Plane And Solid Geometry

PLANE GEOMETRY I.

•

TRIANGLE

Area of Triangle •

•

Given base & altitude: 1 𝐴 𝑇 = 𝑏ℎ 2 ℎ = 𝑎𝑠𝑖𝑛𝜃

•

•

Special Lines in a Triangle

Given three sides a, b, & c: Heron’s Formula 𝐴 𝑇 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 𝑎+𝑏+𝑐 𝑠 2 Equilateral Triangle: √3 2 𝑎 4 𝑎2 𝑠𝑖𝑛60 𝐴𝑇 = 2

•

Special Centers in a Triangle • •

•

𝐴𝑇 =

•

Given 2 sides and their included angle: 𝑎𝑏𝑠𝑖𝑛𝜃 𝐴𝑇 = 2

•

Given 3 angles and one of the sides: 𝑎2 𝑠𝑖𝑛𝛽𝑠𝑖𝑛𝛾 𝐴𝑇 = 2𝑠𝑖𝑛𝛼

•

Triangle with inscribed circle: 𝐴 𝑇 = 𝑟𝑠 (𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 𝑟=√ 2 𝑎+𝑏+𝑐 𝑠= 2 where: r = apothem = radius of inscribed circle

•

•

Triangle inscribed in a circle: 𝑎𝑏𝑐 𝐴𝑇 = 4𝑟 𝑎 𝑏 𝑐 𝑟= = = 2𝑠𝑖𝑛𝐴 2𝑠𝑖𝑛𝐵 2𝑠𝑖𝑛𝐶 r = radius of circumscribed circle Triangle with escribed circle: 𝐴 𝑇 = 𝑟(𝑠 − 𝑎) r = radius of excircle

Median – is a segment from vertex to the midpoint of the opposite side Angle Bisector – is a segment or ray that bisects an angle and extends to the opposite side Altitude – is a segment from vertex perpendicular to the opposite side

• •

Centroid – is a point of intersection of all medians of a triangle. Incenter – is the point of intersection of all the angle bisectors in a triangle. It is also the center of the inscribed circle in a triangle. Circumcenter – is the point of intersection of all perpendicular bisectors of a triangle. It is also the center of the circumscribed circle. Orthocenter – is the point of intersection of all the altitudes of a triangle. Excenter – is the center of the escribed circle.

The line that passes through the incenter and orthocenter of a triangle is called Euler’s line. II. QUADRILATERAL - A plane figure with four straight sides 1. Parallelogram – it is a quadrilateral in which opposite sides are parallel and equal Formula for Area and Perimeter: 𝐴 = 𝑏ℎ 1 𝐴 = 𝑑1 𝑑2 𝑠𝑖𝑛𝜃 2 1 𝐴 = 𝑎𝑏𝑠𝑖𝑛𝜃 2 𝑃 = 2(𝑎 + 𝑏) 2. Square – is a quadrilateral with four equal sides and four right angles. Formula for Area & Perimeter: 𝐴 = 𝑎2 𝑃 = 4𝑎 3. Rectangle – is a quadrilateral in which pairs of opposite sides are parallel and equal angle is 90°. 𝐴 = 𝑎𝑏 𝑃 = 2(𝑎 + 𝑏)

4. Trapezoid or Trapezium – is a quadrilateral with only one pair of opposite sides parallel. Formula for Area & Perimeter: 𝐴=

ℎ (𝑎 + 𝑏) 2

𝑃 =𝑎+𝑏+

ℎ ℎ + 𝑠𝑖𝑛𝛼 𝑠𝑖𝑛𝛽

•

Angle Measurements and Number of Diagonals •

Sum of Interior Angles: 𝑆 = 𝑛𝜃 = (𝑛 − 2)180°

•

Measure of Interior Angle: 𝑆 (𝑛 − 𝑠)180° 𝜃= = 𝑛 𝑛

•

Number of Diagonals: 𝑛 𝐷 = (𝑛 − 3) 2 Sum of Exterior Angles = 360° Measure of Exterior Angle: 360 𝛽= 𝑛

5. Rhombus – is a quadrilateral in which all sides are equal but none of the angles is 90°. Formula for Area & Perimeter: 𝐴 = ℎ𝑠 1 𝐴 = 𝑑1 𝑑2 2 𝐴 = 𝑠 2 𝑠𝑖𝑛𝜃 𝑃 = 4𝑠 6. General Quadrilateral 𝐴 = √(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑) − 𝑎𝑏𝑐𝑑𝑐𝑜𝑠 2 𝜃 𝑎+𝑏+𝑐+𝑑 𝑠= 2 𝐴+𝐶 𝐵+𝐷 𝜃= = 2 2

