Partition without gaps or overlap
An exact polygon can be partitioned into simpler nonoverlapping polygons and their areas added. Subtract holes once. Curved boundaries approximated by straight segments require an error allowance; adding more pieces alone does not guarantee a better measurement.
Areas that repeat: rectangles, trapezoids and rhombuses
A rectangle is length times width. A trapezoid is the average of its two parallel sides times the perpendicular height, and a rhombus is half the product of its two diagonals. In each case the measurement that trips people up is the height: measure straight across, at a right angle to the base, and not along a slanted side.
Circles, rings and arcs
The area of a circle is pi times the radius squared. A ring, or annulus, is the big circle minus the hole, so a ring of outer radius 10 and inner radius 6 has an area of about 201. Arc length and sector area are fractions of the whole circle set by the angle.
Work with the radius, and halve a diameter before you use it.
Convert thickness before calculating volume
Assume a rectangular 10-by-12-foot slab and four-inch thickness. Four inches is one-third foot, so volume is 40 ft³, or 40/27 = 1.48148 yd³ before allowance. Using four feet instead of four inches multiplies the result by twelve.
Geometry does not select concrete strength, reinforcement, subgrade or structural thickness.
Solids: volume and surface area
Volume is how much a shape holds, and surface area is how much covers it. A box, a cylinder, a cone, a sphere and a pyramid each have a formula. A pyramid or a cone holds exactly a third of the box or cylinder with the same base and height.
A hemisphere is half a sphere. Pick volume for fill and capacity, and surface area for paint, wrap and insulation.
Cut-off shapes: frustums and capsules
A bucket, a lampshade or a planter is often a cone with the tip cut off, called a frustum, and a pill-shaped tank is a cylinder with two rounded ends, called a capsule. Both have their own formulas. The common mistake with a capsule is measuring the overall length, which includes the caps, and using it as the length of the straight section.
Angles and triangles
When you know two sides and the angle between them, the law of cosines gives the third side. When you know two angles and one side, the law of sines finds the others. A right triangle is a special case with its own tools: the Pythagorean theorem for the sides and the sine, cosine and tangent for the angles.
Check whether the corner is truly square before you rely on a right triangle.
Units and conversions
Keep every measurement in one unit before you calculate. Mixing inches and feet in a formula gives a wrong answer that looks reasonable. Area units are squared and volume units are cubed, so converting from feet to inches multiplies an area by 144 and a volume by 1,728.
Convert at the end, and only once.
Check the answer with a quick estimate
After any calculation, compare the result with a rough figure. A room of 15 by 12 feet is close to 200 square feet, so an answer of 20 or 2,000 signals a slip. Doubling every edge of a solid multiplies its volume by eight and its surface by four, which is another useful sanity check.
A calculator removes arithmetic errors and leaves measurement errors, so measure twice.
Assumed inputs and calculated outputs
Trapezoid Area Calculator: input sensitivity
| Top base | Area | Midsegment length |
|---|---|---|
| 8 | 28 | 7 |
| 9 | 30 | 7.5 |
| 10 | 32 | 8 |
| 11 | 34 | 8.5 |
| 12 | 36 | 9 |
Open the Trapezoid Area Calculator to use your own numbers →
What goes wrong
Geometry for DIY and design: area, volume and angles: common errors
| The mistake | Why it misleads | Do this instead |
|---|---|---|
| Using a slanted side as the height | The slanted side is longer than the perpendicular height, which overstates the area. | Measure straight across at a right angle to the base. |
| Entering a diameter where a radius is asked | Doubling the radius quadruples an area and multiplies a volume by eight. | Halve a diameter before you enter it, and read the label on each field. |
| Forgetting the one third for a cone or pyramid | Without it the result is the volume of the box or cylinder, three times too large. | Divide by three for a cone or a pyramid. |
| Mixing units in one calculation | Feet and inches in the same formula give a result that looks plausible and is wrong. | Convert every measurement to one unit before you calculate. |
Geometry for DIY and design: area, volume and angles: worked calculation
Using the Trapezoid Area Calculator worked example, moving Top base from 8 to 12 changes the Area from 28 to 36.
| Input | Value | Definition |
|---|---|---|
| Top base | 10 | One of the two parallel sides. |
| Bottom base | 6 | The other parallel side. |
| Height | 4 | The perpendicular distance between the two bases, not the slanted side. |
| Area | 32 |
|---|---|
| Midsegment length | 8 |
| Top base | Area | Midsegment length |
|---|---|---|
| 8 | 28 | 7 |
| 9 | 30 | 7.5 |
| 10 | 32 | 8 |
| 11 | 34 | 8.5 |
| 12 | 36 | 9 |
Which input moves the area most in the trapezoid area calculator?
Ranked by how far each moves the area across the range tested: height (6.4, 20%), top base (4, 13%) and bottom base (2.4, 7.5%).
If I double height in the trapezoid area calculator, does the area double?
Doubling it from 4 to 8 takes the area from 32 to 64, which is 2.00 times the worked-example figure. So the result scales almost exactly in proportion. Halving it to 2 gives 16.
Geometry for DIY and design: area, volume and angles: points to carry forward
- Name the shape and label its parts before you choose a formula.
- Break irregular shapes into rectangles, triangles and circles, and add the areas.
- An assumed isosceles triangle with base 10 and equal sloping sides 13 has perpendicular height 12, so area is 60 square units. Treating a sloping side as height would incorrectly give 65.
- A one-unit-sided cube contains one cubic unit but has six square units of exterior surface. Doubling every side gives eight cubic units and twenty-four square units: capacity and surface coverage scale differently.
- Keep every measurement in one unit, and remember that area units are squared and volume units cubed.
How do I find the area of an irregular shape?
Split it into rectangles, triangles and circles, work out each area, add them and subtract any holes.
What is the formula for the area of a trapezoid?
Half the sum of the two parallel sides times the perpendicular height. The trapezoid area calculator also gives the midsegment.
How do I calculate the volume of a cone or a pyramid?
One third of the base area times the height. It is a third of the cylinder or box with the same base and height.
What is a frustum?
A cone or pyramid with the top sliced off by a plane parallel to the base. Buckets and lampshades are common examples.
How do I find the third side of a triangle?
With two sides and the angle between them, use the law of cosines. With two angles and a side, use the law of sines.
How do I convert square feet to square yards?
Divide by 9. For cubic feet to cubic yards, divide by 27.
What is the difference between area and perimeter?
Perimeter is the distance around a shape, and area is the surface inside it. They use different units and answer different questions.
How do I find the volume of a pool or a tank?
Use the shape's volume formula in feet, then multiply cubic feet by 7.48 for US gallons. A round or oval pool is about 78.5 percent of the rectangle with the same length and width.
Why does doubling every edge multiply volume by eight?
Volume depends on three lengths multiplied together, so doubling each one multiplies the result by two three times over. Surface area depends on two lengths, so it multiplies by four.