Mensuration

Mensuration is the branch of mathematics that deals with measuring the size and length of geometric
figures 1 . It covers both 2D (plane) figures and 3D (solid) figures, focusing on properties like perimeter,
area, volume, and surface area 1 . In simple terms, mensuration uses algebraic formulas to compute these
measures for shapes.

Perimeter and Area of Plane Shapes

The perimeter of a shape is the total length around its boundary 2 . For example, the perimeter of a
square with side a is 4a, and of a rectangle with length l and width w is 2(l + w) 3 .

The area of a shape is the size of the region it covers, measured in square units 4 . Key area formulas
include:

• Square: Area = a² (if a is the side length); Perimeter = 4a.

• Rectangle: Area = l × w; Perimeter = 2(l + w) 3 .
• Triangle: Area = ½ × base × height 5 ; Perimeter = sum of its three sides.
• Parallelogram: Area = base × height 6 ; Perimeter = 2(base + side).
• Trapezoid (Trapezium): Area = ½×(sum of the two parallel sides)×height 7 ; Perimeter = sum of all
four sides.
• Circle: Area = πr²; Circumference (perimeter) = 2πr* 8 9 .

Figure: A triangle with base b and height h. Its area is given by A = ½·b·h 5 .

Each formula above is well-known. For instance, for a triangle of base 10 m and height 6 m, area = ½·10·6 =
30 m². For a rectangle 8 cm by 5 cm, area = 8·5 = 40 cm² and perimeter = 2·(8+5) = 26 cm.

Surface Area and Volume of Solids

In 3D geometry, surface area is the total area of all faces (or surfaces) of a solid, measured in square units,
while volume is the amount of space the solid occupies, in cubic units 10 . Important formulas include:

• Cube: side = a. Volume = a³ 11 ; Surface Area = 6a² 12 .

• Cuboid (Rectangular Prism): sides l, w, h. Volume = l·w·h 13 ; Surface Area = 2(lw + lh + wh) 14 .

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 • Cylinder: radius = r, height = h. Volume = πr²h 15 ; Total Surface Area = 2πr² + 2πrh = 2πr(r+h) 16 .
• Cone (right circular): base radius r, height h, slant l. Volume = (1/3)πr²h 17 ; Total Surface Area = πr(r +
l) 18 (l is slant height).
• Sphere: radius = r. Volume = (4/3)πr³ 19 ; Surface Area = 4πr² 20 .

Figure: A cube of side a. Its volume is a³ 11 and surface area is 6a² 12 .

For example, a cube with side 3 cm has volume 27 cm³ and surface area 54 cm². A cylinder with r = 4 m and
h = 5 m has volume π·4²·5 = 80π m³ and surface area 2π·4(4+5) = 72π m².

Solved Example Problems

1. Triangle Area: Find the area of a triangle with base = 12 cm and height = 5 cm.
Solution: Use A = ½·base·height 5 . So A = ½·12·5 = 30 cm².

2. Rectangle Measurements: A rectangle has length 15 m and width 10 m. Compute area and
perimeter.
Solution: Area = 15·10 = 150 m²; Perimeter = 2·(15+10) = 50 m 3 .

3. Circle Area: Calculate the area and circumference of a circle with radius 7 cm.
Solution: Area = π·7² = 49π cm² 8 ; Circumference = 2π·7 = 14π cm 9 .

4. Cube Volume: A cube has side 5 cm. What are its volume and surface area?
Solution: Volume = 5³ = 125 cm³ 11 ; Surface Area = 6·5² = 150 cm² 12 .

5. Cylinder Surface Area: A cylinder has radius 3 cm and height 10 cm. Find its total surface area.
Solution: TSA = 2πr(r + h) 16 . Here r=3, h=10: TSA = 2π·3·(3+10) = 2π·3·13 = 78π cm².

Each example uses the appropriate mensuration formula cited above.

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Practice Problems
Multiple-Choice:
1. The area of a circle with radius 7 cm is:
A. 14π cm² B. 49π cm² C. 28π cm² D. 7π cm².
2. The volume of a cube of side 2 cm is:
A. 4 cm³ B. 6 cm³ C. 8 cm³ D. 12 cm³.
3. A trapezoid has bases 10 m and 6 m and height 4 m. Its area is:
A. 32 m² B. 64 m² C. 16 m² D. 24 m².

