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Geometry Formula Sheet

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Formula Sheet • 2D, 3D & Polygon Rules

Geometry Formula Sheet

All essential geometry formulas in one place — areas and perimeters of 2D shapes, 3D volumes and surface areas, Pythagoras theorem, and polygon angles.

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A4 / Letter portrait optimized • Pure black-and-white ink saving mode Mode: Comprehensive Study Guide (With Examples)

Area (2D Shapes)

Rectangle
A=l×wA = l \times w

l = length • w = width

Worked Example
A=8×5=40 cm2\displaystyle A = 8 \times 5 = 40\text{ cm}^2
Area Calculator →
Triangle
A=12bhA = \frac{1}{2} b h

b = base • h = perpendicular height

Worked Example
A=12(6)(4)=12 cm2\displaystyle A = \frac{1}{2}(6)(4) = 12\text{ cm}^2
Area Calculator →
Circle
A=πr2A = \pi r^2

r = radius from circle centre to perimeter

Worked Example
A=π(52)=25π≈78.54 cm2\displaystyle A = \pi(5^2) = 25\pi \approx 78.54\text{ cm}^2
Area Calculator →
Trapezoid (Trapezium)
A=a+b2×hA = \frac{a + b}{2} \times h

a, b = parallel horizontal sides • h = height

Worked Example
A=4+102×3=7×3=21 cm2\displaystyle A = \frac{4 + 10}{2} \times 3 = 7 \times 3 = 21\text{ cm}^2
Area Calculator →
Parallelogram
A=b×hA = b \times h

b = base length • h = perpendicular height

Worked Example
A=9×4=36 cm2\displaystyle A = 9 \times 4 = 36\text{ cm}^2
Area Calculator →
Sector of a Circle
A=θ360∘×πr2A = \frac{\theta}{360^\circ} \times \pi r^2

θ = central angle in degrees • r = radius

Worked Example
A=60∘360∘×π(62)=6π≈18.85 cm2\displaystyle A = \frac{60^\circ}{360^\circ} \times \pi(6^2) = 6\pi \approx 18.85\text{ cm}^2
Area Calculator →

Perimeter & Circumference

Polygon / Rectangle Perimeter
P=2(l+w)P = 2(l + w)

Sum of all outer side lengths around the 2D perimeter boundary.

Worked Example
P=2(8+5)=2(13)=26 cm\displaystyle P = 2(8 + 5) = 2(13) = 26\text{ cm}
Perimeter Calculator →
Circle Circumference
C=2πr=πdC = 2\pi r = \pi d

r = radius • d = diameter (d = 2r)

Worked Example
C=2π(4)=8π≈25.13 cm\displaystyle C = 2\pi(4) = 8\pi \approx 25.13\text{ cm}
Perimeter Calculator →
Arc Length of a Sector
L=θ360∘×2πrL = \frac{\theta}{360^\circ} \times 2\pi r

Fraction of full 360° circle circumference spanned by angle θ.

Worked Example
L=90∘360∘×2π(8)=4π≈12.57 cm\displaystyle L = \frac{90^\circ}{360^\circ} \times 2\pi(8) = 4\pi \approx 12.57\text{ cm}
Perimeter Calculator →

Pythagoras' Theorem (Right Triangles)

The Theorem
a2+b2=c2a^2 + b^2 = c^2

a, b = shorter perpendicular legs • c = hypotenuse

Worked Example
32+42=9+16=25=52  ✓\displaystyle 3^2 + 4^2 = 9 + 16 = 25 = 5^2 \;\checkmark
Pythagoras Calculator →
Find Hypotenuse (c)
c=a2+b2c = \sqrt{a^2 + b^2}

Square both short legs, add them together, take the square root.

Worked Example
c=52+122=169=13 cm\displaystyle c = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}
Pythagoras Calculator →
Find Shorter Leg (a or b)
a=c2−b2a = \sqrt{c^2 - b^2}

Square hypotenuse c, subtract other leg squared, take square root.

