Geometry

Basic 2D shapes and their location

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Basic 3D shapes

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Symmetry

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2D shapes- sides and vertices

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3D shapes - faces, vertices and edges

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Perimeter of various shapes

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Compare perimeter and area of various shapes

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Perimeter and area

Perimeter is the distance around the outside of a shape (green line).
Area is the amount of space (measured in square units, e.g. cm2, m2) inside a shape (grey area).



Perimeter and area of rectangles and squares

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Perimeter and area of rectangles and squares

Rectangle and squares are quadrilaterals with four right angles.

With squares, all four sides are the same length, so the perimeter is four times the length of a side a.

$$P = 4a$$
aa

In a rectangle, two pairs of opposite sides are of equal length a, b. The perimeter of a rectangle is given by the formula:

$$P = 2a + 2b$$
ab

The area is given by formulae:

Square:

$$A=a×a=a^2$$

Rectangle:

$$A = a×b $$


Triangle inequality

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Triangle inequality

In each triangle, any two sides together are longer than the remaining third side. So, the following applies for their lengths in each type of triangle:

$$a+b>c$$
$$c+b>a$$
$$a+c>b$$
abAcBC

E.g. if we have the given lengths of two sides of the triangle a = 8, b = 3, c = 6, we can construct the triangle. But if the sides are a = 3, b = 4, c = 9, the triangle cannot be constructed because:

$$3 + 4 <9$$

If we know only two lengths (e.g. a = 3 and b = 7) and we have to find out the possible range of the length of the side c, we proceed as follows:

$$a + b> c ;→;3+7>c$$
$$;→; c <10$$

At the same time

$$ a+c > b ;→; 3+c > 7$$
$$ ;→; c > 4$$

And also:

$$b+c>a ;→;7+c>3$$
$$ ;→; c>0$$

So we know:

$$c<10$$
$$c>4$$
$$c>0$$

In order for c to satisfy all of these inequalities, its length must be between 4 and 10:

$$4


Altitudes (heights) of a triangle

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Altitudes

An altitude (or the height) of a triangle is a line segment through a vertex and perpendicular to (forming a right angle with) a side of the triangle containing the base (the side opposite the vertex). This line containing the opposite side is called the extended base of the altitude. Every triangle has 3 altitudes which intersect at one point called the orthocenter (O).

BCOAacbahhhcb

In an acute triangle, all altitudes and the orthocenter lie within the triangle.

In a right triangle, two sides coincide with the altitudes and the orthocenter coincides with the vertex of the right angle.

BCOA

In an obtuse triangle, two altitudes and the orthocenter lie outside of the triangle.

BCOA


Volume and surface area of cubes and rectangular prisms

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Cube and rectangular prism

A cube is a three-dimensional solid with 6 sides whose length, width, and height are equal. Its angles are right.

aaa

To calculate the volume and surface area, use these formulae:

$$V=a×a×a=a^3$$
$$S=2a×a+2a×a+2a×a=6a^2$$

A rectangular prism is a three-dimensional solid which has six faces that are rectangles. Its angles are right.

abcabbcccb
$$V=a×b×c$$
$$S=2a×b+2b×c+2a×c$$


Volume and surface of cylinders, spheres, and cones

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Sphere

A sphere is a perfectly round solid figure with every point on its surface equidistant from its centre. The radius (r) of a sphere is the distance from the exact center of the sphere (C) to any point on the outside edge of that sphere. We calculate volume and surface area of a sphere using these formulae:

Cr
$$A=4πr^2$$

Surface area is actually equal to areas of four circles of the same radius.

Of all the solids, a sphere has the smallest surface area for a volume.

$$V = 4/3 πr^3$$

Cylinder

A cylinder is a solid that has two parallel (usually circular) bases connected by a curved surface.

rrsshh21

The height (altitude, h) of a cylinder is the perpendicular distance between its bases.

A cylinder can be right or oblique. A right cylinder has bases aligned one directly above the other. In an oblique cylinder, the bases remain parallel to each other, but the sides lean over at an angle that is not 90°. If they have equal height and base area AB, they will have the same volume.

$$V=A_B×height=πr^2h$$

The surface area is the sum of two bases AB(pink) and the lateral area. We need to know the side length (s) to calculate the lateral area. Only for a right cylinder the side length is equal to the height (h=s1).

r2πrh=sπr2πr2

For oblique cylinder it is s2.

$$A=2×A_B+s×2πr=2πr^2+s×2πr$$

Cone

A cone is the solid figure that is generated by a right triangle turning around one of its cathetus. It can be right (the vertex is over the center of its base) or oblique (vertex is not over the center). The base can be a circle or an ellipse.

rhsV

A cone is closely related to a pyramid and the formulae for their volume are similar (volume of a pyramid is one third of a prism with the same width, length and height and a cone is one third of a cylinder with the same base and height).

$$V=1/3×A_B×h$$

The slant height (s) of a right cone is the length of the segment from the vertex of the cone to the circle of the base. We don't define this for oblique cones.

The total surface area of a cone is the sum of the area of its base AB and the lateral (side) surface AL. The lateral surface of a right cone can be calculated as:

$$A_L=πr×s$$

Total surface area of right circular cone is:

$$A=A_B+A_L=πr^2+πrs=$$
$$=πr^2+πrs=πr(r+s)$$
rπr2πrss


   
   

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