Maths Exam- 2D & 3D Shapes

Drawing Pictures of Solids

- Method 1 : Parallel Lines = draw two pairs of parallel lines that cross = draw three lines of equal length straight down from the corners or = chose a point in the middle, below ( or above ) the crossed parallel lines. Draw lines from three corners to this point to make a pyramid Method 2 : Projections - to draw three dimensional objects on paper, artists use different techniques called projections. When drawing a prism, start with the base. This is the same shape as the uniform cross section. Note that the base is not always the face that is closest to the ground. It is important that : = you are able to draw pictures of different solids = you can understand a picture of a solid

Polygon Shapes

- a shape that only has straight sides

Tetrahedron

- a triangle based pyramid

Prisms

- all prisms have a special pair of parallel faces - these two faces are the only prism faces that do not need to be rectangular - if a prism is sliced at any point parallel to these faces, the cross-section is the same. Because of this, the prism is said to have a uniform cross-section - a prism is named according to the shape of it's uniform-cross section. The base of a prism is the same as the uniform cross-section

Pyramids

- all the faces of a pyramid need to be triangular, except for the base. - a pyramid is named according to the shape of its base. A pyramid is also named according to the shape of its non-triangular face. - unlike prisms,pyramids cannot be sliced to always make an identical shape, so pyramids are said to have a non-uniform cross-section

Diagonals

- are lines drawn from one corner of a polygon, across the polygon, to another corner.

Plane Shapes

- are shapes that lie in one plane, or flat surface - they have area but not volume - they are 2D shapes

Quadrilaterals

- family of four sided shapes - each quadrilateral is a special kind of quadrilateral - quadrilaterals are convex, meaning all the diagonals of each shape are inside the shape - non-convex quadrilaterals have at least one diagonal outside the shape, and they also have at least one reflex angle

Euler's Formula

- for any convex polyhedron, F + V = E + 2, where F = number of faces, V = number of vertices and E = number of edges

Solids

- have thickness as well as length and breadth - we say they are 3D ( three-dimensional ) - a solid shape might be full or empty inside - there are two main families of solids ; prisms and pyramids

Triangles

- have three sides and three angles - they can be sorted into different types according to the lengths of their sides or the size of their angles

Looking at Solids from Different Views

- in many technical situations, a solid is represented by being drawn from different views. Usually a solid will be drawn from the front, top and the sides. Together these drawings can be used to completely describe the solid. - From each view, you can obtain two of the three dimensions of the solid: = from the front view you can get the length and height of the solid = from the top view you can get the length and width of the solid = from the side views you can get the width and height of the solid. - dashed or dotted lines are used to show any hidden edges

Nets of Solids

- solid shapes can be made from plane shapes. This can be done by drawing the net of the solid on a piece of paper. The net shows how the faces of the solid are joined to each other. When these faces are folded along their edges, the solid is formed. - when trying to draw a net, you need to answer the following questions : = what types of faces does the solid have? = how many of each type are there? = which faces are joined to each other?

Describing Quadrilaterals

- square, rectangle, parallelogram, rhombus, trapezium and kite are all special types of quadrilaterals - the shapes have many special properties that are related to their sides, angles and diagonals - these properties are used to identify and describe the shapes - when describing the relationships between sides and diagonals, we often use the words parallel and perpendicular - perpendicular lines intersect at right angles

Types of triangles - mixed

- triangles can be of more than one type eg- if a triangle was both isosceles and acute angled, we would call it an acute angled isosceles triangle

Cross Sections

- when a solid is cut by a plane, the shape on the cut face is called a cross-section - a shape can have different cross-section depending on the angle of the cut

Types of triangles - angles

Acute Angled - all angles are acute Right Angled - one right angle Obtuse Angled - one angle is obtuse

Parts of a Solid

Face - a surface of the solid Edge - a line where two faces meet Vertex - a corner where three or more faces meet

Types of triangles - sidesagonals

Scalene - no sides equal - no angles equal Isosceles - two sides equal - angles opposite equal sides are equal Equilateral - all sides equal - three 60 degree angles

Finding the size of angle

Triangle a + b + c = 180 degrees Quadrilateral a + b + c + d = 360 degrees Isosceles a = b Equilateral a = b = c Straight Angle a + b = 180 degrees Angles at a point a + b + c + d = 360 degrees