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Work through fractions, percentages, measurements, shapes, and useful calculations with examples.
Search this topicMultiply pi by the radius squared and by the height. A cylinder with a three-centimeter radius and a ten-centimeter height has volume 90π, approximately 282.74 cubic centimeters. Use the perpendicular height and halve the diameter if that is what you measured. For a container’s capacity, use its internal dimensions.
Calculate the areas of the three different face pairs and double their sum: 2 × (length × width + length × height + width × height). For sides two, three, and four centimeters, this gives 52 square centimeters. An open box excludes the missing face, and wrapping material needs additional overlap allowance.
Treat the diagonal as the hypotenuse of a right triangle. Square the length and width, add them, and take the square root. For a rectangle five by twelve inches, the diagonal is √(25 + 144) = 13 inches. This method requires a right angle; it does not apply unchanged to a slanted quadrilateral.
Add the two known interior angles and subtract that sum from 180 degrees. If they are 47 and 68 degrees, the third is 180 − 115 = 65 degrees. Use interior angles, not an exterior angle beside the triangle. The calculation applies to an ordinary triangle drawn on a flat plane.
Square the hypotenuse, subtract the square of the known leg, then take the square root. With hypotenuse ten and one leg six, the other leg is √(100 − 36) = 8. Identify the hypotenuse opposite the right angle first; it must be longer than either leg.
Work inside grouping symbols first, then exponents. Next do multiplication and division from left to right, followed by addition and subtraction from left to right. Multiplication does not always precede division. For 24 ÷ 6 × 2, the result is 8, whereas 24 ÷ (6 × 2) equals 2.
An exponent tells you how many times to use the base as a factor. For 3⁴, calculate 3 × 3 × 3 × 3 = 81, not 3 × 4. Parentheses determine the base: (2 + 1)² means square the whole sum, giving nine after the addition.
Look for the nonnegative number that multiplies by itself to give the original value. Since 12 × 12 = 144, √144 = 12. For a non-perfect square, use a calculator or estimate between nearby perfect squares. The square-root symbol gives the principal root; solving x² = 144 separately gives both positive and negative twelve.
Find the neighboring perfect squares. Since 49 is less than 60 and 60 is less than 64, √60 lies between seven and eight. Try a decimal and square it to refine the estimate: 7.7² = 59.29, so the root is a little larger than 7.7. Do not just halve sixty.
Subtraction can be rewritten as adding the opposite. The opposite of negative four is positive four, so 6 − (−4) = 6 + 4 = 10. Keep parentheses around negative inputs while rewriting. This rule is about subtraction; two negative signs appearing somewhere in a longer expression do not automatically cancel each other.
For two nonzero numbers, matching signs give a positive result and different signs give a negative result. Thus (−7) × (−3) = 21, while (−21) ÷ 3 = −7. Work out the sign and magnitude separately. Division by zero is undefined, even when the other number is negative.
Divide the first usable group of digits, write that quotient digit, multiply back, subtract, and bring down the next digit. Repeat across the dividend. For 156 ÷ 4, fifteen gives three with remainder three; bringing down six makes thirty-six, giving nine. The answer is 39; check that 39 × 4 = 156.
Multiply the quotient by the divisor and add the remainder; the result should equal the original dividend. For 157 ÷ 4 = 39 remainder 1, check 39 × 4 + 1 = 157. The nonnegative remainder must be smaller than the positive divisor, or another whole group could still be taken out.
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