• •

Area and Perimeter of Regular Polygons •

Given the apothem and perimeter: 1 𝐴 = 𝑃𝑟 2

•

Given the apothem and number of sides: 180 𝐴 = 𝑛𝑟 2 (𝑡𝑎𝑛 ) 𝑛 180 𝑃 = 2𝑛𝑟(𝑡𝑎𝑛 ) 𝑛

•

Given the of the side: 𝑛𝑥 2 180 𝐴= (𝑐𝑜𝑡 ) 4 𝑛 𝑃 = 𝑛𝑥

•

Given R:

7. Cyclic Quadrilateral – is a quadrilateral in which all of its four vertices line on a circle Ptolemy’s Theorem: 𝑑1 𝑑2 = 𝑎𝑐 + 𝑏𝑑 Bramaguptha’s Formula: 𝐴 = √(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑) 𝑟=

√(𝑎𝑏 + 𝑐𝑑)(𝑎𝑐 + 𝑏𝑑)(𝑎𝑑 + 𝑏𝑐) 4𝐴

8. Quadrilateral Circumscribing a Circle 𝐴 = √𝑎𝑏𝑐𝑑 𝐴 = 𝑟𝑠 𝑎+𝑏+𝑐+𝑑 𝑠= 2 III. POLYGONS - It is a closed plane figure bounded by straight line segments as sides

Concave Polygon - is one having at least one interior angle is greater than 180°.

𝑛𝑅 2 360 (𝑠𝑖𝑛 ) 2 𝑛 360 𝑃 = 2𝑛𝑅(𝑠𝑖𝑛 ) 2𝑛

𝐴=

Special Polygons •

Types of Polygon

Pentagram (regular 5-point star) 𝐴 = 1.123𝑟 2 Hexagram (star of David) 𝐴 = √3𝑟 2

•

r = radius of circumscribing circle

Convex Polygon – it is a polygon in which no interior angle is greater than 180°.

•

IV.

PARABOLIC SEGMENT

•

Parabolic Segment 2 𝐴 = 𝑏ℎ 3

Case 2:

Spandrel 𝐴= V.

1 𝑏ℎ 3

𝐴 = 𝐴𝑠𝑒𝑐𝑡𝑜𝑟 + 𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 1 𝐴 = 𝑟 2 (𝜃𝑟 + 𝑠𝑖𝑛α) 2

CIRCLE

Circle Relationships •

• • • • • • • • • •

where:

Circle – is the set of all points in a plane that are at the same distance from a fixed point called the center. Radius – is a line segment joining the center to a point on a circle. Central Angle – is an angle formed by two radii. Inscribed Angle – is an angle whose vertex is on the circle and whose sides are chord. Arc – is a continuous part of a circle. Minor Arc – is an arc that is less than a semicircle. Major Arc – is an arc that is greater than a semi-circle. Chord – is a line segment joining two points of the circumference. Diameter – is a chord through the center of the circle. Secant – is a line that intersects the circle at two points Tangent – is a line that touches the circle at one and only one point.

r = radius 𝜃 = angle (in radians) VI. • •

•

•

•

•

Area of a Circle, Sector and Segment •

Area of a Circle: 𝐴 = π𝑟 2

•

Circumference of a Circle: 𝐶 = 2π𝑟

•

Length of an Arc: 𝑠 = 𝑟𝜃

•

Area of a Circular Segment: Case 1: 𝐴 = 𝐴𝑠𝑒𝑐𝑡𝑜𝑟 − 𝐴𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 1 𝐴 = 𝑟 2 (𝜃𝑟 − 𝑠𝑖𝑛𝜃) 2

Area of a Circular Sector: 1 1 𝐴 = 𝑟𝑠 = 𝑟 2 𝜃 2 2

•

•

SPECIAL PLANE CURVES Cycloid – is a curve described by a point P on a circle or radius “a” rolling on a straight line Catenary – a curve which a heavy uniform flexible chain freely hangs if suspended vertically from its two extremes A and B Epicycloid – is the curved described by a point P on a circle of radius “b” as it rolls on the outside of another fixed circle of radius “a” Cardiod – is a path traced out by a point on a circle of radius “a” rolling around the circumference of another circle having the same radius “a” Hypocycloid – (with 4 cusps) is described by a point P on a circle of radius a/4 as it rolls on the inside of a fixed circle of radius “a” without slipping Trochoid – is a curve described by a point P at a distance “b” from the center of a circle of radius “a” as the circle rolls on the x-axis Involute of a Circle – is a curved described by the endpoint P of a string as it unwinds from a circle of radius “a” while held taut Ovals of Cassini – is the curve described by a point P such that the product of its distance from two fixed points (distance 2a apart) is constant.

