Area and Perimeter Formulas of All Shapes (2D & 3D)

Here are all the area and perimeter formulas a school student needs, from Class 5 to Class 10 — flat (2D) shapes first, then surface area and volume of solids (3D), each with a solved example.

Quick reminder: Perimeter is the boundary length (cm, m); Area is the space inside a flat shape (cm², m²); Volume is the space inside a solid (cm³, m³).

Area and Perimeter of 2D Shapes

Shape	Area	Perimeter
Square (side a)	a²	4a
Rectangle (l, b)	l × b	2(l + b)
Triangle (base b, height h)	½ × b × h	sum of three sides
Triangle, Heron's formula (sides a,b,c)	√[s(s−a)(s−b)(s−c)], s = (a+b+c)/2	a + b + c
Equilateral triangle (side a)	(√3/4) × a²	3a
Circle (radius r)	πr²	2πr
Parallelogram (base b, height h, sides a,b)	b × h	2(a + b)
Rhombus (diagonals d₁,d₂, side a)	½ × d₁ × d₂	4a
Trapezium (parallel sides a,b, height h)	½ × (a+b) × h	sum of four sides

Solved Examples: 2D Shapes

Example 1 — Rectangle: 6 m × 4 m room. Area = 24 m², Perimeter = 20 m.

Example 2 — Heron's formula: sides 13, 14, 15 cm. s = 21; Area = √(21×8×7×6) = √7056 = 84 cm²

Example 3 — Circle: radius 7 cm (π = 22/7). Area = 154 cm², Circumference = 44 cm

Example 4 — Rhombus: diagonals 10 cm and 24 cm. Area = 120 cm²; side = √(5² + 12²) = 13 cm; Perimeter = 52 cm

Example 5 — Trapezium: parallel sides 10 and 6 cm, height 4 cm. Area = ½ × 16 × 4 = 32 cm²

Surface Area and Volume of 3D Solids

Solid	Curved/Lateral SA	Total SA	Volume
Cube (side a)	4a²	6a²	a³
Cuboid (l,b,h)	2h(l+b)	2(lb+bh+hl)	l×b×h
Cylinder (r,h)	2πrh	2πr(r+h)	πr²h
Cone (r,h, slant l)	πrl	πr(l+r)	⅓πr²h
Sphere (radius r)	4πr²	4πr²	(4/3)πr³
Hemisphere (radius r)	2πr²	3πr²	(2/3)πr³

For a cone, the slant height is l = √(r² + h²).

Solved Examples: 3D Solids

Example 6 — Cuboid: 10 × 6 × 4 cm. Volume = 240 cm³, TSA = 2(60+24+40) = 248 cm²

Example 7 — Cylinder: radius 7 cm, height 10 cm. Volume = 1540 cm³, CSA = 440 cm²

Example 8 — Cone: radius 3 cm, height 4 cm. Slant = 5 cm; CSA = 15π ≈ 47.14 cm²; Volume = 12π ≈ 37.71 cm³

Example 9 — Sphere: radius 21 cm. Volume = 4/3 × 22/7 × 21³ = 38808 cm³

Tips to Remember the Formulas

• Perimeter adds lengths; area multiplies two lengths; volume multiplies three — that's why units are cm, cm², cm³.
• A cone holds exactly ⅓ the volume of a cylinder with the same base and height.
• Hemisphere TSA = 3πr² because it's the curved part (2πr²) plus the flat circle (πr²).
• Check your units — if an area answer comes out in cm³, something has gone wrong.

Frequently Asked Questions

What is the difference between area and perimeter?
Perimeter is the total boundary length; area is the surface inside. A fence measures perimeter, a carpet measures area.

When should I use Heron's formula?
When you know all three sides of a triangle but not its height.

What is the difference between CSA and TSA?
Curved/lateral surface area covers only the sides; total surface area also includes the top and bottom faces.