Solid geometry formulas PDF

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formulas on solid geometry...

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Plane and Solid Geometry Formulas ASIAN DEVELOPMENT FOUNDATION COLLEGE Tacloban City

The content of this material is one of the intellectual properties of Engr. Romel Tarcelo F. Verterra of Asian Development Foundation College. Reproduction of this copyrighted material without consent of the author is punishable by law.

Given four sides a, b, c, d, and sum of two opposite angles:

10 sides 11 sides 12 sides 15 sides 16 sides

A= (s −a)(s − b)(s − c)(s − d) − abcdcos2 θ a + b+ c + d 2 ∠ A + ∠C ∠B + ∠D θ= or θ = 2 2

s=

Parallelogram

Number of diagonals, D The diagonal of a polygon is the line segment joining two non-adjacent sides. The number of diagonals is given by: n D = (n − 3) 2

C d1

d2

θ

b

D a Given diagonals d1 and d2 and included angle θ: A = ½ d 1 × d2 × sin θ

PLANE AREAS

Given two sides a and b and one angle A:

B

A = ab sin A Rhombus

C

D d1

θ

C

A B

Given base b and altitude h A = ½ bh

s=

a

A = ½ d 1 × d2

a+b 2

h h

b

Area =

C c d1 D

d2 a d A

( s − a)(s − b)( s − c)(s − d)

a + b+ c + d 2

s=

“For any cyclic quadrilateral, the product of the diagonals equals the sum of the products of the opposite sides” d1 × d2 = ac + bd

Perimeter, P = 2(a + b) Diagonal, d = a 2 + b 2

d

There are two basic types of polygons, a convex and a concave polygon. A convex polygon is one in which no side, when extended, will pass inside the polygon, otherwise it called concave polygon. The following figure is a convex polygon. β4 θ4 β3

a

Perimeter, P = 4a a

Diagonal, d = a 2 General quadrilateral b

C

B

β5

d1

β2

d2 D

A

θ5

θ3

c

θ

d

Given diagonals d1 and d2 and included angle θ:

A = ½ d 1 × d2 × sin θ

θ2 θ1

β1

= = = =

A circle is escribed about a triangle if it is tangent to one side and to the prolongation of the other two sides. A triangle has three escribed circles.

c a

AT s− a

AT

; rc =

s−c

Arc C = r × θradians =

πrθ 180°

Area = ½ r 2 θradians =

πr 2θ 360 °

A circle is circumscribed about a quadrilateral if it passes through the vertices of the quadrilateral.

b r

C

θ

r

d

(ab + cd)(ac + bd)(ad + bc ) 4 Aquad (s − a)( s − b)(s − c)(s − d)

r

Circle incribed in a quadrilateral b A circle is inscribed in a quadrilateral r if it is tangent to the three a sides of the quadrilateral.

c

O

C

Area = Asect or – At riangle Area = ½ r 2 θr – ½ r 2 sin θ Area = ½ r 2 (θr – sin θ)

s

d ;

s = ½(a + b + c + d)

abcd

SOLID GEOMETRY

α = 360 - θ r r θ

2 bh 3

A quad

r

O

Area = Asect or + At riangle Area = ½ r 2 αr + ½ r 2 sin θ Area = ½ r 2 (αr + sin θ)

r=

Aquad =

θ

r

θr = angle in radians

POLYHEDRONS

A polyhedron is a closed solid whose faces are polygons.

h

Ellipse

b

Area = π a b Perimeter, P

P = 2π

c

a

D

Segment of a circle

Area =

s −b

Circle circumscribed about a quadrilateral

r=

r

AT

; rb =

Circumference = 2π r = πD π 2 D Area, A = π r 2 = 4

β6

triangle quadrangle or quadrilateral pentagon hexagon

ra =

Aquad =

θ6

Polygons are classified according to the number of sides. The following are some names of polygons. 3 sides 4 sides 5 sides 6 sides

Circles escribed about a triangle (Excircles)

s = ½(a + b + c + d)

Parabolic segment

C

b

Note: 1 radian is the angle θ such that C = r.

Square

Area, A = a2

r

Circle

Area = ½ C × r POLYGONS

r

b

ra

Perimeter, P = n × x n− 2 × 180° Interior angle = n Exterior angle = 360° / n

Ptolemy’s theorem

a r

ra

Sector of a circle

b

c

A

x

Area, A = ½ R2 sin θ × n = ½ x r × n

B

∠A + ∠C = 180° ∠B + ∠D = 180°

AT s s = ½(a + b + c)

ra

θ = 360° / n

b

A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle.

Circle inscribed in a triangle (Incircle) A circle is inscribed in a triangle if it is tangent to the three sides of the triangle. B Incenter of the triangle

Inscribed circle

x = side θ = angle subtended by the side from the center R = radius of circumscribing circle r = radius of inscribed circle, also called the apothem n = number of sides

a

Cyclic Quadrilateral

Rectangle

a

c b

abc r= 4A T

x

Trapezoid

The area under this condition can also be solved by finding one side using sine law and apply the formula for two sides and included angle.

d

R R θ θ θ θ θ Apothem

A = a 2 sin A

A=

a 2 sinB sinC A= 2 sin A

x

Given side a and one angle A:

2

Given three angles A, B, and C and one side a:

a

x

r

s (s − a)(s − b)(s − c )

Area, A = ab

Polygons whose sides are equal are called equilateral polygons. Polygons with equal interior angles are called equiangular polygons. Polygons that are both equilateral and equiangular are called regular polygons. The area of a regular polygon can be found by considering one segment, which has the form of an isosceles triangle. Circumscribing x circle x

A

a

a + b+ c

The area under this condition can also be solved by finding one angle using cosine law and apply the formula for two sides and included angle.

r

a

r=

Regular polygons

Given diagonals d1 and d2:

Given two sides a and b and included angle θ: A = ½ ab sin θ

A=

d2

90°

b

Given three sides a, b, and c: (Hero’s Formula)

Circumcenter of the triangle

AT = area of the triangle

PLANE GEOMETRY

h

A circle is circumscribed about a triangle if it passes through the vertices of the triangle.

A = ½ ab sin B + ½ cd sin D

A

a

Circle circumscribed about a triangle (Cicumcircle)

Sum of exterior angles The sum of exterior angles β is equal to 360°. ∑β = 360°

Divide the area into two triangles

B

c

RADIUS OF CIRCLES

decagon undecagon dodecagon quindecagon hexadecagon

Sum of interior angles The sum of interior angles θ of a polygon of n sides is: Sum, Σθ = (n – 2) × 180°

Given four sides a, b, c, d, and two opposite angles B and D:

Part of: Plane and Solid Geometry by RTFVerterra © October 2003

Triangle

= = = = =

RTFV e rte rra

a 2 + b2

PRISM

a

b b

a

A prism is a polyhedron whose bases are equal polygons in parallel planes and whose sides are

7 sides 8 sides 9 sides

= heptagon or septagon = octagon = nonagon

2

parallelograms. Prisms are classified according to their bases. Thus, a hexagonal prism is one whose base is a...