Volume:

SOLID GEOMETRY I.

𝑉 = 𝐵ℎ 𝑉 = π𝑟 2 ℎ

CUBE -is a polyhedron whose six faces are all squares

Where: C = circumference B = area of the base r = radius of the cylinder L = slant height H = height or altitude

Total Surface Area: 𝐴𝑠 = 6𝑎2 Volume: 𝑉 = 𝑎3

IV.

Space Diagonal: 𝐷 = √3𝑎

PRISM -is a polyhedron having two identical and parallel faces (usually referred to as the “ends” or “bases”) and whose sides are parallelogram

Face Diagonal: 𝑑 = √2𝑎 II.

•

Oblique Prism -a prism with an axis that is not at right angle to the base 𝑉 = 𝐴𝑅 𝐿 𝐴𝑠 = 𝑃𝑅 𝐿

•

Right Prism -a right prism is one whose axis is perpendicular to the base 𝑉 = 𝐴𝐵 ℎ 𝐴𝑠 = 𝑃𝐵 ℎ

•

Truncated Prism

RECTANGULAR PARALLELEPIPED -is a polyhedron whose six faces are all rectangles Area of a Rectangular Parallelepiped: 𝐴𝑠 = 2(𝑎𝑏 + 𝑏𝑐 + 𝑎𝑐) Volume: 𝑉 = 𝑎𝑏𝑐 Space Diagonal:

𝑉 = 𝐴𝑅 (

𝑑 = √𝑎2 + 𝑏 2 + 𝑐 2 Face Diagonal:

Where: V = volume AR = area of the right section AS = lateral area AB = area of the base PB = perimeter of the base h = height n = no. of sides

𝑑𝑙 = √𝑎2 + 𝑏 2 𝑑𝑠 = √𝑏 2 + 𝑐 2 III.

•

•

CYLINDER -is a solid bounded by closed cylindrical surface and two parallel planes Oblique Cylinder Lateral Surface Area: 𝐴 = 2π𝑟𝐿 Volume: 𝑉 = 𝐵ℎ Right Circular Cylinder Lateral Surface Area: 𝐴 = 𝐶ℎ 𝐴 = 2π𝑟ℎ

ℎ1 + ℎ2 + ℎ3 … + ℎ𝑛 ) 𝑛

V.

•

CONE -is a tree dimensional shape formed by a straight line when one end is moved around a simple closed curved, while the other end of the line is kept fixed at a point which is not in the plane of curve Oblique Cone 1 𝑉 = 𝐵ℎ 3

𝐴𝐿 = π𝑟𝐿 •

Right Circular Cone 1 𝑉 = 𝐵ℎ 3 𝐴𝐿 = π𝑟𝐿

•

VI.

•

Frustum of a cone ℎ 𝑉 = (𝑏 + 𝐵 + √𝑏𝐵) 3 𝐶𝐵 + 𝐶𝑏 𝐴𝐿 = ( )𝐿 2 Where: AL = lateral surface area V = volume B = area of the lower base b = area of the upper base h = height L = slant height PYRAMID -is a polyhedron having any polygons as one face (base) with all other faces (sides) being triangles meeting at a common vertex Right Pyramid

h = perpendicular distance between the two bases Volume of Special Prismatoid •

A wedge cut from a cylinder of radius r by two planes, one perpendicular to the axis of the cylinder and the other intersecting the first plane at an angle 𝜃 along a diameter 2√3𝑟 3 𝑉= 𝑖𝑓 𝜃 = 30° 9 2𝑟 3 𝑉= 𝑖𝑓 𝜃 = 45° 3

•

Solid with circular base of radius r and every plane section perpendicular to a certain diameter is an equilateral triangle 4𝑟 3 𝑉= √3

•

Solid with circular base of radius r and every section perpendicular to a certain diameter is an isosceles triangle of altitude h 𝜋𝑟 2 ℎ 𝑉= 2

•

Solid with circular base of radius r and every perpendicular to a certain diameter is an isosceles triangle with altitude equal to onehalf of its base. 4𝜋𝑟 3 𝑉= 3

•

Solid with circular base of radius r and every plane section perpendicular to a certain diameter is a square 16𝑟 3 𝑉= 3

•

Solid with circular base of radius r and every plane section perpendicular to a fixed diameter is a semi-circle 2𝜋𝑟 3 𝑉= 3 Solid common to two identical cylinders with their axis meeting at 90° 16𝑟 3 𝑉= 3

1 𝑉 = 𝐵ℎ 3 𝐴𝐿 = 𝑃𝐿 •

VII.

