Polygons can be regular or irregular. Regular polygons have all sides and angles equal, while irregular polygons do not. Common polygons include triangles, quadrilaterals, pentagons, and hexagons. Polygons are 2D shapes with straight sides. 3D shapes include polyhedra with flat faces that are polygons, as well as non-polyhedral shapes like cylinders, cones, and spheres. Polyhedra include the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) as well as prisms and pyramids. Prisms have two identical polygon bases and parallelogram sides, while pyramids have one polygon base and triangular sides

1. Polygons 1
Polygon: A 2-dimensional shape. They are made of straight lines (except the circle) and the shape is "closed"
Regular polygon= all angles and sizes are equal
Irregular polygon= angles and sizes aren’t equal
Regular Irregular
Point : A location in space
Line : Connects two points via the shortest path and continues indefinetely in both directions Perpendicular
Line Segment : Part of a line between two points.
Perpendicular Line Segment: Line Segments that intersect (cross) at an angle of 90°
Parallel Line Segments : Line segments that never intersect
Vertex : Point at which two line segments intersect Parallel
Angle:It’s the surface between two sides that has a common vertex. Angles measures by grades.
Right Angle An angle that measures 90°
Acute Angle An angle that measures less than 90°
Obtuse Angle An angle that measures more than 90°
Perimeter : The distance around a two-dimensional shape.
The perimeter of this rectangle is 7+3+7+3 = 20
Area: The size of a surface. To work out the area of a polygon we must use the correct formula.
Apothem : The distance from the center of a regular polygon to the midpoint of a side.
Circle: It’s a curved shape whose all points are at the same distance from the centre
Circumference : It’s the perimeter of a circle.
Diameter : The distance between two points of the circumference that joins the centre.
Radio: The distance from the center to any point of the circumference. Diameter
It’s the half of a diameter.
Three sides polygons: Triangles
Four sides polygons: Parallelograms
Equilateral Triangle
Three equal sides
Three equal angles, always 60°
Isosceles Triangle
Two equal sides
Two equal angles
Scalene Triangle
No equal sides
No equal angles
Acute Triangle
All angles are less than 90°
Right Triangle
Has a right angle (90°)
Obtuse Triangle
Has an angle more than 90°

2. Names of Polygons - 2
Name Sides Shape
Area
Triangle 3
h
b
A= (b x a):2 A= (b. h):2
Square 4 a
A= l x l= l2
A= a2
Rectangle
4
h
a
A= b x a A= a . h
Trapezoid
Trapezium
Parallelogram
4
4
a
b h
h
A = (B + b) . a A=(b +a).h
2 2
A= b . a A= b. h
Pentagon 5
b
A= perimeter.apothem
2
Hexagon 6 Idem all regular polygons
Circle
A= 2. Π. r2
Circumference L= 2.π.r
REMEMBER
h = height = alto
w= width = ancho
d= depth= fondo
b= base = base
l = side= lado

3. POLIEDROS = POLYHEDRAS
Polyhedra: A polyhedron is a solid with flat faces which are polygons. It hasn`t got curved surfaces.
Faces = Flat surfaces (superficies planas) = caras
Edges (bordes) = aristas
Vertices = corner of the edges , corner points = vértices
1. Poliedros Regulares = Platonic solids
Un poliedro regular es aquel cuyas caras son polígonos regulares iguales y en cada uno de sus vértices
concurren el mismo número de caras. Sólo existen cinco tipos de poliedros regulares:
Name Area Volumen
Tetraedro
Tethahedrom
Surface Area = √3 × (Edge
Length)2
Volume = (√2)/12 × (Edge
Length)3
Cubo
Cube
Surface Area = 6 × (Edge
Length)2
Volume = (Edge Length)3
Octaedro
Octahedrom
Surface Area = 2 × √3 × (Edge
Length)2
Volume = (√2)/3 × (Edge
Length)3
Dodecaedro
Dodecahedrom
Surface Area = 3×√(25+10×√5) ×
(Edge Length)2
Volume = (15+7×√5)/4 × (Edge
Length)3
Icosaedro
Icosahedrom
Surface Area = 5×√3 × (Edge
Length)2
Volume = 5×(3+√5)/12 × (Edge
Length)3
Platonic Solids = poliedros regulares
Prisms = prismas
Pyramids = pirámides

4. 2. Prismas= Prismas
Poliedro limitado por dos polígonos iguales, llamados bases, situados en planos paralelos, y por varios
paralelogramos, llamados caras laterales.
Un prisma se llama triangular, cuadrangular, pentagonal… según que sus bases sean triángulos,
cuadriláteros, pentágonos…
A lateral = perímetro de la base · altura
A total = área lateral + 2 · área de la base
V = área de la base · altura
Lateral area= base perimeter . lenght
Surface area= lateral area + 2.base area
Volume = base area . length
3-. Pirámides= Pyramids
apex
CUERPOS CILÍNDRICOS = NON - POLYHEDRA
CILINDRO-CYLINDER
Á. Total = A. Lateral + 2· A. Base
Volumen = A. Base · Altura
CONO-CONE
Area total= A.Lateral + A.Base A Base= Π r2
Volumen = A.Base · Altura ALateral = Π·r·g
3
ESFERA- SPHERE
Área= A= 4·Π·r2
Volumen= 4/3 Π ·r3
The Volume of a Pyramid
 1/3 × [Base Area] × Height
 V= (area base . a): 3
The Surface Area of a Pyramid
When all side faces are the same:
 [Base Area] + [Lateral Area]
 V = Area base + Area lateral .

5. 1-. Rectangular Prisms : area and volume
V= 576 cm3
V= 80m2
V= 200 m2
V=8
V=24

6. 2-. Calculate the volumes
volume = 45 cm3
volume = 24000 cm3
V = 565.2 cm3
V = 98,125 cm3
V= 12 cm3

7. PROBLEMS
1-. These diagram shows a cylinder of height 4 cm and diameter 4 cm. What is its
total surface area? What is its volume?
A= 65,36 cm and V= 50,24 cm2
2-. A big building has got four water drums with these measures. If a m3 of water equals a liter . How
many liters are there in total? (6160 liters)
(YOU MUST READ METRE INSTEAD IN)
3-. A cone diameter 20cm and height 20 cm is filled with ice cream. How many ice cream
does it contain? (2094,4 cm3)
4-. A piece of cheese has got these measures.
10 mm
15
18

8. €