Unit 7, lesson 59

Geometry Formulas and Units

How to pick the right lengths from a picture, use the standard formulas for circles, triangles, rings and cylinders, and keep units consistent so that lengths, areas and volumes come out right.

  • Theory part 117
  • Practice part 118
  • Geometry and Models

A water tank, a washer, the cross section of a pipe. Each of these is a shape you can measure, and each one has a short formula for how much space it covers or holds. The formulas are easy. The mistakes come from using a diameter where a radius belongs, a slanted side where a perpendicular height belongs, or centimeters mixed with meters. Calculus builds volumes and rates out of exactly these formulas, so getting them clean now saves a lot of trouble later.

Key terms

Length

Definition A length is a nonnegative real number measuring the size of a segment or curve in one dimension, given in a unit of length such as meters or centimeters. The distance between points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in the plane is

d=(x2−x1)2+(y2−y1)2.d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

A length of 00 describes a degenerate segment, a single point.

In plain words Length is how far, along one direction. It is never negative, even when a coordinate difference is.

Example The distance from (1,2)(1, 2) to (4,6)(4, 6) is

(4−1)2+(6−2)2=9+16=5.\begin{aligned} \sqrt{(4 - 1)^2 + (6 - 2)^2} &= \sqrt{9 + 16} \\[0.4em] &= 5. \end{aligned}

Radius and diameter

Definition A circle with center CC and radius r>0r > 0 is the set of points at distance rr from CC. A diameter is a segment through the center with both ends on the circle, and its length is d=2rd = 2r. The circumference, the length of the circle, is

C=2πr=πd.C = 2\pi r = \pi d.

In plain words The radius goes from the center to the edge. The diameter goes all the way across, so it is twice as long.

Example A pipe has a diameter of 1010 cm, so its radius is 55 cm. Its circumference is 10π≈31.4210\pi \approx 31.42 cm. Using 1010 as the radius in an area formula would make the answer four times too big.

Area

Definition Area measures the size of a region in the plane, in square units, where a square with side 11 unit has area 11 square unit. Area is nonnegative, and the area of a region split into pieces that do not overlap is the sum of the areas of the pieces. The standard formulas are

Arectangle=ℓw,Acircle=πr2,Atriangle=12bh,\begin{gathered} A_{\text{rectangle}} = \ell w, \qquad A_{\text{circle}} = \pi r^2, \\[0.6em] A_{\text{triangle}} = \frac{1}{2}bh, \end{gathered}

where ℓ\ell and ww are the side lengths of the rectangle, rr is the radius of the circle, and bb and hh are a base and the perpendicular height of the triangle.

In plain words Area counts how many unit squares fit inside a region. For a shape built from simpler pieces, add the pieces.

Example A window is a 44 m by 22 m rectangle with a half disk on top whose diameter is the 44 m side. The half disk has radius 22 m, so

A=4⋅2+12π⋅22=8+2π≈14.28 square meters.\begin{aligned} A &= 4 \cdot 2 + \frac{1}{2}\pi \cdot 2^2 \\[0.4em] &= 8 + 2\pi \\[0.4em] &\approx 14.28 \text{ square meters.} \end{aligned}

Perpendicular height

Definition For a triangle with a chosen base, the perpendicular height hh is the distance from the opposite vertex to the line containing the base, measured along a segment perpendicular to that line. Any side can serve as the base, and each choice has its own height, but the product bhbh is the same for every choice.

In plain words Height means straight up from the base line at a right angle, not along a slanted side.

Example The triangle with vertices (0,0)(0, 0), (10,0)(10, 0) and (3,4)(3, 4) has base 1010 along the xx axis. The third vertex is 44 units above that axis, so h=4h = 4 and the area is 12⋅10⋅4=20\frac{1}{2} \cdot 10 \cdot 4 = 20. The side from (0,0)(0, 0) to (3,4)(3, 4) has length 55, and using it as the height would give the wrong area 2525.

Annulus

Definition An annulus is the region between two concentric circles with radii 0≤r≤R0 \le r \le R, including the points on both circles. When r=0r = 0 it is the whole disk, and when r=Rr = R it is just a circle with area 00. Its area is the area of the large disk minus the area of the small one,

A=πR2−πr2=π(R2−r2).A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right).

In plain words An annulus is a ring, like a flat washer. Find its area by taking away the hole.

Example A washer has outer radius 55 cm and inner radius 33 cm. Its area is

π(52−32)=16π≈50.27square centimeters.\begin{gathered} \pi\left(5^2 - 3^2\right) = 16\pi \approx 50.27 \\[0.4em] \text{square centimeters.} \end{gathered}

Note that this is not π(5−3)2=4π\pi(5 - 3)^2 = 4\pi. Subtract the areas, not the radii.