Frustum of a Pyramid ℎ 𝑉 = (𝑏 + 𝐵 + √𝑏𝐵) 3 𝑃𝐵 + 𝑃𝑏 𝐴𝐿 = ( )𝐿 2 PRISMATOID

General Prismatoid ℎ 𝑉 = (𝐴1 + 4𝐴𝑚 + 𝐴2 ) 6 Where: V = volume A1 = area of the lower base Am = area of the mid section A2 = area of the upper base

•

VIII. SIMILAR SOLIDS • Relationship between Area and Altitude: 𝐴2 ℎ2 = ( )2 𝐴1 ℎ1 •

•

Relationship between Volume and Altitude: 𝑉2 ℎ2 = ( )3 𝑉1 ℎ1 Relationship between Volume and Area: 𝑉2 𝐴2 = (√ )3 𝑉1 𝐴1

IX. SPHERE • Formula for Volume and Area: 4 𝑉 = π𝑟 3 3 𝐴 = 4π𝑟 2 • Spherical Zone: (𝑜𝑛𝑒 𝑏𝑎𝑠𝑒) 𝐴𝑠 = 2π𝑟ℎ (𝑡𝑤𝑜 𝑏𝑎𝑠𝑒𝑠) 𝐴𝑠 = 2π𝑟ℎ • Spherical Segment: 𝜋ℎ2 (3𝑟 − ℎ) 𝑉= (𝑜𝑛𝑒 𝑏𝑎𝑠𝑒) 3 𝜋ℎ [3(𝑎2 + 𝑏 2 ) + ℎ2 ] 𝑉= (𝑡𝑤𝑜 𝑏𝑎𝑠𝑒𝑠) 6

•

•

• •

•

𝜋𝑟 2 𝐸 180 𝜋𝑟 3 𝐸 𝑉= 540

(𝜃 𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)

•

Prolate Spheroid -volume generated when an ellipse is revolved around the major axis 4 𝑉 = π𝑎𝑏 2 3 sin−1 𝑒 𝐴 = 2π𝑏(𝑏 + 𝑎 ) 𝑒 Where: √𝑎2 + 𝑏 2 𝑒= 𝑎

•

Oblate Spheroid -volume generated when an ellipse is revolved about the minor axis 4 𝑉 = π𝑎2 𝑏 3 𝜋𝑏 2 1+𝑒 𝐴 = 2π𝑎2 + [ ln ( )] 𝑒 1−𝑒 Where: √𝑏 2 − 𝑎2 𝑒= 𝑎

XI. PARABOLOID OF REVOLUTION • One base: 1 𝑉 = π𝑟 2 ℎ 2 4𝜋𝑟 𝑟2 𝑟 𝐴 = 2 [(ℎ2 + ) − ( )3 ] 3ℎ 4 2 •

Two bases: 𝑉=

Spherical Pyramid: 𝐴=

(𝜃 𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)

X. ELLIPSOID • General Ellipsoid / Spheroid -a three-dimensional figure all planar crosssections of which are either ellipses or circles 4 𝑉 = π𝑎𝑏𝑐 3

Where: a = radius of the small base b = radius of the big base Spherical Sector: 2 𝑉 = π𝑟 2 ℎ 3 Spherical Polygon: -a closed geometric figure on the surface of a sphere formed by the arcs of great circles. 𝜋𝑟 2 𝐸 𝐴= 180 Where: E = spherical excess E = [sum of all angles] – (n-2)180° r = radius of the sphere n = number of sides

Spherical Lune: 𝜋𝑟 2 𝜃 𝐴= 90 Spherical Wedge: 𝜋𝑟 3 𝜃 𝐴= 270

XII.

𝜋ℎ 2 (𝑟 + 𝑟2 2 ) 2 1

TORUS 𝐴 = 2𝜋 2 𝑟𝑅 𝑉 = 2𝜋 2 𝑟 2 𝑅

XIII.

