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About Area and Perimeter TABLE of CONTENTS About Area and Perimeter TABLE OF CONTENTS About Area and Perimeter ............................................................................................................................ 1 What is AREA AND PERIMETER? ............................................................................................................... 1 Square/Rectangle .......................................................................................................................................... 1 Square ....................................................................................................................................................... 1 Rectangle .................................................................................................................................................. 2 Triangles ........................................................................................................................................................ 2 Right Triangle ............................................................................................................................................ 2 Non-Right Triangles ................................................................................................................................... 3 Equilateral Triangles .................................................................................................................................. 3 Parallelogram ................................................................................................................................................ 4 Trapezoid ...................................................................................................................................................... 5 Circle ............................................................................................................................................................. 6 Complex Shapes ............................................................................................................................................ 6 Glossary ......................................................................................................................................................... 7 References .................................................................................................................................................... 9 About Area and Perimeter What is AREA AND PERIMETER? ● For a two-dimensional figure, the area is the amount of space enclosed by the figure, and the perimeter is the distance around the figure. Any closed curve has an area and a perimeter, but, except for simple figures like squares and triangles, you need calculus to calculate the area and perimeter exactly. ● This unit is about how to calculate areas and perimeters for simple figures and how to estimate them for more complex figures. This is sufficient for many everyday applications, such as building a fence or painting a room. Area is always in square units: square centimetres, square meters, square kilometres. Perimeter is always in one-dimensional units: centimetres, meters, kilometres. Square/Rectangle Square ● The area of a square with sides of length s is: A = s 2. Its perimeter is: P = 4s. 1 Rectangle ● The area of a rectangle with length l and width w is: A = lw. Its perimeter is: P = 2l + 2w. Triangles Right Triangle ● Given a right triangle with sides a, b and hypotenuse c, the area is: A = ½ ab The perimeter is: P = a + b + c 2 Non-Right Triangles ● Most commonly, we are given a non-right triangle, for which we know the length of one of its sides (called the base b) and its height h. To measure the height, draw a line perpendicular to the base from the angle opposite the base. This line doesn't always hit the base; sometimes, you need to extend the line containing the base. The length of the line described is the height h. The area is: A = ½ bh. Perimeter is difficult to determine, as the triangles below – each with equivalent area - demonstrate. ● If you do not know the height, sometimes you can figure it out from other information, using trigonometry. This is due to the fact that the height is always part of a right triangle. ● If you know the lengths of all the sides, you can use Heron's Formula (a.k.a. Hero's Formula): A = √ s(s - a)(s - b)(s - c); s is the semiperimeter (half the perimeter) and a, b, and c are the lengths of the sides. Equilateral Triangles ● Equilateral triangles are important, having the largest area for any given perimeter and the smallest perimeter for any given area, of all possible triangles. 3 ● The area of an equilateral triangle with side of length x is: A = ( √3 /4) x2 and the perimeter is: P = 3x Parallelogram ● The formula for the area of a parallelogram directly relates to the formula for the area of a rectangle. Imagine cutting off a triangle from one side of the parallelogram. Flip this triangle over and it fits perfectly on the other side, creating a rectangle. The area of this rectangle is the area of the parallelogram. As with the formula for the area of a triangle, you don't need to know the lengths of all the sides to find the area; you need to know the length of one side (the base b) and the height h. Area is given by: A = bh ● The two parallelograms given below have the same area; however, they clearly have different perimeters. The perimeter is related to the steepness of the angle between the base and side. The perimeter of a parallelogram is given by P = 2b + 2x where b is the base and x is the length of a side not parallel to the base. 4 ● If you are given the base, the height, and one angle of the parallelogram, you can find the perimeter. Each parallelogram has two acute angles θ and two obtuse angles Φ. θ = 180° - Φ. Consider the right triangle created by drawing the height. sin( θ) = h/x so x = h/sin θ . Hence, another formula for the perimeter is: P = 2b + 2h/sin θ. Trapezoid ● A trapezoid is a four-sided shape having exactly one pair of parallel sides, with the exception of a parallelogram being debatable. ● The bases of a trapezoid b1 and b 2 are the two parallel sides. The height h is the distance between the bases. The area of a trapezoid is the sum of the area of two triangles, one with base b 1 and height h, and the other with base b 2 and height h. So, the area is A = ½ h(b 1 + b 2). If you know b 1, b 2, h and the measure of two angles θ1 and θ2, you can find the perimeter: P = b 1 + b 2 + h/sin( θ1) + h/sin( θ2) (Use the acute angles; if you are given the obtuse angles, you can find the acute ones by subtracting from 180°). 5 Circle ● The area of a circle is A = πr2 and the perimeter (called the circumference) is C = 2 πr. ● To find the area of an ellipse, you need to know the length of the major and minor axes. If a is half the length of the major axis, and b is half the length of the minor axis, then the area is given by: A = πab. ● Finding the perimeter of an ellipse is much more difficult. One good formula was invented by Srinivasa Ramanujan, a great 20th- century mathematician from India. His formula says the perimeter is approximately: π(a + b)(3 - √ 4 - h) where h = (a - b) 2 (a + b) 2 Complex Shapes ● If you need to find the area of a shape that is not a simple geometric shape, you can draw it on graph paper and count the squares. You need to estimate how much the partial squares add up to. 6 Glossary Area: a measure of the space contained within the sides of a 2D shape. Circle: a shape whose centre is equidistant to each point lying on the shape itself. Cube: a shape formed by six equal squares, connected to one another at right angles. Ellipse: a stretched circle. Equilateral Triangle: a triangle with all sides and all angles being equal. Isosceles triangle: a triangle with two equal sides. Parallelogram: a 4-sided shape, with two sets of sides being of equal length and parallel. Perimeter: distance around a shape. Rectangle: a 4-sided shape, with 4 interior right angles and 2 pairs of sides of equal length (a.k.a. a right parallelogram). Right Triangle: triangle with a right angle. Semiperimeter: half the perimeter. 7 Surface area: a measure of the total space enclosed within the faces of a shape, which is not to be confused with volume (a measure of the interior space of a shape). Square: a 4-sided shape, with 4 interior right angles and all sides equal. Trapezoid: a 4-sided shape, with only two parallel sides. Triangle: a 3-sided shape. Volume: a measure of the interior space of a 3D shape. 8 References 7 ways to find the area of a triangle (using trig, etc.) by Henry Bottomley, a government statistician for the UK. http://www.btinternet.com/ se16/hgb/triangle.htm Triangles with the same area and perimeter (Henry Bottomley) http://www.btinternet.com/ se16/hgb/triangleareaperimeter.htm Math Open Reference (free interactive textbook) http://www.mathopenref.com/trianglearea.html Trapezoid Perimeter and Area: dynamic http://argyll.epsb.ca/jreed/math9/strand3/trapezoid area per.htm 9 .
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