Geometry

Surface areas of 3D shapes

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

Cube: \$A = L^3\$

\$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.

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

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.

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

Volumes

\$V = L^3\$

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

\$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.

\$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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