Tools
Shapes and angles
Work out the sides and angles of triangles and slopes, the area of flat shapes and the volume of solids, with a drawing to check which size is which.
How it works
How the calculator works
Pick a shape and enter the sizes you know. The tool works out the rest with the standard formulas below and draws the shape to one scale, with the sizes you entered in blue and the ones it worked out in black. Where a shape takes any two figures, such as a right-angled triangle or a slope, it uses the two you changed last.
About right-angled triangles
The longest side c, opposite the right angle, comes from Pythagoras: c² = a² + b². The angle A at the base, opposite the height a, comes from sine, cosine or tangent: sin A = a ÷ c, cos A = b ÷ c and tan A = a ÷ b. The two angles other than the right angle add up to 90°. Any two of the height, the base, the longest side and the angle give the rest.
About any triangle
With three sides, each angle comes from the cosine rule, cos A = (b² + c² − a²) ÷ 2bc, and the area from Heron's formula: half the perimeter s, then area = √(s(s − a)(s − b)(s − c)). With two sides and the angle C between them, the third side is √(a² + b² − 2ab cos C) and the area is ½ab sin C. Three sides only meet if the longest is shorter than the other two added together.
About slopes and gradients
A slope is a right-angled triangle on its side: the rise, the run measured level, and the length along the slope. The gradient in percent is 100 × rise ÷ run, the angle is tan⁻¹(rise ÷ run), and 1 in X is run ÷ rise, so 1 in 12 is 8.33% or 4.76°. The tool measures 1 in X along the level (run ÷ rise). Measured along the slope instead, the figure differs by less than 1% for slopes gentler than 1 in 7.
About areas
- Circle: π × diameter² ÷ 4, and the distance round it π × diameter.
- Rectangle: length × width; the diagonal is √(length² + width²).
- Ring: π × (outer² − inner²) ÷ 4.
- Trapezium: the two parallel sides added together, halved, times the height between them.
About volumes
- Box: length × width × height.
- Cylinder: π × diameter² ÷ 4 × length.
- Hollow tube: π × (outside² − inside²) ÷ 4 × length, where the inside diameter is the outside less twice the wall: the volume of the material itself.
- Cone: a third of a cylinder of the same base and height.
- Sphere: π × diameter³ ÷ 6.
Volumes in metres or millimetres are also given in litres: one cubic metre is 1,000 litres. To turn a volume into a weight, multiply by the density of the material, as our weight estimator does.
About units and rounding
Choose millimetres, metres, inches or feet; switching converts the figures you have typed. Inches and feet are converted at their exact definitions, 25.4 mm and 0.3048 m. Answers are rounded only for showing: lengths to 0.1 mm, 0.001 m, 0.01 in or 0.001 ft, angles to 0.01°, gradients to 0.001%, and areas and volumes to four significant figures.
FAQs
Common questions
How do I use Pythagoras to work out a sling leg length?
The leg is the longest side of a right-angled triangle. The other two sides are the height from the lifting point up to the hook, and the distance across from the lifting point to the point under the hook. Square both, add them and take the square root: 1.5 m across and 2 m up gives √(1.5² + 2²) = 2.5 m. Our rigging calculator works out the legs, their angles and the loads in them for a whole rig.
How do I work out the angle of a slope?
Divide the rise by the run and take the inverse tangent: angle = tan⁻¹(rise ÷ run). A rise of 1 m over a run of 10 m gives tan⁻¹(0.1) = 5.71°. The run is measured level, as on a plan. If you only know the length along the slope, the angle is sin⁻¹(rise ÷ length).
What is a 1 in 12 ramp in degrees?
About 4.76°. A 1 in 12 slope rises 1 for every 12 along the level, a gradient of 8.33% (1 ÷ 12 × 100), and tan⁻¹(1 ÷ 12) = 4.76°. A 1 in 20 slope is 5% and 2.86°.
How do I work out the area of a circle?
Area = π × diameter² ÷ 4, or π × radius². A circle 1 m across has an area of 3.1416 × 1 ÷ 4 = 0.785 m². If you only know the distance round it, divide that by π to get the diameter first.
How do I work out the volume of a cylinder or tank?
Volume = π × diameter² ÷ 4 × length. A cylinder 1 m across and 2 m long holds 1.571 m³, which is 1,571 litres. Use the inside diameter for what a tank holds; for the metal in its shell, choose Hollow tube. Dished ends hold more again, and our weight estimator allows for them.
Diameter or radius: which do I measure?
The diameter is the full width across the middle of a circle; the radius is half of it, from the centre to the edge. On site the diameter is easier to measure, or measure round the outside with a tape and divide by π (3.1416). Mixing them up makes an area four times too big or too small, so check which one a drawing gives.
How do I work out the length of a ramp?
The length along the ramp is the longest side of a right-angled triangle: √(rise² + run²). For a 1 m rise at 1 in 12, the run is 12 m and the length along the ramp is √(1 + 144) = 12.04 m. If you know the angle instead, the length is the rise ÷ sin(angle).
What is the difference between a gradient in percent and in degrees?
A gradient in percent is the rise for every 100 along the level: 100 × rise ÷ run. Degrees are the angle itself, tan⁻¹(rise ÷ run). The two are not in proportion: 100% is 45°, and a vertical face has no figure in percent at all.
Planning guidance only. This works out the geometry from the figures you enter. Measure on site, and the appointed person confirms the sizes, angles and weights used in the lift plan.
© Synergy Lifting Ltd. The shapes and angles tool and its code are our own work. You are welcome to use it to plan your lifts, but please do not copy, reproduce or reuse the tool, its code or its results without our written permission.
Get a quote
Ready for a quote?
Send us the details and we will come back with a quote, usually the same day for a desktop survey.
Or call 0800 008 6096
