Mensuration Formulas: All Area and Volume Formulas

How to use this sheet

The formulas are grouped into flat shapes and solids. In every formula a letter stands for a length: a, b and c for sides, l for a length or a slant height, h for a height measured at right angles, r and R for radii, and p and q for diagonals. All lengths must be in the same unit. Areas then come out in square units and volumes in cubic units. Tap a shape's name for its figure, the reason behind each formula and worked examples.

The sheet covers 16 flat shapes and 10 solids. Where a shape has more than one useful measure, such as the curved and the total surface of a cylinder, each is listed on its own line. Square roots are written with the √ sign, and π stands for the usual constant, about 3.1416 or 22/7.

A formula sheet is for revision, not for learning. Formulas stay in the mind when you know where they come from, so read the "Why the formulas work" part of a shape once before you rely on the sheet.

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Flat shapes: area and perimeter

Shape	Formulas
Square	Area = a² Perimeter = 4a Diagonal = a√2
Rectangle	Area = l × b Perimeter = 2(l + b) Diagonal = √(l² + b²)
Triangle	Area = ½ × b × h Perimeter = a + b + c Area = √(s(s − a)(s − b)(s − c)), where s is half the perimeter
Right-angled triangle	Hypotenuse c = √(a² + b²) Area = ½ × a × b Perimeter = a + b + c
Equilateral triangle	Area = (√3 ⁄ 4) a² Perimeter = 3a Height = (√3 ⁄ 2) a
Isosceles triangle	Height = √(a² − b² ⁄ 4) Area = ½ × b × height Perimeter = 2a + b
Parallelogram	Area = b × h Perimeter = 2(a + b)
Rhombus	Area = ½ × p × q Side = ½ √(p² + q²) Perimeter = 4 × side
Trapezium	Area = ½ (a + b) × h
Kite	Area = ½ × p × q
Circle	Area = πr² Circumference = 2πr Diameter = 2r
Semicircle	Area = ½ πr² Curved edge = πr Perimeter = πr + 2r
Sector of a circle	Area = (θ ⁄ 360) × πr² Arc length = (θ ⁄ 360) × 2πr Perimeter = arc length + 2r
Ellipse	Area = πab Perimeter ≈ π [3(a + b) − √((3a + b)(a + 3b))]
Ring (annulus)	Area = π(R² − r²) Outer edge + inner edge = 2π(R + r) Width = R − r
Regular polygon	Area = na² ⁄ (4 tan(180° ⁄ n)) Perimeter = na Interior angle = (n − 2) × 180° ⁄ n Apothem = a ⁄ (2 tan(180° ⁄ n))

Solids: surface area and volume

Shape	Formulas
Cube	Volume = a³ Total surface area = 6a² Lateral surface area = 4a² Diagonal = a√3
Cuboid	Volume = l × b × h Total surface area = 2(lb + bh + hl) Lateral surface area = 2h(l + b) Diagonal = √(l² + b² + h²)
Cylinder	Volume = πr²h Curved surface area = 2πrh Total surface area = 2πr(r + h)
Cone	Slant height l = √(r² + h²) Volume = ⅓ πr²h Curved surface area = πrl Total surface area = πr(r + l)
Sphere	Volume = (4 ⁄ 3) πr³ Surface area = 4πr²
Hemisphere	Volume = (2 ⁄ 3) πr³ Curved surface area = 2πr² Total surface area = 3πr²
Square pyramid	Slant height l = √(h² + a² ⁄ 4) Volume = ⅓ a²h Lateral surface area = 2al Total surface area = a² + 2al
Triangular prism	End area by Heron's formula Volume = end area × L Lateral surface area = (a + b + c) × L Total surface area = lateral area + 2 × end area
Frustum of a cone	Slant height l = √(h² + (R − r)²) Volume = ⅓ πh(R² + Rr + r²) Curved surface area = π(R + r)l Total surface area = π(R + r)l + π(R² + r²)
Hollow cylinder (pipe)	Volume = π(R² − r²)h Curved surface area = 2π(R + r)h Total surface area = 2π(R + r)h + 2π(R² − r²) Capacity = πr²h

Patterns worth noticing

Half of something. A triangle is half a parallelogram, and a rhombus or kite is half the rectangle drawn round its diagonals.
Base area times height. Cube, cuboid, prism and cylinder all use it for volume.
One third. A cone is a third of its cylinder and a pyramid a third of its prism.
Powers match dimensions. A length formula has lengths to the first power, an area formula has two lengths multiplied, and a volume formula three. If your formula for an area has only one length in it, something is missing.

For the value of π see what is π, and for units see units of area and volume.

All mensuration formulas

How to use this sheet

Flat shapes: area and perimeter

Solids: surface area and volume

Patterns worth noticing