Formulas for 3D Shapes

Some of the useful math geometry formulas for 3D shapes are discussed below.

(i) Area of a Triangle: Let ABC be any triangle. If AD be perpendicular to BC and BC = a, CA = b, AB = c then the area of the triangle ABC (to be denoted by ⊿) is given by,

area of a triangle


⊿ = ¹/₂ × base × altitude. 

= ¹/₂ ∙ BC ∙ AD 

(b) ⊿ = √[s(s - a)(s - b)(s - c)] 

Where 2x = a + b + c = perimeter of the ⊿ ABC. 

(c) If the a be the length of a side of an equilateral triangle then its height = (√3/2) a and it’s area = (√3/4) a²

(ii) If a be the length and b, the breadth of a rectangle then its area = a ∙ b, length of its diagonal = √(a² + b² ) and its perimeter = 2 ( a + b). 


(iii) If a be the length of a side of a square, then its area = a² length of its diagonal = a√2 and perimeter = 4a.


(iv) If the lengths of two diagonals of a rhombus be a and b respectively then its area = (1/2) ab and length of a side = (1/2) √(a² + b²)


(v) If a and b be the lengths of two parallel sides of a trapezium and h be the distance between the parallel sides then the area of the trapezium = (1/2) (a + b) ∙ h.


(vi) Area of a Regular Polygon: The area of a regular polygon of n sides = (na²/4) cot (π/n) where a is the length of a side of the polygon. In particular, if a be the length of a side of a regular hexagon then its area

= (6a²/4) ∙ cot (π/6) = (3√3/2) ∙ a²


(vii) The length of circumference of a circle of radius r is 2πr and
its area = πr²


(viii) Rectangular Parallelopiped: If a, b and c be the length, breadth and height respectively of a rectangular parallelopiped then,

Rectangular Parallelopiped


(a) the area of its surfaces = 2 ( ab + bc + ca) 

(b) its volume = abc and 

(c) the length of diagonal = √(a² + b² + c² ).


(ix) Cube: If the length of the side of a cube be a then,

Cube surface area


(a) the area of its surfaces = 6a²,

(b) its volume = a³ and

(c) the length of the diagonal = √3a.


(x) Cylinder: Let r (= OA) be the radius of the base and h (=OB) be the height of a right circular cylinder ; then

curved surface area of a cylinder


(a) area of its curved surface = perimeter of base × height = 2πrh

(b) area of the whole surface = area of its curved surface + 2 × area of circular base
                                                = 2πrh + 2πr²
                                                = 2πr(h + r)

(c) volume of cylinder = area of base × height
                                    = πr²h


(xi) Cone: Let r (= OA) be the radius of the base, h (= OB), the height and I, the slant height of a right circular cone ; then

curved surface area of a cone


(a) l² = h² + r²

(b) area of its curved surface

= (1/2) × perimeter of the base × slant height = (1/2)∙ 2πr ∙ l = πrl

(c) area of its whole surface = area of the curved surface + area of the circular base

= πrl + πr² = πrl + πr(l + r).

(d) volume of the cone = (1/3) × area of the base × height = (1/3)πr²h

 Mensuration



11 and 12 Grade Math 

From Formulas for 3D Shapes to HOME PAGE


New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.



Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Adding 5-digit Numbers with Regrouping | 5-digit Addition |Addition

    Mar 18, 24 02:31 PM

    Adding 5-digit Numbers with Regrouping
    We will learn adding 5-digit numbers with regrouping. We have learnt the addition of 4-digit numbers with regrouping and now in the same way we will do addition of 5-digit numbers with regrouping. We…

    Read More

  2. Adding 4-digit Numbers with Regrouping | 4-digit Addition |Addition

    Mar 18, 24 12:19 PM

    Adding 4-digit Numbers with Regrouping
    We will learn adding 4-digit numbers with regrouping. Addition of 4-digit numbers can be done in the same way as we do addition of smaller numbers. We first arrange the numbers one below the other in…

    Read More

  3. Worksheet on Adding 4-digit Numbers without Regrouping | Answers |Math

    Mar 16, 24 05:02 PM

    Missing Digits in Addition
    In worksheet on adding 4-digit numbers without regrouping we will solve the addition of 4-digit numbers without regrouping or without carrying, 4-digit vertical addition, arrange in columns and add an…

    Read More

  4. Adding 4-digit Numbers without Regrouping | 4-digit Addition |Addition

    Mar 15, 24 04:52 PM

    Adding 4-digit Numbers without Regrouping
    We will learn adding 4-digit numbers without regrouping. We first arrange the numbers one below the other in place value columns and then add the digits under each column as shown in the following exa…

    Read More

  5. Addition of Three 3-Digit Numbers | With and With out Regrouping |Math

    Mar 15, 24 04:33 PM

    Addition of Three 3-Digit Numbers Without Regrouping
    Without regrouping: Adding three 3-digit numbers is same as adding two 3-digit numbers.

    Read More