Most of school geometry never comes back, but a handful of formulas turn up every time you paint a room, order concrete, compare pizzas or cut a roof rafter. This article works through those jobs with real numbers from our geometry calculator, points out the unit mistakes that turn a correct formula into a wrong order, and explains the one triangle case that catches people out: two sides and an angle that is not between them.
Areas: paint, flooring and pizza
A wall. A wall 4 m wide and 2.7 m high has an area of 10.8 m². Take off a 1.2 × 1.0 m window and 9.6 m² is left to paint. Paint tins state coverage per litre per coat, so two coats double it. The rectangle formula is trivial; what goes wrong is forgetting the second coat or the openings.
A pizza. Is one 14-inch pizza more food than two 10-inch pizzas? Pizza sizes are diameters, so the radius is half. One 14-inch: π × 7² = 153.9 in². Two 10-inch: 2 × π × 5² = 157.1 in². Almost exactly the same, even though "14" sounds smaller than "20". Area grows with the square of the size: a pizza 41% wider has twice the area.
A circular rug, a round table, a garden bed: the same πr² with the radius, not the diameter. Using the diameter by mistake gives four times the area.
Volumes: concrete, gravel and balls
A post hole. A round hole 30 cm across and 80 cm deep is a cylinder: π × 0.15² × 0.8 = 0.0565 m³, or 56.5 litres of concrete. Ten posts need about 0.57 m³, which is how ready-mixed concrete is sold. Enter the dimensions in cm and ask for the result in m: the calculator converts the volume with the cube of the factor, so 56,549 cm³ becomes 0.0565 m³.
A gravel pile. Gravel tipped on the ground forms a cone. A pile 3 m across and 1.2 m high holds π × 1.5² × 1.2 / 3 = 2.83 m³: a third of the cylinder with the same base, not a half.
A ball. A football of radius 11 cm has a volume of 4/3 × π × 11³ = 5,575 cm³, about 5.6 litres of air.
Units: the expensive mistakes
Length units convert by one factor, areas by its square and volumes by its cube:
| Length | Area | Volume | |
|---|---|---|---|
| m to cm | × 100 | × 10,000 | × 1,000,000 |
| ft to in | × 12 | × 144 | × 1,728 |
| in to mm | × 25.4 | × 645.16 | × 16,387.064 |
Converting an area from m² to cm² by multiplying by 100 is out by a factor of 100. The calculator keeps input and output units separate for that reason, and the unit converter covers litres, gallons and the rest. One litre is exactly 1,000 cm³, or 0.001 m³.
A roof: solving a triangle from SAS
A gable roof has two rafters of 4.2 m meeting at the ridge at 120°. How wide is the building? That is two sides and the angle between them (SAS): by the law of cosines, c² = a² + b² − 2ab·cos C = 4.2² + 4.2² − 2 × 4.2 × 4.2 × cos 120°, so c = 7.27 m. The calculator also gives the other two angles (30° each, the roof pitch), the height of the ridge above the wall plates (the height to side c, 2.1 m) and the area of the gable end for cladding.
The SSA trap: one measurement, two triangles
Now a surveying-style problem. You know side b = 10 m, side a = 7 m, and the angle A = 35° opposite side a. How long is the third side? The law of sines gives
sin B = b · sin A / a = 10 × sin 35° / 7 = 0.819.
Two angles between 0° and 180° have that sine: B = 55.0° and B = 125.0°. Both leave a positive third angle (90.0° and 20.0°), so two different triangles fit the same measurements, with the third side c = 12.20 m or c = 4.18 m. Picture side a swinging from the end of b: it is long enough to reach the base line twice.
That is why SSA is not a congruence rule. The calculator detects the case, shows both triangles side by side and draws them. The other outcomes:
- If b · sin A / a is greater than 1, side a is too short to reach: no triangle.
- If it equals 1, side a just touches: one right triangle.
- If a ≥ b, or if A is obtuse, only one triangle is possible (when any).
In real work, the second triangle is resolved by another fact: the rough shape on site, a third measurement, or the knowledge that an angle must be acute.
Common mistakes
- Diameter for radius in πr²: four times the area, eight times a sphere's volume.
- Area units by the length factor: 1 m² is 10,000 cm², not 100.
- Cone as half a cylinder: it is a third.
- Degrees in a radians formula: the sector area θr²/2 needs θ in radians (60° = π/3).
- Assuming SSA has one answer: check whether the second triangle is also possible.
- Too many digits: a hole measured to the nearest centimetre does not have a volume known to the millilitre. Set the precision to 3 significant figures for measured inputs.
Keep the formulas for the jobs you do, and let a calculator do the arithmetic and the unit conversion: that is where errors actually happen.