← Tutti gli articoli

Geometry Formulas Cheat Sheet: Area and Volume

Pubblicato il 2 novembre 2026 · Di Sumzo team

A geometry formulas cheat sheet: area and perimeter, circles, Pythagoras and trigonometry, angles, and the volume and surface area of 3D shapes.

This geometry formulas cheat sheet lists the area, perimeter, volume and angle formulas used most in school geometry, each with a quick example. Learn the few ideas behind them — area is base times height, volume is area times length — and most of the list follows.

Area and perimeter of 2D shapes

• Rectangle: $A = lw$, $P = 2(l + w)$. A 5 by 3 rectangle: area 15, perimeter 16.
• Square: $A = s^2$, $P = 4s$.
• Triangle: $A = \frac{1}{2}bh$. Base 10, height 6: area 30.
• Parallelogram: $A = bh$ (perpendicular height).
• Trapezoid (trapezium): $A = \frac{1}{2}(a + b)h$. Sides 6 and 10, height 4: area 32.
• Equilateral triangle: $A = \frac{\sqrt{3}}{4}s^2$.

Circles

• Circumference: $C = 2\pi r = \pi d$.
• Area: $A = \pi r^2$. Radius 5: area $25\pi \approx 78.54$.
• Arc length: $\frac{\theta}{360^\circ} \times 2\pi r$.
• Sector area: $\frac{\theta}{360^\circ} \times \pi r^2$. A 90° sector of radius 4: $\frac{1}{4} \times 16\pi = 4\pi$.

Right triangles

• Pythagoras: $a^2 + b^2 = c^2$, with $c$ the hypotenuse. Legs 6 and 8: $c = 10$.
• Trigonometry (SOH CAH TOA): $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$.
• Special triangles: 45–45–90 sides $1 : 1 : \sqrt{2}$; 30–60–90 sides $1 : \sqrt{3} : 2$.

Any triangle

• Angle sum: $180^\circ$.
• Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$.
• Cosine rule: $c^2 = a^2 + b^2 - 2ab\cos C$.
• Area from two sides and the angle between them: $\frac{1}{2}ab\sin C$.
• Heron’s formula: $\sqrt{s(s - a)(s - b)(s - c)}$ with $s = \frac{a + b + c}{2}$. Sides 5, 6, 7: $s = 9$, area $\sqrt{216} = 6\sqrt{6}$.

Angles

• Angles on a straight line add to $180^\circ$; around a point to $360^\circ$.
• Vertically opposite angles are equal.
• Interior angles of an $n$-sided polygon add to $(n - 2) \times 180^\circ$: a hexagon, $720^\circ$.
• Exterior angles of any polygon add to $360^\circ$.

Volume of 3D shapes

• Cube: $V = s^3$. Cuboid: $V = lwh$.
• Prism: area of cross-section × length. A triangular prism with a triangle of base 6 and height 4, 10 long: $12 \times 10 = 120$.
• Cylinder: $V = \pi r^2 h$. Radius 2, height 7: $28\pi$.
• Cone: $V = \frac{1}{3}\pi r^2 h$. Radius 3, height 4: $12\pi$.
• Pyramid: $V = \frac{1}{3} \times \text{base area} \times h$.
• Sphere: $V = \frac{4}{3}\pi r^3$.

Surface area

• Cube: $6s^2$. Cuboid: $2(lw + lh + wh)$.
• Cylinder: $2\pi r^2 + 2\pi rh$.
• Cone: $\pi r^2 + \pi r l$, with $l$ the slant height.
• Sphere: $4\pi r^2$.

Coordinate geometry

• Distance: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Midpoint: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Circle with centre $(h, k)$ and radius $r$: $(x - h)^2 + (y - k)^2 = r^2$.

Worked example: combining formulas

A window is a rectangle 10 cm tall and 6 cm wide, with a semicircle on top whose diameter is the 6 cm width.

• Area: rectangle $10 \times 6 = 60$, plus half a circle of radius 3: $\frac{1}{2}\pi \times 3^2 = 4.5\pi$. Total $60 + 4.5\pi \approx 74.14\text{ cm}^2$.
• Perimeter: the three straight sides $10 + 6 + 10 = 26$ (the top edge of the rectangle is inside the shape, so it isn’t counted), plus half the circumference, $\pi \times 3 = 3\pi$. Total $26 + 3\pi \approx 35.42$ cm.

Splitting a shape into pieces you know is the key move in most composite-shape questions. The same idea works in 3D: a hemisphere of radius 3 is half a sphere, so its volume is $\frac{1}{2} \times \frac{4}{3}\pi \times 27 = 18\pi \approx 56.55$.

How to use this sheet

The pattern behind the formulas saves memorising: pointed shapes (cones, pyramids) have $\frac{1}{3}$, prisms and cylinders don’t; areas are in square units and volumes in cubic units. To practise, see the lessons on triangles, the area of a trapezium and prisms and cylinders, and check answers with the area calculator or volume calculator.

Circle theorems

• The angle in a semicircle is $90^\circ$.
• The angle at the centre is twice the angle at the circumference standing on the same arc.
• Angles in the same segment are equal.
• Opposite angles of a cyclic quadrilateral add up to $180^\circ$.
• A tangent meets the radius at $90^\circ$.

Similar shapes

If lengths are multiplied by a scale factor $k$, areas are multiplied by $k^2$ and volumes by $k^3$. Doubling every length ($k = 2$) makes the area 4 times as big and the volume 8 times as big.

Units

• 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
• 1 litre = 1,000 cm³.
• Always convert lengths to the same unit before using a formula.

Transformations

• Translation: slide by a vector, such as 3 right and 2 up.
• Reflection: in a line, such as the y-axis: $(x, y) \to (-x, y)$.
• Rotation: by an angle about a point — $90^\circ$ anticlockwise about the origin sends $(x, y)$ to $(-y, x)$.
• Enlargement: by a scale factor from a centre.

Frequently asked questions

What geometry formulas should I know? The area of rectangles, triangles and circles, the circumference of a circle, Pythagoras, the angle sum of a triangle, and the volume of prisms and cylinders.

What is the difference between area and volume formulas? Area measures a flat surface, in square units; volume measures space, in cubic units. Volume formulas usually multiply an area by a length.

Why do cones and pyramids have one third? A cone holds exactly one third of the cylinder with the same base and height — and a pyramid one third of the matching prism.

Prova Sumzo AI