4 ESO Academics - UNIT 06 - AREAS AND VOLUMES. SIMILARITY.

1. Unit 06 January
1. PERIMETERS AND AREAS OF SHAPES 2D.
1.1. POLYGON AREAS AND PERIMETERS.
For Regular Polygons we have:
Axel Cotón Gutiérrez Mathematics 4º ESO 4.6.1

2. Unit 06 January
MATH VOCABULARY: Area, Perimeter, Surface, Triangle, Square, Trapezoid, Rectangle,
Rhombus, Parallelogram, Polygon, Apothem, Radius, Center, Equilateral Triangle,
Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon.
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3. Unit 06 January
1.2. CIRCUMFERENCE AREAS AND LENGTH.
For Arcs and Sectors we have:
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4. Unit 06 January
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5. Unit 06 January
For Annulus we have:
MATH VOCABULARY: Circumference, Diameter, Length, Circle, Arc, Sector Angle,
Segment, Chord, Annulus.
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6. Unit 06 January
2. AREAS OF SHAPES 3D.
MATH VOCABULARY: Tetrahedron, Square Pyramid, Hexagonal Pyramid, Cube, Cuboid,
Triangular Prism, Octahedron, Pentagonal Prism, Hexagonal Prism, Dodecahedron,
Sphere, Ellipsoid, Icosahedron, Cone, Cylinder, Face, Edges, Vertices, Polyhedron.
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7. Unit 06 January
2.1. PRISM´S SURFACE AREA.
In geometry, a Prism is a Polyhedron comprising an n-sided polygonal base, a
second base which is a translated copy of the first, and 𝒏𝒏 other faces (necessarily all
parallelograms) joining corresponding sides of the two bases.
The Surface Area will be always: “2 x (Area of Regular Polygon Face) +
[(number of sides of regular polygon) x (Length of Prism)]”
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8. Unit 06 January
2.2. PYRAMID´S SURFACE AREA.
In geometry, a Pyramid is a polyhedron formed by connecting a polygonal base
and a point, called the Apex. Each base edge and apex form a triangle, called a Lateral
Face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1
vertices, n + 1 faces, and 2n edges.
MATH VOCABULARY: Apex, Slant Height, Lateral Face.
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9. Unit 06 January
2.3. SPHERE´S SURFACE AREA.
The Spherical Dome is the figure resulting from having made a flat cut in a
sphere.
The Spherical Crescent is the part of the surface of a sphere in between two
planes that cut at the diameter of the sphere.
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10. Unit 06 January
MATH VOCABULARY: Spherical Dome, Spherical Crescent.
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11. Unit 06 January
3. VOLUME OF SHAPES 3D.
Volume is the quantity of three-dimensional space enclosed by a closed
surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape
occupies or contains.
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12. Unit 06 January
4. SIMILARITY.
Two figures are Similar if they are identical in shape, but not in size.
If one figure can become another using Resizing (also called dilation,
contraction, compression, enlargement or even expansion), then the figures are
similar. When you resize a figure it gets bigger or smaller, but it still looks similar.
For any pair of similar figures:
• Corresponding Angles are equal.
• Corresponding sides are in the same ratio. This ratio is called Similarity Ratio or
Scale Factor.
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13. Unit 06 January
If the Similarity Ratio of two similar figures is 𝒌𝒌, then:
• The ratio of their perimeters is also k
• The ratio of their areas is k2
• The ratio of their volumes is k3
4.1. TRIANGLES.
There are three ways to find if two triangles are similar:
• If two triangles have two of their angles equal, the triangles are similar.
• If two triangles have two pairs of sides in the same ratio and the included angles are
also equal, then the triangles are similar.
• If two triangles have three pairs of sides in the same ratio, then the triangles are
similar
If a line is drawn Parallel to one side of a triangle, the smaller triangle and the
larger triangle are similar.
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14. Unit 06 January
You can say that the triangles AB0 and A’B’0 are in Thales’ Position.
4.2. OTHER 2D AND 3D SHAPES.
All of the shapes behave as triangles.
MATH VOCABULARY: Similarity, Resize, Angle, Similarity Ratio, Scale Factor, Parallel.
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