Worked Example
a=102−62=64=8 cm\displaystyle a = \sqrt{10^2 - 6^2} = \sqrt{64} = 8\text{ cm}
Pythagoras Calculator →

Volume (3D Shapes)

Cuboid / Rectangular Prism
V=l×w×hV = l \times w \times h

l = length • w = width • h = height

Worked Example
V=4×3×5=60 cm3\displaystyle V = 4 \times 3 \times 5 = 60\text{ cm}^3
Volume Calculator →
Cylinder
V=πr2hV = \pi r^2 h

r = circular base radius • h = cylinder height

Worked Example
V=π(32)(8)=72π≈226.19 cm3\displaystyle V = \pi(3^2)(8) = 72\pi \approx 226.19\text{ cm}^3
Volume Calculator →
Sphere
V=43πr3V = \frac{4}{3} \pi r^3

r = radius from sphere centre to surface

Worked Example
V=43π(33)=36π≈113.10 cm3\displaystyle V = \frac{4}{3}\pi(3^3) = 36\pi \approx 113.10\text{ cm}^3
Volume Calculator →
Cone
V=13πr2hV = \frac{1}{3} \pi r^2 h

Exactly one third the volume of a cylinder with identical r and h.

Worked Example
V=13π(32)(5)=15π≈47.12 cm3\displaystyle V = \frac{1}{3}\pi(3^2)(5) = 15\pi \approx 47.12\text{ cm}^3
Volume Calculator →
Pyramid
V=13×Abase×hV = \frac{1}{3} \times A_{\text{base}} \times h

A = polygon base area • h = apex perpendicular height

Worked Example
V=13(30)(9)=90 cm3\displaystyle V = \frac{1}{3}(30)(9) = 90\text{ cm}^3
Volume Calculator →
General Uniform Prism
V=Across-section×lV = A_{\text{cross-section}} \times l

Multiply constant cross-sectional area by length or prism depth l.

Worked Example
V=15 cm2×10=150 cm3\displaystyle V = 15\text{ cm}^2 \times 10 = 150\text{ cm}^3
Volume Calculator →

Total Surface Area

Cylinder (Total Surface Area)
A=2πrh+2πr2A = 2\pi r h + 2\pi r^2

Curved side area (2πrh) plus both top and bottom circle faces (2πr²).

Worked Example

Cylinder with radius r = 3 cm, height h = 5 cm:

Calculation
A=2π(3)(5)+2π(32)=48π≈150.8 cm2\displaystyle A = 2\pi(3)(5) + 2\pi(3^2) = 48\pi \approx 150.8\text{ cm}^2
Geometry Tools →
Sphere (Surface Area)
A=4πr2A = 4\pi r^2

The total surface area of a sphere is exactly 4 times its great circle area.

Worked Example

Sphere with radius r = 3 cm:

Calculation
A=4π(32)=36π≈113.10 cm2\displaystyle A = 4\pi(3^2) = 36\pi \approx 113.10\text{ cm}^2
Geometry Tools →
Cone (Curved Surface Area)
A=πrl(l=r2+h2)A = \pi r l \quad (l = \sqrt{r^2 + h^2})

r = radius • l = slant height (from base rim to apex)

Worked Example

Cone with radius r = 3 cm, slant height l = 5 cm:

Calculation
A=π(3)(5)=15π≈47.12 cm2\displaystyle A = \pi(3)(5) = 15\pi \approx 47.12\text{ cm}^2
Geometry Tools →

Polygon Angle Rules

Sum of Interior Angles
Sum=(n−2)×180∘\text{Sum} = (n - 2) \times 180^\circ

n = number of polygon sides

Worked Example
Pentagon (n=5):(5−2)×180∘=540∘\displaystyle \text{Pentagon }(n=5): (5 - 2) \times 180^\circ = 540^\circ
Each Interior Angle (Regular)
Angle=(n−2)×180∘n\text{Angle} = \frac{(n - 2) \times 180^\circ}{n}

Applicable to equiangular, regular polygons with equal side lengths.

Worked Example
Hexagon (n=6):4×180∘6=120∘\displaystyle \text{Hexagon }(n=6): \frac{4 \times 180^\circ}{6} = 120^\circ
Each Exterior Angle
Exterior=360∘n\text{Exterior} = \frac{360^\circ}{n}

Exterior angles of any convex polygon always sum strictly to 360°.

Worked Example
Octagon (n=8):360∘8=45∘\displaystyle \text{Octagon }(n=8): \frac{360^\circ}{8} = 45^\circ
Curriculum & Revision Note

About These Geometry Formulas

Geometry accounts for a significant proportion of secondary school maths marks. This sheet brings together area and perimeter formulas for 2D shapes, 3D volume and surface area equations, right-angle Pythagoras rules, and polygon angle relationships.

Need step-by-step solutions? Open Pythagoras Calculator →