = 3 for 0 \leq x < 2, and g (x) = 1 for 2 \leq x \leq 5 . Statement P: The total area under the curve from x = 0 to x = 5 is 9 square units. Statement Q: The width of the second rectangular region is 5 units. Which of the following is correct?

Question 6

Level 1: Warm-up

Consider a cube of edge length a . Statement P: The ratio of the body diagonal to the edge length is 3 : 1 . Statement Q: The total surface area of the cube is 4 a^{2} . Which of the following is correct?

Question 7

Level 1: Warm-up

Two similar triangles have a scale factor of 1:3. The area of the smaller triangle is 5 cm². A student claims the area of the larger triangle is 15 cm². What is the minimum area (in cm²) the larger triangle must actually have?

Question 8

Level 1: Warm-up

A rectangular garden of 10 m by 8 m has a rectangular pond of 4 m by 3 m inside it. To minimize the remaining area of the garden, the pond must lie entirely within it. What is this minimum remaining area of the garden (in m²)?

Question 9

Level 1: Warm-up

What is the minimum number of centimeters needed to draw a line connecting two opposite vertices of a cuboid with dimensions 3 cm, 4 cm, and 12 cm?

Question 10

Level 1: Warm-up

A solid toy is made by mounting a hemisphere of radius 7 cm exactly on top of a solid cylinder of base radius 7 cm and height 10 cm. What is the total exposed surface area of the toy (in cm², use π = \frac{22}{7})?

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Geometry, Mensuration and Area Short Notes for GATE CS

Geometry, Mensuration and Area short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Quick Revision: Similarity and Area

Similarity Same shape, different size. Corresponding angles equal, sides in proportion.

Scale Factor (

k

) Ratio of any corresponding linear dimensions (sides, heights, perimeters).

Area Ratio Always

k^{2}

Nested Triangles A line parallel to the base creates a small triangle similar to the large triangle.

Trapezium Area Area of large triangle minus area of small triangle.

Golden Rule Never mix linear ratios with area ratios without squaring.

Quick Revision: Area Computation

Additivity Area of a complex shape = Sum of areas of its non-overlapping standard parts.

Decomposition Use auxiliary lines to split irregular shapes into rectangles and triangles.

Inclusion-Exclusion

Area (A \cup B) = Area (A) + Area (B) - Area (A \cap B)

Geometric Invariance If a triangle is half of Shape X and half of Shape Y, then

Area (X) = Area (Y)

Step Functions Area under a piecewise constant curve = Sum of (width

\times

height) for each rectangular segment.

Golden Rule Never double count an overlapping region.

Quick-Recall: Solid Geometry Formula Sheet

Solid	Volume	Surface
Cuboid	$l bh$	LSA $= 2 h (l + b)$ , TSA $= 2 (l b + bh + h l)$
Cube	$a^{3}$	TSA $= 6 a^{2}$ , diagonal $= a 3$
Cylinder	$π r^{2} h$	CSA $= 2 π r h$ , TSA $= 2 π r (r + h)$
Cone	$\frac{1}{3} π r^{2} h$	CSA $= π r l$ , TSA $= π r (r + l)$
Sphere	$\frac{4}{3} π r^{3}$	SA $= 4 π r^{2}$
Hemisphere	$\frac{2}{3} π r^{3}$	CSA $= 2 π r^{2}$ , TSA $= 3 π r^{2}$

Cube diagonal angle.

cos θ = \frac{1}{3}

with any edge.
Sheet → cylinder. Joined edges → height; other edge → circumference.
Sheet → closed cube.

6 a^{2} = sheet area

Geometry, Mensuration and Area: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Quantitative Aptitude MCQ

Assertion (A): If a triangle is half the area of a square and also half the area of a rectangle, then the square and rectangle have equal areas.

Reason (R): The area of a triangle is always measured in cubic units.

Choose the correct option:

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: This is a geometric invariance problem combined with a units check. Step 1: Analyze Assertion (A). - Let the triangle's area be

T

. - Triangle is half the square:

T = \frac{1}{2} \times Area_{square}

, so

Area_{square} = 2 T

. - Triangle is half the rectangle:

T = \frac{1}{2} \times Area_{rectangle}

, so

Area_{rectangle} = 2 T

. - Therefore,

Area_{square} = Area_{rectangle} = 2 T

. - Assertion (A) is true. Step 2: Analyze Reason (R). - Area is a two-dimensional measurement. - Area is measured in square units (e.g., cm², m²), not cubic units. - Cubic units are for volume (three-dimensional). - Reason (R) is false. Answer: A is true but R is false

Question 2 · Quantitative Aptitude MCQ

Assertion (A): If a triangle and a parallelogram share the same base and lie between the same parallel lines, the area of the triangle is half the area of the parallelogram.

Reason (R): The area of a triangle is measured in cubic units when the base and height are in linear units.

Choose the correct option:

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: This is an assertion-reason question testing geometric properties and the correct units for area.

Step 1: Analyze Assertion (A).