Volume

Definition Volume measures the size of a solid region in cubic units, where a cube with edge 11 unit has volume 11 cubic unit. Like area, it is nonnegative and adds over pieces that do not overlap. A box with edges ℓ\ell, ww and hh has volume ℓwh\ell w h. More generally, a prism or cylinder whose cross sections parallel to the base all match the base has volume equal to the base area times the perpendicular height.

In plain words Volume counts how many unit cubes fit inside. For a straight solid, it is the area of one layer times how high the layers stack.

Example A box measuring 22 m by 33 m by 0.50.5 m has volume

2⋅3⋅0.5=3 cubic meters.2 \cdot 3 \cdot 0.5 = 3 \text{ cubic meters.}

Cylinder

Definition A right circular cylinder with radius r>0r > 0 and height h>0h > 0 is the solid whose base is a disk of radius rr and whose sides rise perpendicular to that base for a distance hh. Its volume and curved side area are

V=πr2h,Sside=2πrh.V = \pi r^2 h, \qquad S_{\text{side}} = 2\pi r h.

With r=0r = 0 the solid would shrink to a segment, and with h=0h = 0 to a flat disk, both with volume 00.

In plain words A cylinder is a stack of identical disks. Its volume is the area of one disk times the height of the stack.

Example A can has radius 44 cm and height 1010 cm. Its volume is

π⋅42⋅10=160π≈502.65cubic centimeters.\begin{gathered} \pi \cdot 4^2 \cdot 10 = 160\pi \approx 502.65 \\[0.4em] \text{cubic centimeters.} \end{gathered}

Units

Definition Every measured quantity is a number together with a unit. Only quantities with the same unit may be added, subtracted or compared. Multiplying quantities multiplies their units, so length times length gives square units and length times area gives cubic units. A formula for an area must have total units of length squared, and a formula for a volume must have length cubed.

In plain words Units follow the arithmetic. Checking them is a quick way to catch a formula used in the wrong place.

Example For a circle of radius rr meters, 2πr2\pi r has units of meters, so it is a length, the circumference. The expression πr2\pi r^2 has units of square meters, so it is an area. If a computation for a volume ends in square meters, a factor is missing.

Unit conversion

Definition A unit conversion multiplies a quantity by a ratio equal to 11, such as 100 cm1 m\frac{100 \text{ cm}}{1 \text{ m}}. Because area and volume carry squared and cubed units, the conversion factor is squared or cubed with them.

1 m=100 cm,1 m2=104 cm2,1 m3=106 cm3,1 L=1000 cm3.\begin{gathered} 1 \text{ m} = 100 \text{ cm}, \\[0.4em] 1 \text{ m}^2 = 10^4 \text{ cm}^2, \\[0.4em] 1 \text{ m}^3 = 10^6 \text{ cm}^3, \\[0.4em] 1 \text{ L} = 1000 \text{ cm}^3. \end{gathered}

Convert every length to one common unit before putting them into a formula.

In plain words Pick one unit and change everything into it first. Then use the formula.

Example A cylindrical tank has radius 2020 cm and height 1.51.5 m. Convert the height to 150150 cm, then

V=π⋅202⋅150=60000π≈188496 cubic centimeters,\begin{aligned} V &= \pi \cdot 20^2 \cdot 150 \\[0.4em] &= 60000\pi \\[0.4em] &\approx 188496 \text{ cubic centimeters,} \end{aligned}

which is about 188.5188.5 liters. Mixing the units and computing π⋅202⋅1.5\pi \cdot 20^2 \cdot 1.5 would give a number that is not in any single unit.

Check yourself

  1. A ring has outer radius 77 cm and an inner hole of diameter 66 cm. Find its area.
Show the answer

The inner radius is 33 cm, so the area is

π(72−32)=40π≈125.66square centimeters.\begin{gathered} \pi\left(7^2 - 3^2\right) = 40\pi \approx 125.66 \\[0.4em] \text{square centimeters.} \end{gathered}
  1. A cylinder has radius 0.50.5 m and height 8080 cm. Find its volume in liters.
Show the answer

In centimeters the radius is 5050 and the height is 8080, so

V=π⋅502⋅80=200000π cm3=200π L≈628.3 L.\begin{aligned} V &= \pi \cdot 50^2 \cdot 80 \\[0.4em] &= 200000\pi \text{ cm}^3 \\[0.4em] &= 200\pi \text{ L} \approx 628.3 \text{ L.} \end{aligned}
  1. How many square centimeters are in 33 square meters?
Show the answer

Each square meter is 100⋅100=104100 \cdot 100 = 10^4 square centimeters, so 33 square meters equal 3000030000 square centimeters.

How this lesson connects