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Area, Volume and Surface Area

Practice MCQs

None

Mathematics Mensuration Area, Volume & Surface Area

Area, Volume and Surface Area are important topics in mensuration. They help us measure flat surfaces, space occupied by solid objects, and the total outer surface of three-dimensional shapes.

What are Area, Volume and Surface Area?

Area is the measure of the space covered by a two-dimensional figure. It is measured in square units such as \(cm^2\), \(m^2\), or \(km^2\).

Volume is the measure of the space occupied by a three-dimensional object. It is measured in cubic units such as \(cm^3\), \(m^3\), or \(litres\).

Surface Area is the total area of all the outer surfaces of a three-dimensional object.

Quick idea: Area is for flat shapes, volume is for solid shapes, and surface area is the total outer covering of a solid.
Concept Meaning Common Unit
Area Space covered by a flat figure \(cm^2, m^2\)
Volume Space occupied by a solid object \(cm^3, m^3\)
Surface Area Total area of the outer surface of a solid \(cm^2, m^2\)
Perimeter Total distance around a flat figure \(cm, m\)

“Mensuration helps us measure shapes in daily life, construction, packaging, farming, and engineering.”

Mathematics Tip
Key points
  • Area is measured in square units.
  • Volume is measured in cubic units.
  • Surface area is also measured in square units.
  • Always use the same unit before applying a formula.
  • Write final answers with correct units.
  • Identify the shape before choosing the formula.
area volume surface area mensuration

Units Used in Mensuration

Units are very important in area, volume, and surface area problems. A correct numerical answer without the correct unit is incomplete.

Length Units

Used for sides, radius, height, length, breadth, and perimeter.

  • \(mm\)
  • \(cm\)
  • \(m\)
  • \(km\)
Area Units

Used for flat surfaces and surface area.

  • \(cm^2\)
  • \(m^2\)
  • \(km^2\)
  • \(hectare\)
Volume Units

Used for solid shapes and capacity.

  • \(cm^3\)
  • \(m^3\)
  • \(litre\)
  • \(1 litre = 1000 cm^3\)
Rule: Do not mix units. Convert all measurements to the same unit before solving.

Area of Common Plane Figures

Area formulas are used to find the space covered by two-dimensional figures.

Shape Formula Meaning of Symbols
Square \(Area = a^2\) \(a\) = side
Rectangle \(Area = l \times b\) \(l\) = length, \(b\) = breadth
Triangle \(Area = \frac{1}{2} \times b \times h\) \(b\) = base, \(h\) = height
Parallelogram \(Area = b \times h\) \(b\) = base, \(h\) = height
Circle \(Area = \pi r^2\) \(r\) = radius
Trapezium \(Area = \frac{1}{2}(a+b)h\) \(a,b\) = parallel sides, \(h\) = height
Important: For circle-based questions, use \(\pi = \frac{22}{7}\) or \(\pi = 3.14\) as given in the question.

Volume and Surface Area of Solid Figures

Solid figures have length, breadth, and height. Their volume measures the space inside them, while surface area measures the total outer covering.

Solid Volume Total Surface Area Curved / Lateral Surface Area
Cube \(V = a^3\) \(TSA = 6a^2\) \(LSA = 4a^2\)
Cuboid \(V = lbh\) \(TSA = 2(lb + bh + lh)\) \(LSA = 2h(l+b)\)
Cylinder \(V = \pi r^2h\) \(TSA = 2\pi r(r+h)\) \(CSA = 2\pi rh\)
Cone \(V = \frac{1}{3}\pi r^2h\) \(TSA = \pi r(l+r)\) \(CSA = \pi rl\)
Sphere \(V = \frac{4}{3}\pi r^3\) \(SA = 4\pi r^2\) Same as total surface area
Hemisphere \(V = \frac{2}{3}\pi r^3\) \(TSA = 3\pi r^2\) \(CSA = 2\pi r^2\)

Here, \(a\) = side, \(l\) = length, \(b\) = breadth, \(h\) = height, \(r\) = radius, and \(l\) in cone also means slant height.

Total Surface Area vs Curved Surface Area
Total Surface Area
Total area of all the surfaces of a solid object. For a cylinder: \(TSA = 2\pi r(r+h)\)
Curved Surface Area
Area of only the curved part of a solid object. For a cylinder: \(CSA = 2\pi rh\)
Lateral Surface Area
Area of side faces, usually excluding top and bottom faces. For a cuboid: \(LSA = 2h(l+b)\)

Tip: If painting the entire solid, use total surface area. If wrapping only the side, use curved or lateral surface area.

Area volume surface area concept
Visual placeholder. Replace this with your own mensuration concept image if needed.

Solved Examples

Question Method Answer
Find the area of a square of side \(6 cm\). \(Area = a^2 = 6^2\) \(36 cm^2\)
Find the area of a rectangle of length \(12 m\) and breadth \(5 m\). \(Area = l \times b = 12 \times 5\) \(60 m^2\)
Find the area of a triangle with base \(10 cm\) and height \(8 cm\). \(Area = \frac{1}{2} \times 10 \times 8\) \(40 cm^2\)
Find the volume of a cube of side \(4 cm\). \(V = a^3 = 4^3\) \(64 cm^3\)
Find the volume of a cuboid with \(l = 8 cm\), \(b = 5 cm\), \(h = 3 cm\). \(V = lbh = 8 \times 5 \times 3\) \(120 cm^3\)
Find the curved surface area of a cylinder with \(r = 7 cm\) and \(h = 10 cm\). \(CSA = 2\pi rh = 2 \times \frac{22}{7} \times 7 \times 10\) \(440 cm^2\)
Find the total surface area of a cube of side \(5 cm\). \(TSA = 6a^2 = 6 \times 5^2\) \(150 cm^2\)
Find the volume of a cylinder with \(r = 3 cm\) and \(h = 7 cm\). \(V = \pi r^2h = \frac{22}{7} \times 3^2 \times 7\) \(198 cm^3\)