A triangle and a parallelogram on the same base and between the same parallels have a known area relationship.
$Area (Parallelogram) = base \times height$ .
$Area (Triangle) = \frac{1}{2} \times base \times height$ .
Therefore, the triangle's area is exactly half of the parallelogram's area.
Assertion (A) is true.

Step 2: Analyze Reason (R).

Area is a two-dimensional measurement.
When linear dimensions (like base and height) are multiplied, the resulting unit is squared (e.g., m², cm²).
Cubic units (e.g., m³) are used for volume, a three-dimensional measurement.
Reason (R) is false.

Answer: A is true but R is false

Question 3 · Quantitative Aptitude MCQ

Assertion (A): The angle $θ$ between the body diagonal of a cube and any of its edges satisfies $cos θ = \frac{1}{3}$ .

Reason (R): The body diagonal of a cube with edge $a$ has a length of $a 3$ , which is measured in cubic units.

Choose the correct option:

A.
Both A and R are true and R is the correct explanation of A
B.
Both A and R are true but R is NOT the correct explanation of A
C.
A is true but R is false
D.
A is false but R is true

Correct Answer:

Step-by-Step Solution

Key idea: This is an assertion-reason question testing both a geometric formula and the fundamental concept of physical units.

Step 1: Analyze Assertion (A).

In a cube of edge $a$ , the body diagonal is $a 3$ .
The angle $θ$ with an edge forms a right triangle where the edge $a$ is adjacent and the body diagonal is the hypotenuse.
$cos θ = \frac{adjacent}{hypotenuse} = \frac{a}{a 3} = \frac{1}{3}$ .
Assertion (A) is true.

Step 2: Analyze Reason (R).

The length of the body diagonal is indeed $a 3$ .
However, length is a 1-dimensional measurement. It must be measured in linear units (e.g., cm, m), not cubic units.
Cubic units (e.g., cm³) are strictly for volume.
Reason (R) is false.

Answer: A is true but R is false

Question 4 · Quantitative Aptitude MCQ

Consider the statements about the area under a step function:

$f (x) = 4$ for $0 \leq x < 1$

$f (x) = 2$ for $1 \leq x \leq 3$

Statement P: The width of the second rectangular region is 3 units.

Statement Q: The total area under $f (x)$ from $x = 0$ to $x = 3$ is 8 square units.

Which of the following is correct?

A.
Only P is true
B.
Only Q is true
C.
Both P and Q are true
D.
Neither P nor Q is true

Correct Answer:

Step-by-Step Solution

Key idea: For a step function, the area under the curve is the sum of rectangular areas. Each rectangle's width is the length of its interval.

Step 1: Analyze Statement P.

The second piece is defined for $1 \leq x \leq 3$ .
Width = upper bound - lower bound = $3 - 1 = 2$ units.
Statement P claims width is 3 units, which is false.

Step 2: Analyze Statement Q.

First rectangle: width = $1 - 0 = 1$ , height = 4. Area = $1 \times 4 = 4$ .
Second rectangle: width = $3 - 1 = 2$ , height = 2. Area = $2 \times 2 = 4$ .
Total area = $4 + 4 = 8$ square units.
Statement Q is true.

Answer: Only Q is true

Question 5 · Quantitative Aptitude MCQ

Consider a step function $g (x)$ defined as $g (x) = 3$ for $0 \leq x < 2$ , and $g (x) = 1$ for $2 \leq x \leq 5$ .

Statement P: The total area under the curve from $x = 0$ to $x = 5$ is 9 square units.

Statement Q: The width of the second rectangular region is 5 units.

Which of the following is correct?

A.
Only P is true
B.
Only Q is true
C.
Both P and Q are true
D.
Neither P nor Q is true

Correct Answer:

Step-by-Step Solution

Key idea: The area under a step function is the sum of the areas of the individual rectangular blocks.

Step 1: Analyze the first rectangular region.

Interval: $0 \leq x < 2$ .
Width = $2 - 0 = 2$ units.
Height = $g (x) = 3$ .
Area = $2 \times 3 = 6$ square units.

Step 2: Analyze the second rectangular region.

Interval: $2 \leq x \leq 5$ .
Width = $5 - 2 = 3$ units.
Height = $g (x) = 1$ .
Area = $3 \times 1 = 3$ square units.

Step 3: Evaluate Statement P.

Total area = $6 + 3 = 9$ square units.
Statement P is true.

Step 4: Evaluate Statement Q.

The width of the second region is $5 - 2 = 3$ units, not 5.
Statement Q is false.

Answer: Only P is true

Question 6 · Quantitative Aptitude MCQ

Consider a cube of edge length $a$ .

Statement P: The ratio of the body diagonal to the edge length is $3 : 1$ .

Statement Q: The total surface area of the cube is $4 a^{2}$ .

Which of the following is correct?

A.
Only P is true
B.
Only Q is true
C.
Both P and Q are true
D.
Neither P nor Q is true

Correct Answer:

Step-by-Step Solution

Key idea: This is a statement truth problem testing the specific properties and formulas of a cube.

Step 1: Analyze Statement P.