POLYHEDRON -it is a solid bounded by flat surfaces with each surface bounded by straight sides. A regular polyhedron is a solid figure each of whose sides is a regular polygon (or the same size) and each of whose angles is formed by the same number of sides

•

•

Tetrahedron: 𝐴𝑠 = √3𝑎2 √2 3 𝑉= 𝑎 12 Hexahedron: 𝐴𝑠 = 6𝑎2 𝑉 = 𝑎3

•

•

Octahedron: 𝐴𝑠 = 2√3𝑎2 √2 3 𝑉= 𝑎 3 Dodecahedron: 𝐴𝑠 = 20.65𝑎2 𝑉 = 7.66𝑎3

•

Icosahedron: 𝐴𝑠 = 5√3𝑎2 𝑉 = 2.18𝑎3

PAST BOARD PROBLEMS: 1. Each side of a cube is increased by 10%. By what percent is the volume of the cube increased? a. 33.1% c. 0.0031% b. 3.31% d. 13.31% 2. A rectangle ABCD which measures 18x24 cm, is folded once, perpendicular to diagonal AC, so that the opposite vertices A and C coincide. Find the length of the fold. a. 22.50 cm c. 21.5 cm b. 18.75 cm d. 19.5 cm 3. A reservoir is shaped like a square prism. If the area of its base is 225 square centimeters, how many liters of water will it hold? a. 337.5 L c. 3375 L b. 3.375 L d. 33.75 L 4. A regular octagon is inscribed in a circle of radius 10. Find the area of the octagon.

a. 228.2 b. 288.2

c. 238.2 d. 282.8

5. One side of a regular octagon is 2. Find the area of the region inside the octagon. a. 31.0 c. 19.3 b. 21.4 d. 13.9 6. A piece of wire is shaped to enclose the square whose area is 169 sq. cm. it is then reshaped to enclose the rectangle whose length is 15 cm. What is the area of the rectangle? a. 156 sq. cm c. 175 sq. cm b. 165 sq. cm d. 170 sq. cm 7. The distance between the centers of the three circles which are mutually tangent to each other externally are 10, 12 and 14 units. The area of the largest circle is? a. 72 π c. 64 π b. 23 π d. 16 π 8. If the sides of a parallelogram and an included angle are 6, 10 and 100° respectively, find the length of the shorter diagonal. a. 10.63 c. 10.73 b. 10.37 d. 10.23 9. A trapezoid has an area of 360 m2 and an altitude of 20 m. its two bases have ratio of 4:5. What are the lengths of the bases? a. 12, 15 c. 8, 10 b. 7, 11 d. 16, 20 10. The angel of a sector is 30° and the radius is 15 cm. what is the area of the sector in cm2? a. 59.8 c. 58.9 b. 89.5 d. 85.9 11. Two triangles have equal bases. The altitude of one triangle is 3 units more than its base and the altitude of the other is 3 units less than its base. Find the altitudes, if the areas of the triangle differ by 21 sq. units. a. 4 and 10 c. 6 and 12 b. 3 and 9 d. 5 and 11 12. Each angle of a regular dodecagon is equal to ________ degrees. a. 135 c. 125

b. 150

d. 105

13. A metal washer 1 inch in diameter is pierced by ½ inch hole. What is the volume of the washer if it is 1/8 inch thick? a. 0.074 c. 0.028 b. 0.047 d. 0.082 14. If an equilateral triangle is circumscribed about a circle of radius 10 cm, determine the side of the triangle. a. 64.21 cm c. 32.10 cm b. 36.44 cm d. 34.64 cm 15. Find the approximate change in the volume of a cube of side “x” inches caused by increasing its side by 1%. a. 0.3x3 in3 c. 0.10x3 in3 3 3 b. 0.02x in d. 0.03x3 in3 16. What is the distance in cm between two vertices of a cube which are farthest from each other, if an edge measures 8 cm? a. 13.86 c. 16.93 b. 11.32 d. 14.33 17. A piece of wire of length 50 m is cut into two parts. Each part is then bent to form a square. It is found that the total area of the square is 100 sq. m. Find the difference in length of the sides of the two squares. a. 6.62 c. 5.44 b. 5.32 d. 6.61 18. A pyramid whose altitude of 5 ft weight 800 lbs. At what distance from the vertex must it be cut by a plane parallel to its base so that the two solids of equal weight will be formed? a. 2.52 ft c. 2.96 ft b. 3.97 ft d. 4.96 ft 19. A right triangle is inscribed in a circle in such that one side of the triangle is the diameter of a circle. If one of the acute angels of the triangle measure 60° and the side opposite that angle has length 15, what is the area of the circle? a. 175.15 b. 235.62

c. 223.73 d. 228.61

20. What is the apothem of a regular polygon having an area of 225 and a perimeter of 60? a. 6.5 c. 5.5 b. 8.5 d. 7.5