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Geometry

Surface areas of 3D shapes

Mathematics
The angle in a semi-circle is a right-angle

Cube and Prism

Cube: $A = L^3$

Sphere

$A = 4πr^2$

Pyramid

$A = B + {PL}/2$, where B is the area of the base, P is the perimeter of the base, and L is the height of the slant $L= √{{h^2+r^2}}$ where h is the pyramid height and r is the inradius of the base.

Same arc angles
Angles subtended on a circle by an arc are half the size of angles subtended at the centre by the same arc

Cone

$A = πr(r + l)$, where r is the radius of the base, and l is the lateral length, given by $l = √{r^2 + h^2}$, where h is the height of the cone.

Cylinder

$A = πd(r + h)$, where d is the diameter of the base, r the radius of the base and h the height of the cylinder.

Circle

Arcs

The length of an arc = $({θ}/{2π})(2πr) = rθ$

This formula assumes the angle is given in radians.

Sectors

The area of a sector = $({θ}/{2π})(πr^2)={θr^2}/2$

where r the radius of the circle, and the sector subtends the central angle θ.

Length of chord: $2r$sin$(θ/2)$, where the chord is subtended by central angle $θ$ of a circle with radius $r$.

Sagitta = perpendicular line from the centre of a chord to the circumference of a circle. The length of the sagitta is $r - r$cos$C/2$, where $r$ is the radius of the circle and C the angle at the centre subtended by the chord.

Segment

Area of a circle segment
Area of a circle segment

The area of a circle segment is:

$A_{seg} = 1/2(L_ar-L_c(r-h))=1/2r^2(θ-sin(θ))$

where $θ$ is the central angle in radians, $r$ is the radius, $L_a$ is the length of the arc, and $L_c$ is the length of the chord.

Radians

A radian is the size of the angle subtended by an arc the same length as the radius of the circle.

Since the circumference of a circle is 2πr, there are 2π radians in 360°.

One radian is equal to ${360°}/{2π} = 57.2957795°$. Since π is irrational, the radian cannot be expressed exactly in degrees.

Angle between a tangent and a chord
The angle between a tangent and a chord is equal to an angle subtended by the chord

Volumes

Cube

$V = L^3$

Sphere

$V = 4/3πr^3$

Cone and Pyramid

$V = 1/3BH$, where B is the area of the base, and H is the height, measured perpendicularly from the base to the apex.

Cylinder

$V = πr^2H$, where H is the height and r is the radius of the base circle.

Volume of a Cone

$V = 1/3 π r^2 h$

where V is the volume, r is the radius of the base circle, and h is the height of the cone.

Conic Sections

Conic Sections
Conic Sections: slicing a cone forms the shapes of four special functions
Hyperbola
Hyperbola: $x^2 - y^2 = 1$

Circle

$x^2+y^2=r^2$

Ellipse

${x/a}^2+{y/b}^2=r^2$, where $a$ and $b$ are the major and minor axes.

${x/a}^2+{y/b}^2= $sin$^2θ + $cos$^2θ$, where $θ$ is the angle to a point on the ellipse from the centre, and x and y are the Cartesian coordinates of the point, where $x=0$ and $y=0$ is the centre.

Parabola

$ax^2+bx+c$

Hyperbola

${x/a}^2-{y/b}^2=1$

Content © Andrew Bone. All rights reserved. Created : March 30, 2015

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