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

Common Mistakes and How to Avoid Them

Common Mistakes
  • Using area formula when volume is asked.
  • Forgetting square units or cubic units.
  • Mixing centimetres and metres without conversion.
  • Using diameter instead of radius in circle formulas.
  • Confusing curved surface area with total surface area.
  • Forgetting to include top and bottom faces in total surface area.
Useful Shortcuts
  • If diameter is given, find radius using \(r = \frac{d}{2}\).
  • For cube, remember one side is enough for all formulas.
  • For cuboid, identify length, breadth, and height first.
  • For cylinders, use \(\pi = \frac{22}{7}\) when radius is multiple of 7.
  • Write formula before substituting values.
  • Check final unit carefully.
Exam approach: Identify the shape, write the correct formula, substitute values, calculate carefully, and write the correct unit.

Quick Formula Revision

Square

\(Area = a^2\)

\(Perimeter = 4a\)

\(TSA\ of\ cube = 6a^2\)

Rectangle / Cuboid

\(Area = lb\)

\(V = lbh\)

\(TSA = 2(lb+bh+lh)\)

Circle / Cylinder

\(Area = \pi r^2\)

\(V = \pi r^2h\)

\(CSA = 2\pi rh\)

Cone / Sphere

\(V_{cone} = \frac{1}{3}\pi r^2h\)

\(V_{sphere} = \frac{4}{3}\pi r^3\)

\(SA_{sphere} = 4\pi r^2\)

Memory tip: Area has power \(2\), volume has power \(3\). So area uses square units and volume uses cubic units.

Practice

A) Multiple Choice Questions
  1. What is the area of a square of side \(8 cm\)?
    \(16 cm^2\) \(32 cm^2\) \(64 cm^2\) \(80 cm^2\)
  2. Which formula gives the area of a circle?
    \(2\pi r\) \(\pi r^2\) \(\pi d\) \(r^2h\)
  3. Volume is measured in:
    square units cubic units linear units degree units
  4. Total surface area of a cube is:
    \(a^2\) \(4a^2\) \(6a^2\) \(a^3\)
  5. Curved surface area of a cylinder is:
    \(2\pi rh\) \(\pi r^2h\) \(2\pi r(r+h)\) \(4\pi r^2\)
B) Solve the Problems
  1. Find the area of a rectangle with length \(15 m\) and breadth \(6 m\). (Hint: Use \(Area = l \times b\).)
  2. Find the area of a triangle with base \(12 cm\) and height \(9 cm\). (Hint: Use \(Area = \frac{1}{2}bh\).)
  3. Find the volume of a cube of side \(6 cm\). (Hint: Use \(V = a^3\).)
  4. Find the volume of a cylinder with radius \(7 cm\) and height \(5 cm\). (Hint: Use \(V = \pi r^2h\).)
  5. Find the total surface area of a cube of side \(9 cm\). (Hint: Use \(TSA = 6a^2\).)
C) Match the Shape with the Correct Formula
Shape / Concept Correct Formula
Area of Square \(a^2\)
Area of Rectangle \(l \times b\)
Area of Circle \(\pi r^2\)
Volume of Cube \(a^3\)
Volume of Cuboid \(lbh\)
CSA of Cylinder \(2\pi rh\)
Mensuration Reminder

Area, volume, and surface area are formula-based topics. The most important skill is to identify the shape correctly and choose the right formula.

Task: Make a formula chart for square, rectangle, triangle, circle, cube, cuboid, cylinder, cone, and sphere.

Show Suggested Answers
Multiple Choice
  1. \(64 cm^2\)
    Area of square \(= a^2 = 8^2 = 64 cm^2\).
  2. \(\pi r^2\)
    Area of a circle is \(\pi r^2\).
  3. Cubic units
    Volume is measured in units such as \(cm^3\) or \(m^3\).
  4. \(6a^2\)
    Total surface area of cube is \(6a^2\).
  5. \(2\pi rh\)
    Curved surface area of cylinder is \(2\pi rh\).
Solved Problems
  1. Rectangle area: \(15 \times 6 = 90 m^2\)
  2. Triangle area: \(\frac{1}{2} \times 12 \times 9 = 54 cm^2\)
  3. Cube volume: \(6^3 = 216 cm^3\)
  4. Cylinder volume: \(\pi r^2h = \frac{22}{7} \times 7^2 \times 5 = 770 cm^3\)
  5. Total surface area of cube: \(6 \times 9^2 = 486 cm^2\)
Concept Matching
  1. Area of Square → \(a^2\)
  2. Area of Rectangle → \(l \times b\)
  3. Area of Circle → \(\pi r^2\)
  4. Volume of Cube → \(a^3\)
  5. Volume of Cuboid → \(lbh\)
  6. CSA of Cylinder → \(2\pi rh\)
Clue Explanation

Area formulas usually involve square units, while volume formulas involve cubic units. Surface area problems ask for the outer covering of a solid object.

Exam tips
  • Read whether the question asks area, volume, or surface area.
  • Convert all units before calculation.
  • Use radius, not diameter, in circle formulas.
  • Remember \(1 litre = 1000 cm^3\).
  • Write formula first, then substitute values.
  • Always add the correct unit in the final answer.
Practice MCQs