Descriptive:
4. Calculate the perimeter and area of a triangle with sides 3 cm, 4 cm, 5 cm. (Hint: It’s a right triangle.)
5. Find the surface area of a cylinder with radius 2 cm and height 7 cm.
6. Compute the volume of a cone with base radius 3 cm and height 12 cm.

Practice Solutions
• 1: Area = π·7² = 49π cm² → B 8 .
• 2: Volume = 2³ = 8 cm³ → C 11 .
• 3: Area = ½(10+6)·4 = ½·16·4 = 32 m² → A . 7

• 4: Perimeter = 3+4+5 = 12 cm. Area = ½·3·4 = 6 cm² 5 .

• 5: TSA of cylinder = 2πr(r+h) = 2π·2·(2+7) = 2π·2·9 = 36π cm² 16 .
• 6: Volume of cone = (1/3)π·3²·12 = (1/3)π·9·12 = 36π cm³ 17 .

Each solution follows directly from the formulas given. Use the formula that fits the shape and plug in the
values, as illustrated above.

Key Formulas (Summary):

Area Perimeter/ Volume

Shape Surface Area Formula
Formula(s) Circumference Formula Formula

Square (side a) A = a² P = 4a – SA = 6a² 12

SA = 2(lw + lh + wh)
Rectangle (l, w) A = l·w P = 2(l+w) 3 –
14 (as cuboid)

Triangle (b, h) A = ½·b·h 5 P = sum of sides – –

Parallelogram
A = b·h 6 P = 2(b + side) – –
(b, h)

Trapezoid (a, b, A = ½·(a+b)·h

P = a + b + other sides – –
h) 7

Circle (r) A = πr² 8 C = 2πr 9 – –

Cube (a) – – V = a³ 11 SA = 6a² 12

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 Area Perimeter/ Volume
Shape Surface Area Formula
Formula(s) Circumference Formula Formula

SA = 2(lw + lh + wh)
Cuboid (l, w, h) – – V = l·w·h 13
14

Cylinder (r, h) – – V = πr²h 15 TSA = 2πr(r + h) 16

V = (1/3)πr²h
Cone (r, h, l) – – TSA = πr(r + l) 18
17

V = (4/3)πr³
Sphere (r) – – SA = 4πr² 20
19

Formulas in the table are cited from reputable sources. Use these formulas to solve mensuration problems:
plug in the dimensions of the shape into the appropriate formula. Detailed examples above illustrate the
steps.

Sources: Definitions and formulas are based on standard geometry texts and reference materials 3 5

8 7 12 14 16 15 17 18 .

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 1 Mensuration - Formulas, Meaning, Examples
https://www.cuemath.com/measurement/mensuration-and-measurement/

2 PERIMETER Definition & Meaning - Merriam-Webster

https://www.merriam-webster.com/dictionary/perimeter

3 Rectangle - Wikipedia
https://en.wikipedia.org/wiki/Rectangle

4 Area - What is Area? Definition, Area of Shapes Formula

https://www.cuemath.com/measurement/area/

5 Area of a triangle - Wikipedia

https://en.wikipedia.org/wiki/Area_of_a_triangle

6 byjus.com
https://byjus.com/maths/area-of-parallelogram/

7 Area of a Trapezoid - Math Steps, Examples & Questions

https://thirdspacelearning.com/us/math-resources/topic-guides/geometry/area-of-a-trapezoid/

8 9 Area of a circle - Wikipedia

https://en.wikipedia.org/wiki/Area_of_a_circle

10 Volume - Formula, Definition, Calculate, Examples

https://www.cuemath.com/measurement/volume/

11 13 Kids Math: Finding the Volume of a Cube or Box

https://www.ducksters.com/kidsmath/finding_the_volume_of_a_cube_or_box.php

12 byjus.com
https://byjus.com/maths/surface-area-of-cube/

14 Surface Area of Cuboid - Formulas of TSA and CSA (LSA), Examples

https://www.cuemath.com/measurement/surface-area-of-cuboid/

15 Volume of a Cylinder - Formula | How to Find Volume of Cylinder?

https://www.cuemath.com/measurement/volume-of-cylinder/

16 Surface Area of a Cylinder - Formula, TSA and CSA

https://www.cuemath.com/measurement/surface-area-of-cylinder/

17 Volume of Cone - Formula, Derivation, Examples, FAQs

https://www.cuemath.com/measurement/volume-of-cone/

18 Surface Area of a Cone | GeeksforGeeks

https://www.geeksforgeeks.org/surface-area-of-cone/

19 20 byjus.com
https://byjus.com/maths/volume-of-sphere/