KEY FORMULAS FOR SHAPES / POLYGON / AREA / PERIMETER / VOLUME / SURFACE AREA

Number of Diagonals (N_d)

(N_d) = (n/2) (n-3)

Measure of a Single Interior angle (θ_int)

(θ_int) = ((n-2)* (180°)) / n

Sum of the interior angle (S_int)

(S_int) = (n-2)*180°

Legend: What is n,r, and b?

n = number of sides

r = radius of circle

b = length of the side

Area of the polygon

A=(1/4)((n)(b)²cot(180°/n))

Area of polygon inscribed in a circle

A= (1/2)(n*r²*sin(360°/n)

Area of polygon circumscribed in a circle

A = n*r²*tan(180°/n)

Area of a circle (A_circle)

A_circle = pi*r²

Circumference of a circle (C_circle)

C_circle = pi*2r = pi*D

Arc Length of a Circle (s)

s =rθ

Formula: Sector of a circle (A_sector)

(laman ng pizza triangle)

A_sector = (1/2)*r²*θ

Segment of a circle (A_segment)

A_segment = A_sector - A_triangle = (1/2)*r²*(θ - sinθ)

Area of Rectangle

A = Base x Height

Perimeter of a Rectangle

P = 2(a+b)

Area of Square 

A = s²

Perimeter of Square

P = 4s

It is a quadrilateral whose opposite sides are parallel

Parallelogram

Area of Parallelogram

A = absinθ = (0.5)(d_1)(d_2)sinβ = bh

Perimeter of Parallelogram

P = 2(a+b)

It is a parallelogram of equal sides

Rhombus

Area of Rhombus

A = a² * sin θ = (0.5)(d_1)(d_2) = ah

Perimeter of Rhombus

P = 4a

It is a quadrilateral with one pair of sides parallel to each other.

Trapezoid

Area of Trapezoid

A = (0.5)(a+b)(h)

Median of Trapezoid 

Median = (a+b) / 2

It is also known as general quadrilateral or polygon of four sides or no sides parallel.

Trapezium

Area of Trapezium / Bretschneider’s Formula

A = sqrt of ((s-a)(s-b)(s-c)(s-d) - abcdcos²(θ))

Where:

S= (a+b+c+d)/2

θ = (A+C)/2 = (B+D)/2

Area of Trapezium

A = (0.5)(d_1)(d_2)sin(beta)

Area of Cyclic Quadrilateral / Brahmagupta’s Formula

A = sqrt of ((s-a)(s-b)(s-c)(s-d))

It is a quadrilateral that lies in a circle.

Cyclic Quadrilateral

Radius of Cyclic Quadrilateral

r = ( sqrt of ((ab + cd)(ac + bd)()(ad + bc) ) / 4A

Area of Quadrilateral Circumscribing a Circle

A = rs

Where:

s = (0.5)(a+b+c+d)

Area of Ellipse

A = pi*a*b

Perimeter of Ellipse

P = 2*pi*sqrt of ((a²+b²)/2))

Area of Parabolic Segment

A = (2/3)*b*h

Perimeter of Parabolic Segment

P = (c/2) + (b²/8h) * In(4h+c/b) + b

Where:

c= sqrt of (b² + 16h²)

It is closed space bounded by planes.

Polyhedron

Rectangular parallelepiped

Volume of Rectangular

V = L*W*h

Total Surface Area of a Rectangular

A_total surface = 2(ab+bc+ac)

Lateral Area of a Rectangular

A = 2(bh + wh)

It is a polyhedron with six faces which are all squares.

Cube

Volume of Cylinder

V = pi*r²*h

Total Surface Area of Cylinder

A_total_surface = 2*pi*r*h + 2*pi*r²

Lateral Area of Cylinder

A_lateral_area = 2*pi*r*h

Volume of a Cube

V = a³

Total Surface Area of Cube (A_total_surface)

A_total_surface = 6a²

Lateral Area of Cube (A_lateral)

A_lateral = 4a²

Volume of Cone

V = 1/3*pi*r²*h

Lateral Area of Cone (A_lateral)

A_lateral = (0.5)( c )(l) = pi*r*l = pi*r*sqrt of (r² + h²)

Where: l = sqrt of (r² + h²)

c = 2*pi*r

Volume of Pyramid

V = (1/3)*A*h

Volume of Frustum of a Cone

V = (h/3)(A1 + A2 + sqrt of (A1A2)

Volume of Frustum of a Cone

V = [(pi)*(h)/3] (R³ + r² + Rr)

Formula for finding slant of height(l) of Frustum of a Cone

Height_slant (l) = sqrt of (h² + (R-r)²)

Lateral Area of Frustum of a Cone

Area_lateral = pi*(R+r)*l = pi*(R+r) * sqrt of (h² + (R-r)²)

Volume of Sphere

V = (4/3)*pi*R³ = (pi/6)*D³

Surface Area of Sphere

A_surface = 4*pi*R² = pi*D²

Volume of Spherical Wedge (Solid) and Lune (Surface)

V = (2/3)*r³*θ

Area of Spherical Wedge (Solid) and Lune (Surface)

A = 2*r²*θ

Volume of Spherical Segment (One Base)

V = [(pi*h²)/3 ] (3R-h)

Volume of Spherical Segment (One Base)

V = [(pi*h)/6](3a² + 3b² + h²)

Volume of Ellipsoid 

V = (4/3)*pi*a*b*c

Volume of Paraboloid of Revolution

V = (0.5)*pi*b²*a

Regular Polyhedrons

Faces are congruent regular polygons are whose polyhedral angles are all equal. Also known as Platonic Solids

Area of the Triangle / Heron’s Formula

A = sqrt of ((s(s-a)(s-b)(s-c))