The body diagonal of a cube is $a 3$ .
The edge length is $a$ .
The ratio is $a 3 : a$ , which simplifies to $3 : 1$ .
Statement P is true.

Step 2: Analyze Statement Q.

A cube has 6 identical square faces.
The area of one face is $a^{2}$ .
The Total Surface Area (TSA) is $6 \times a^{2} = 6 a^{2}$ .
Statement Q claims the TSA is $4 a^{2}$ (which is actually the Lateral Surface Area).
Statement Q is false.

Answer: Only P is true

Question 7 · Quantitative Aptitude MCQ

Two similar triangles have a scale factor of 1:3. The area of the smaller triangle is 5 cm². A student claims the area of the larger triangle is 15 cm². What is the minimum area (in cm²) the larger triangle must actually have?

A.
15
B.
25
C.
45
D.
135

Correct Answer:

Step-by-Step Solution

Key idea: The ratio of areas of similar triangles equals the square of the ratio of their corresponding linear dimensions.

Step 1: Linear scale factor = $1 : 3$ , so $k = 3$ (larger to smaller).

Step 2: Area ratio = $k^{2} = 3^{2} = 9$ .

Step 3: Area of larger triangle = Area of smaller $\times 9$ .

Step 4: Area = $5 \times 9 = 45$ cm².

The student's claim of 15 cm² is incorrect because they multiplied by 3 (the linear factor) instead of 9 (the area factor).

Answer: 45 cm²

Question 8 · Quantitative Aptitude MCQ

A rectangular garden of 10 m by 8 m has a rectangular pond of 4 m by 3 m inside it. To minimize the remaining area of the garden, the pond must lie entirely within it. What is this minimum remaining area of the garden (in m²)?

A.
92
B.
68
C.
80
D.
12

Correct Answer:

Step-by-Step Solution

Key idea: This is a geometric decomposition problem where the area of a hole must be subtracted from the total bounding area.

Step 1: Calculate the total area of the rectangular garden. $Area_{garden} = 10 \times 8 = 80$ m².

Step 2: Calculate the area of the rectangular pond. $Area_{pond} = 4 \times 3 = 12$ m².

Step 3: To minimize the remaining area of the garden, the pond must be fully inside the garden, meaning its entire area is removed.

Step 4: Subtract the pond's area from the garden's total area: $Remaining Area = 80 - 12 = 68$ m².

Answer: 68

Question 9 · Quantitative Aptitude MCQ

What is the minimum number of centimeters needed to draw a line connecting two opposite vertices of a cuboid with dimensions 3 cm, 4 cm, and 12 cm?

A.
13
B.
12
C.
5
D.
144

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction problem where students might mistakenly pick a face diagonal or an edge. The shortest path through the interior connecting opposite vertices is the body diagonal.

Step 1: Identify the dimensions: $l = 3$ , $b = 4$ , $h = 12$ .

Step 2: Recall the formula for the body (longest) diagonal of a cuboid: $d = l^{2} + b^{2} + h^{2}$ .

Step 3: Substitute the values: $d = 3^{2} + 4^{2} + 1 2^{2}$ .

Step 4: Calculate the squares: $d = 9 + 16 + 144 = 169$ .

Step 5: Evaluate the square root: $d = 13$ cm.

Answer: 13

Question 10 · Quantitative Aptitude MCQ

A solid toy is made by mounting a hemisphere of radius 7 cm exactly on top of a solid cylinder of base radius 7 cm and height 10 cm. What is the total exposed surface area of the toy (in cm², use $π = \frac{22}{7}$ )?

A.
1056
B.
902
C.
748
D.
594

Correct Answer:

Step-by-Step Solution

Key idea: This is a casework problem involving the surface area of a composite solid. The touching faces are hidden and must not be counted.

Step 1: Identify the exposed surfaces. The toy consists of a cylinder and a hemisphere. The flat base of the hemisphere and the top circular face of the cylinder are joined together and are therefore hidden.

Step 2: The total exposed surface area is the sum of:

Curved Surface Area (CSA) of the cylinder.
Curved Surface Area (CSA) of the hemisphere.
Area of the bottom circular base of the cylinder.

Step 3: Calculate the CSA of the cylinder: $2 π r h = 2 \times \frac{22}{7} \times 7 \times 10 = 440$ cm².

Step 4: Calculate the CSA of the hemisphere: $2 π r^{2} = 2 \times \frac{22}{7} \times 7^{2} = 308$ cm².

Step 5: Calculate the area of the bottom base: $π r^{2} = \frac{22}{7} \times 7^{2} = 154$ cm².

Step 6: Sum the exposed areas: $440 + 308 + 154 = 902$ cm².

Answer: 902

geometry mensuration and area short notes

Quick Revision: Similarity and Area

Quick Revision: Area Computation

Quick-Recall: Solid Geometry Formula Sheet

Try a question

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Geometry, Mensuration and Area Short Notes for GATE CS

Quick Revision: Similarity and Area

Quick Revision: Area Computation

Quick-Recall: Solid Geometry Formula Sheet

Geometry, Mensuration and Area: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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