ASIAN DEVELOPMENT FOUNDATION COLLEGE

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

Prepared by: RTFVerterra

10 sides 11 sides 12 sides 15 sides 16 sides = = = = = decagon undecagon dodecagon quindecagon hexadecagon RADIUS OF CIRCLES Circle circumscribed about a triangle (Cicumcircle)

A=

(s − a)(s − b)(s − c)(s − d) − abcdcos2 θ

s=

a+b+c+d 2 ∠A + ∠C ∠B + ∠D θ= or θ = 2 2

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

Sum of interior angles The sum of interior angles θ of a polygon of n sides is: Sum, Σθ = (n – 2) × 180° Sum of exterior angles The sum of exterior angles β is equal to 360°. ∑β = 360°

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

Given four sides a, b, c, d, and two opposite angles B and D: Divide the area into two triangles

Circumcenter of the triangle

a b

r c

r=

A = ½ ab sin B + ½ cd sin D

abc 4A T

AT = area of the triangle

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 Regular polygons 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

B d1 A

C

θ

d2

b

PLANE GEOMETRY

PLANE AREAS Triangle

D a Given diagonals d1 and d2 and included angle θ: A = ½ d1 × d2 × sin θ Given two sides a and b and one angle A:

r=

B a h c

A = ab sin A

Rhombus

C d1 d2 a

D

C

θ

b

A = ½ bh

A B

90° a A

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 R R θ θ θ r θ θ Apothem x x Inscribed circle

AT s s = ½(a + b + c)

c r r b r

a

A

C

Circles escribed about a triangle (Excircles)

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. ra ra c a ra

Given base b and altitude h

Given diagonals d1 and d2:

Given two sides a and b and included angle θ: A = ½ ab sin θ Given three sides a, b, and c: (Hero’s Formula) A= s= s( s − a)(s − b)(s − c )

a+b+c 2

A = ½ d1 × d2

Given side a and one angle A:

x

x

A = a2 sin A

Trapezoid

a h b b B d1 d2 a A d D C 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. Given three angles A, B, and C and one side a: a 2 sin B sin C A= 2 sin A

a+b h A= 2

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 θ = 360° / n

b AT AT AT ra = ; rc = ; rb = s −b s−a s−c

Circle circumscribed about a quadrilateral

Cyclic Quadrilateral

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

Area, A = ½ R2 sin θ × n = ½ x r × n Perimeter, P = n × x n−2 × 180° Interior angle = n Exterior angle = 360° / n Circle

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

b a

r c d

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

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....