Geometry Equations

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Geometry Equations

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Triangles
Area
Surface Area
Perimeter & Circumference
Volume
Miscellaneous

Geometry is a branch of pure mathematics that deals with the measurement, properties, and relationships of points, lines, angles, and two- and three-dimensional figures.

Geometry Facts

The sum of the interior angles of a triangle are equal to 180°
The sum of the interior angles of a quadrilateral are equal to 360°
The sine of the angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
The cosine of the angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
The tangent of the angle is the ratio of the length of the side opposite the angle to the length of side adjacent to the angle.
The cotangent of the angle is the ratio of the length of the side adjacent to the angle to the length of the side opposite the angle.
The secant of the angle is the ratio of the length of the hypotenuse to the length of the side adjacent to the angle
The cosecant of the angle is the ratio of the length of the hypotenuse to the length of the opposite side

Triangles

	Name	Description
	Right Angled	A Right Angled triangle has one 90° angle.
	Obtuse	An Obtuse triangle has one angle that is greater than 90°.
	Acute	An Acute triangle has all three angles less than 90°.
	Equilateral	An Equilateral triangle has all three sides the same length. All internal angles will be 60°.
	Isosceles	An Isosceles triangle has two sides with the same length.
	Scalene	A Scalene triangle has all three sides different lengths.

Area

Shape	Summary	Explanation
Square	X²	Multiply the base measurement by itself
Rectangle	X*Y	Base multiplied by height
Parallelogram	X*Y	Base multiplied by height
Trapezoid	½(A+B)*Y	Add the lengths of the two parallel sides (A+B) Divide this by two Multiply by the distance between the 2 parallel sides (height Y)
Triangle	½B*H	Half the Base length multiplied by the height. NB: To calculate the area of a triangle where the height is unknown see Heron's Formula below
Pentagon	½BH * 5	Half the length of one of the sides multiplied by the height, then multiplied by 5 NB - this is the calculation for a regular pentagon. For an irregular Pentagon the area of each of the triangles needs to be calculated separately, then all the areas added together
Circle	ΠR²	Pi (n=3.14 approx) multiplied by the square of the radius (Radius = half the diameter)
Sector	^θ/₃₆₀ΠR²	Divide the angle of the sector (θ) by 360, multiply by Pi (n=3.14 approx), then multiply by the square of the radius (R*R) (Radius = half the diameter)
Ellipse	ΠAB	Pi (n=3.14 approx) multiplied by half the width (A) multiplied by half the height (B)

Perimeter and Circumference

Shape	Summary	Explanation
Square	4X	Multiply the length of one side by 4
Rectangle	2(X + Y)	Add the length of one side to the height then multiply by 2
Parallelogram	2(X + Y)	Add the length of one side to the height then multiply by 2
Trapezoid	A + B + C + D	Add the length of all four sides
Triangle	A+B+C	Add the length of all three sides.
Circle	2ΠR	Pi (n=3.14 approx) multiplied by the radius multiplied by 2 (or Pi multiplied by the diameter)
Sector	θ ^π/₁₈₀R + 2R	Multiply the angle of the sector (θ) by Pi (n=3.14 approx) over 180 (=0.017 approx) Multiply this figure by by the radius (R) then add twice the radius
Ellipse	2Π √(^A²+B²/₂)	Pi (n=3.14 approx) multiplied by 2, multiplied by the square root of A squared (AA) plus B squared (BB) divided by 2

Surface Area

Shape	Summary	Explanation
Cube	6X²	Multiply the base measurement by itself (area of a square) Multiply this number by six
Rectangular Prism	2(XY) + 2(XZ) + 2(Y*Z)	Calculate the area of two sides (LengthHeight) and multiply by 2 Calculate the area of adjacent sides (LengthWidth) and multiply by 2 Calculate the area of ends (Height*Width) and multiply by 2 Add the three areas together
Trapezoidal Prism	(A+B)*H + AZ + BZ + 2YZ	Calculate the area of both ends by adding the lengths of the two parallel sides (A+B) and multiply by the distance between them (H). Calculate the area of the base (Length Z multiplied by Width A). Calculate the area of the top (Length Z multiplied by Width B). Calculate the area of the slanted sides (Length Z multiplied by Height Y) and multiply by 2 Add all the figures together. NB. For an irregular trapezoidal prism the area of all sides must be calcuated separately (as shown above) then added together.
Triangular Prism	2(B*½H)+BL+CL+AL	Calculate the area of two ends (Height * half the Base length) and multiply by 2 Calculate the area of the 3 sides (Length*Width) Add them all together
Triangular Pyramid	Regular ¹/₂BA + ³/₂BS	Calcualte the area of the base (Width B multiplied by Height A) then divide by 2. Calculate the area of the three slanted sides (Width multiplied by half the Slant Height S) multiply by 3. Add both figures together.
Triangular Pyramid	Irregular ½BA + ½BS + ½CS + ½DS	Calculate the area of all four triangles separately (as shown above) then add together.
Pyramid	XY + ½[(2X + 2Y)*S]	Calculate the area of the base (same as area of a rectangle/square). Calculate the length of the base perimeter (2X+2Y), multiply this figure by the slant height of the slope (see Pythagorus' theorem below), divide this figure by 2. Add the two figures together
Cone	ΠR² + ΠRL = ΠR(R + L)	Calculate the area of the base, Pi (n=3.14 approx) multiplied by the square of the radius Calculate the area of the side, Pi multiplied by the radius (R) multilplied by the slant height (L) Add the two figures together
Cylinder	2ΠR²+2ΠRH = 2ΠR(R + H)	Calculate the area of the base, Pi (n=3.14 approx) multiplied by the square of the radius, multiply this by 2 Calculate the area of the side, 2 times Pi multiplied by the radius, multiply this by the height Add the two figures together
Sphere	4ΠR²	Four times Pi (n=3.14 approx) multiplied by the square of the radius (Radius = half the diameter)

Volume

Shape	Summary	Explanation
Cube	X³	Multiply width by height by length.
Rectangular Prism	XYZ	Multiply width by height by length.
Trapezoidal Prism	½(A+B)YZ	Calculate the area of one end by adding the lengths of the two parallel sides (A+B), divide this by two then multiply by the distance between the two parallel sides (height Y) Multiply by the length (Z)
Triangular Prism	B½HL	Multiply width by height by length then divide by 2
Triangular Pyramid	¹/₆ABH	Calculate the area of the base by multiplying the width (B) by the height (A) then dividing by 2. Multiply the base area by the height (H) then divide by 3
Pyramid	¹/₃(XYH)	Calculate the area of the base (X*Y), multiply by the height (H) then divide by 3
Cone	¹/₃ΠR²H	Calculate the area of the base; Pi (n=3.14 approx) multiplied by the square of the radius. Multiply this figure by the height then divide by 3
Cylinder	ΠR²H	Calculate the area of the base; Pi (n=3.14 approx) multiplied by the square of the radius (R*R). Multiply this figure by the height.
Sphere	⁴/₃ΠR³	Pi (n=3.14 approx) multiplied by the cube of the radius (RRR) Multiply this figure by 4 then divide by 3

Miscellaneous

Pythagoras' Theorem		Pythagoras' Theorem states that: The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides, a² + b² = c² or c = √(a² + b²) This is the formula for calculating the length of the longest side (c) in a right angled triangle. In simple terms it means that if you add the squares of the two shorter sides together (aa + bb), the longest side (c) is the square root of that number.
Heron's Formula		Heron's Formula is used to calculate the area of a triangle when the length of all three sides (a, b and c) are known but the height of the triangle is not.