Formula Sheet

List of all mathematical formulas on the site, organized by topic with quick search.

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Area

Area of a Rectangle
A=wร—hA = w \times h
Area of a Square
A=s2A = s^2
Area of a Triangle
A=bร—h2A = \frac{b \times h}{2}
Area of a Right-angled Triangle
A=ab2A = \frac{ab}{2}
Pythagorean Theorem
a2+b2=c2a^2 + b^2 = c^2
For a right-angled triangle: the square of the hypotenuse (c) is equal to the sum of the squares of the legs (a and b).
Altitude of an Isosceles Triangle
h=b2โˆ’(a2)2h = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}
Height of an Equilateral Triangle
h=a32h = a \frac{\sqrt{3}}{2}
Area of an Equilateral Triangle
A=a234A = a^2 \frac{\sqrt{3}}{4}
Area of a 30-60-90 Triangle
A=a232A = a^2 \frac{\sqrt{3}}{2}
where a is the shortest leg.
Ratio of Inscribed Square to Circle
AsquareAcircle=2ฯ€\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi}
Ratio of Inscribed Circle to Square
AcircleAsquare=ฯ€4\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4}
Area of a Circle
A=ฯ€r2A = \pi r^2

Percentage

Percentage Formula
p=PartWholeร—100%p = \frac{\text{Part}}{\text{Whole}} \times 100\%
Total Amount Formula
T=A+B+โ€ฆT = A + B + \dots
Percentage Increase Formula
Vnew=Vorigร—(1+p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 + \frac{p}{100}\right)
Percentage Rate of Increase Formula
p=(VnewVorigโˆ’1)ร—100p = \left(\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right) \times 100
Percentage Decrease Formula
Vnew=Vorigร—(1โˆ’p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 - \frac{p}{100}\right)
Percentage Rate of Decrease Formula
p=(1โˆ’VnewVorig)ร—100p = \left(1 - \frac{V_{\text{new}}}{V_{\text{orig}}}\right) \times 100
Working Backward from Increase
Vorig=Vnew1+p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 + \frac{p}{100}}
Original Value from Increase Amount
Vorig=Vinc(p100)V_{\text{orig}} = \frac{V_{\text{inc}}}{\left(\frac{p}{100}\right)}
Working Backward from Decrease
Vorig=Vnew1โˆ’p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 - \frac{p}{100}}
Original Value from Decrease Amount
Vorig=Vdec(p100)V_{\text{orig}} = \frac{V_{\text{dec}}}{\left(\frac{p}{100}\right)}
General Percentage Change Formula
Vnew=Vorigร—(1ยฑp100)V_{\text{new}} = V_{\text{orig}} \times \left(1 \pm \frac{p}{100}\right)
Use + for increase and โˆ’ for decrease.
General Percentage Rate Formula
p=โˆฃVnewVorigโˆ’1โˆฃร—100p = \left|\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right| \times 100
General Working Backward Formula
Vorig=Vnew1ยฑp100V_{\text{orig}} = \frac{V_{\text{new}}}{1 \pm \frac{p}{100}}
Use + for increase and โˆ’ for decrease.
Total Multiplier
M=(1ยฑp1100)ร—(1ยฑp2100)ร—โ€ฆM = \left(1 \pm \frac{p_1}{100}\right) \times \left(1 \pm \frac{p_2}{100}\right) \times \dots
New Amount
Vnew=Vorigร—MV_{\text{new}} = V_{\text{orig}} \times M
Net Percentage
pnet=(Mโˆ’1)ร—100p_{\text{net}} = (M - 1) \times 100

Formula Sheet

List of all mathematical formulas on the site, organized by topic.

