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#notetype:Basic
#deck:Math::Area and volume formulas
#tags:math geometry area-volume
#columns:Front	Back	Tags
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Rectangle, length l and width w: perimeter and area?	Perimeter 2l + 2w. Area lw.	formula
Triangle, base b and altitude h: area?	One half b h.	formula
Circle of radius r: circumference and area?	Circumference 2 pi r, which is pi d. Area pi r^2.	formula
Closed box l by w by h: volume and surface area?	Volume lwh. Surface 2lh + 2wh + 2lw, three pairs of matching rectangles.	formula
Sphere of radius r: volume and surface area?	Volume (4/3) pi r^3. Surface 4 pi r^2.	formula
Closed right circular cylinder, radius r and height h: volume and surface area?	Volume pi r^2 h. Surface 2 pi r^2 + 2 pi r h: two circles and the side.	formula
What is the altitude of a triangle?	The height measured at a right angle to the base, from the base to the opposite corner. Not a side, unless the triangle is right.	formula
Why is the area of a triangle half of bh?	A triangle is half of a rectangle with the same base and height.	formula
Unrolled, the side of a cylinder is what shape, and how big?	A rectangle 2 pi r wide and h tall, so its area is 2 pi r h.	formula
How do you tell a length, an area and a volume formula apart without memorising?	Count the lengths multiplied in each term: one is a length, two an area, three a volume. Constants do not count.	dimension
Double every length of a shape. What happens to perimeter, area, volume?	Perimeter times 2, area times 4, volume times 8.	dimension
Scale every length by k. What happens to an area and to a volume?	Area times k^2, volume times k^3.	dimension
Every side of a box is tripled. Volume?	27 times as large, 3^3.	dimension
A formula for a volume grows by 4 when lengths double. What is wrong?	It is an area in disguise: a volume must grow by 8.	dimension
Why do the constants 2, one half, pi and 4/3 not count in the dimension test?	They are pure numbers without units, so they cannot change feet into square feet.	dimension
Which fact did Archimedes prove about the sphere's surface?	It equals the side of the cylinder that just fits around it: 2 pi r times 2r = 4 pi r^2.	formula
How is the surface area of a sphere related to its volume?	It is what the volume gains per unit of radius: a skin of thickness t has volume about 4 pi r^2 t.	dimension
Circle of radius 5 m: area and circumference?	25 pi, about 78.54 square meters. 10 pi, about 31.42 meters.	problems
Sphere of radius 3 ft: volume and surface area?	36 pi cubic feet and 36 pi square feet: the same number in different units.	problems
Cylinder r 9, h 8 against r 8, h 9: which holds more?	r 9, h 8: 648 pi against 576 pi. The radius is squared, the height is not.	problems
Square of side 2 inside a circle, corners on the circle: area of the circle?	2 pi. The diagonal is the square root of 8, so r^2 = 8/4 = 2.	problems
Circle inside a square of side 2, touching all sides: area outside the circle?	4 - pi, about 0.86.	problems
Square of side 6 with a border 2 wide all round: area of the border?	64. The outer side is 6 + 2 + 2 = 10, and 100 - 36 = 64.	problems
Wheel of diameter 16 in, 4 turns: distance?	64 pi inches, about 201 inches. One turn is pi d.	problems
Disk of diameter 4 ft rolled 20 ft: how many turns?	20 / (4 pi) = 5/pi, about 1.59.	problems
Norman window: rectangle 4 wide and 6 tall, semicircle on top. Area?	24 + 2 pi, about 30.28 square feet. The semicircle has radius 2, area half of 4 pi.	problems
Ornament: triangle base 4, height 6, semicircle of diameter 4 on top. Area?	12 + 2 pi, about 18.28 square centimeters.	problems
Trap: the volume of a sphere is 4 pi r^3?	No. (4/3) pi r^3. 4 pi r^3 is bigger than the 8r^3 cube around the sphere.	trap
Trap: the surface area of a sphere is (4/3) pi r^2?	No. 4 pi r^2. The dimension test cannot catch this one: only the constant is wrong.	trap
Trap: a wheel of diameter d rolls 2 pi d in one turn?	No, pi d. The 2 in 2 pi r is there because r is half of d.	trap
Trap: triangle with base 10 and two slanted sides 13. Area one half 10 times 13?	No. The altitude is 12, from 13^2 - 5^2 = 144, so the area is 60.	trap
Trap: a 16 inch pizza for twice the price of a 10 inch one is a bad deal?	No. Area grows with the square: 2.56 times the pizza for 2 times the price.	trap
Trap: a circle of radius 2 has area equal to its circumference, both 4 pi. Are they equal?	Only as numbers. One is square units, the other units; change the unit and they differ.	trap
Trap: a closed can and an open can of the same size have the same surface area?	No. The open can is missing one lid, pi r^2.	trap
Trap: border 2 wide around a square of side 6 makes an outer square of side 8?	No, 10. The border is on both sides.	trap