Perimeter

Total Perimeter (with Holes)
Ptotal=Pouter+PinnerP_{\text{total}} = P_{\text{outer}} + P_{\text{inner}}
Perimeter of a Rectangle
P=2(w+h)P = 2(w + h)
Perimeter of a Square
P=4sP = 4s
Perimeter of a Right-angled Triangle
P=a+b+a2+b2P = a + b + \sqrt{a^2 + b^2}

Area

Area of a Rectangle
A=wร—hA = w \times h
Area of a Square
A=s2A = s^2
Area of a Triangle
A=bร—h2A = \frac{b \times h}{2}
Area of a Right-angled Triangle
A=ab2A = \frac{ab}{2}
Pythagorean Theorem
a2+b2=c2a^2 + b^2 = c^2
For a right-angled triangle: the square of the hypotenuse (c) is equal to the sum of the squares of the legs (a and b).
Altitude of an Isosceles Triangle
h=b2โˆ’(a2)2h = \sqrt{b^2 - \left(\frac{a}{2}\right)^2}
Height of an Equilateral Triangle
h=a32h = a \frac{\sqrt{3}}{2}
Area of an Equilateral Triangle
A=a234A = a^2 \frac{\sqrt{3}}{4}
Area of a 30-60-90 Triangle
A=a232A = a^2 \frac{\sqrt{3}}{2}
where a is the shortest leg.
Ratio of Inscribed Square to Circle
AsquareAcircle=2ฯ€\frac{A_{\text{square}}}{A_{\text{circle}}} = \frac{2}{\pi}
Ratio of Inscribed Circle to Square
AcircleAsquare=ฯ€4\frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{\pi}{4}
Area of a Circle
A=ฯ€r2A = \pi r^2

Angle Fundamentals

Sum of Angles in a Triangle
a+b+c=180โˆ˜a + b + c = 180^\circ

Percentage

Percentage Formula
p=PartWholeร—100%p = \frac{\text{Part}}{\text{Whole}} \times 100\%
Total Amount Formula
T=A+B+โ€ฆT = A + B + \dots
Percentage Increase Formula
Vnew=Vorigร—(1+p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 + \frac{p}{100}\right)
Percentage Rate of Increase Formula
p=(VnewVorigโˆ’1)ร—100p = \left(\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right) \times 100
Percentage Decrease Formula
Vnew=Vorigร—(1โˆ’p100)V_{\text{new}} = V_{\text{orig}} \times \left(1 - \frac{p}{100}\right)
Percentage Rate of Decrease Formula
p=(1โˆ’VnewVorig)ร—100p = \left(1 - \frac{V_{\text{new}}}{V_{\text{orig}}}\right) \times 100
Working Backward from Increase
Vorig=Vnew1+p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 + \frac{p}{100}}
Original Value from Increase Amount
Vorig=Vinc(p100)V_{\text{orig}} = \frac{V_{\text{inc}}}{\left(\frac{p}{100}\right)}
Working Backward from Decrease
Vorig=Vnew1โˆ’p100V_{\text{orig}} = \frac{V_{\text{new}}}{1 - \frac{p}{100}}
Original Value from Decrease Amount
Vorig=Vdec(p100)V_{\text{orig}} = \frac{V_{\text{dec}}}{\left(\frac{p}{100}\right)}
General Percentage Change Formula
Vnew=Vorigร—(1ยฑp100)V_{\text{new}} = V_{\text{orig}} \times \left(1 \pm \frac{p}{100}\right)
Use + for increase and โˆ’ for decrease.
General Percentage Rate Formula
p=โˆฃVnewVorigโˆ’1โˆฃร—100p = \left|\frac{V_{\text{new}}}{V_{\text{orig}}} - 1\right| \times 100
General Working Backward Formula
Vorig=Vnew1ยฑp100V_{\text{orig}} = \frac{V_{\text{new}}}{1 \pm \frac{p}{100}}
Use + for increase and โˆ’ for decrease.
Total Multiplier
M=(1ยฑp1100)ร—(1ยฑp2100)ร—โ€ฆM = \left(1 \pm \frac{p_1}{100}\right) \times \left(1 \pm \frac{p_2}{100}\right) \times \dots
New Amount
Vnew=Vorigร—MV_{\text{new}} = V_{\text{orig}} \times M
Net Percentage
pnet=(Mโˆ’1)ร—100p_{\text{net}} = (M - 1) \times 100
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