Length and Area Charles Delman April 20, 2010
What is the length? Unit
Solution Unit 5 (linear) units
What is the length? Unit
Solution Unit 5 2 = 2 1 2 (linear) units
What is the perimeter of the shaded rectangle? Unit
Solution 12 (linear) units. 2 + 5 + 2 + 5 = 2 2 + 2 5 = 12.
What is the shaded area? Unit
Solution 10 square units. The units may be counted by multiplying the length times the height.
What is the shaded area? Unit
Solution Each small rectangle is 1 10 unit. There are 5 23 = 115 little rectangles. So the area is 115 10 = 11.5 units. 5 2 23 5 = 110 10 = 11.5. Multiplying length by height works for fractional units as well.
What is the shaded area? Unit
Solution 10 square units. The area occupied by the elephant compensates for the missing elephant. So the shaded area is the same as that of the rectangle obtained by using the elephant to fill in the hole.
What is the shaded area? Unit
Solution Unit 8 square units. The right triangle at left may be used to fill in the missing triangle on the right, creating a rectangle with height of two linear units and width of four linear units.
General solution for parallelograms We see that, in general, the area of a parallelogram is the product of the base and the height, because it can be dissected and reassembled into a rectangle with the same base and height. Remember that the height must be measured perpendicular to the base. It is not the length of the other side (unless the parallelogram happens to be a rectangle to begin with).
What is the shaded area? Unit
Solution 9 square units. The shaded area is half of a rectangle with area of 18 square units.
What is the shaded area? Unit
Solution 4 square units. The shaded triangle is half of a parallelogram with area of 8 square units. Note that the base and height of the corresponding parallelogram are the same as those of the triangle.
General solution for triangles Thus, in general, we see that the area of a triangle is one-half the product of its base and height, because every triangle makes up half of a parallelogram with the same base and height. (As we saw, the area of the parallelogram is the product of its base and height.) Remember that the height must be measured perpendicular to the base.
What is the shaded area? Unit
Solution 4 square units. The base of the triangle is still 4 linear units, and the height of the triangle is still 2 linear units.
Another explanation It is also easy to see that the triangle occupies half the area of a rectangle with base of 4 linear units and height of two linear units.
What is the shaded area? Unit
Solution 8 square units. We can dissect the trapezoid into two triangles having the same height, namely the height of the trapezoid (measured perpendicular to its parallel sides). The bases of these triangles are the parallel sides of the trapezoid (sometimes called its bases). Since the height of the trapezoid is 2 linear units and its bases have length 6 units and 2 units, the areas of the triangles are 6 square units and 2 square units. 6 + 2 = 8.
General solution for trapezoids So we see that, in general, the area of a trapezoid is half of one base times the height plus half of the other base times the height. That is, if the bases have length b 1 and b 2, and the height is h linear units, then the area is 1 2 b 1h + 1 2 b 2h square units. Using the distributive law, we see that the area of a trapezoid is the product of the height and the average of the bases: 1 2 (b 1 + b 2 )h square units.
Another way to see the area of a trapezoid h b b 1 2 A trapezoid is half of a parallelogram with the same height and base equal to the sum of the two bases of the trapezoid.
What is the shaded area? Unit
Solution 6 square units. The shaded region is obtained by removing a triangle of area 2 square units (base of 4 linear units, height of 1 linear unit) from a parallelogram of area 8 square units (base of 4 linear units, height of 2 linear units). 8 2 = 6.
Properties of Measure, I All measures share certain properties. If the set to be measured is divided into disjoint parts, the measure of the whole is the sum of the measures of the parts. In practice, it is often useful to apply this property subtractively as well as additively. Example: What is the area of the quadrilateral below?
Solution Subtract the four triangles from the rectangle. 18 3 3/2 2 2 = 9.5 square units.
Properties of Measure, II Congruent figures have the same area. Thus, we may rearrange disjoint regions in order to better perceive the total area.
Properties of measure, III Every system of measurement must have a unit. This unit must be appropriate to the type of set being measured. For example: linear units to measure the length of a segment, square units to measure the area of a region on a surface, cubic units to measure the volume of a solid region. A square whose sides measure one linear unit has area of one square unit. A cube whose sides measure one linear unit has volume of one cubic unit.
The Pythagorean Theorem The Pythagorean Theorem, which has been known since ancient times, gives an important relationship between length and area. The Pythagorean Theorem states that: For a right triangle with legs of length a and b and hypotenuse of length c, a 2 + b 2 = c 2.
The Converse of the Pythagorean Theorem Since triangles are rigid - the angles of a triangle are determined by the sides - the converse of the above statement must also be true: In a triangle with sides of length a, b, and c, if a 2 + b 2 = c 2, then the angle opposite the side of length c (the longest side) is a right angle.
Why the Pythagorean Theorem is true 2 2 a + b = c 2 a c b c a b a c b cc a b
The Pythagorean Theorem holds only in a flat plane The Pythagorean Theorem is true only in a flat plane. The Pythagorean Theorem is not true on a sphere or other curved surface. In fact, there is a right triangle on a sphere that is equilateral and has three right angles! There are no squares on a sphere, so the picture that proves the theorem in the plane will does not fit together on a sphere.
The area of a circle r The area of a circle is clearly proportional to the square of its radius. That is, A = kr 2. Clearly, k < 4. Why? And k > 2. Why?
In fact, dissection of the regular dodecagon shows that k > 3.
The area of a circle of radius r is A = πr 2 In fact, k = π (as you probably remember). Remember that π is defined in terms of linear measurements; it is the ratio of circumference to diameter. Thus, we have another deep relationship between length and area! C 2r = π = A r 2 Why does π, the ratio of circumference to diameter, also turn out to be the ratio of the area of the circle to the area of a square on the radius? Is it just a miracle, or can we understand the reason?
Why A = πr 2 h b As the number of sides, n, increases, the area of the inscribed n-gon approaches the area of the circle.
Why A = πr 2, continued h b Each triangle has area 1 2 bh.
Why A = πr 2, continued h b So the area of the inscribed polygon is n 2 bh. (There are n triangles.)
Why A = πr 2, continued h nb nb is the perimeter of the polygon. As n, nb C, the circumference of the circle, and h r, the radius of the circle. Remember that C = 2πr.
Why A = πr 2, conclusion Thus, as n, the area of the inscribed polygon, (nb)h, 2 approaches 2πr r = πr 2. 2
What is the (exact) area of the shaded region? Unit
Solution Each side of a square unit is one linear unit. Therefore, the radius of the circle is one linear unit. So the area of the circle is π square units.
What is the (exact) area of the shaded region? Unit
Solution The radius of the circle is 2 linear units. Therefore, the area of the circle is π(2 2 ) = 4π square units.
What is the (exact) area of the shaded region? Unit
Solution The shaded area is the area left when the small circle is removed from the large circe. Thus the area of the shaded region may be obtained by subtracting the area of the small circle from the area of the large circle. Therefore, the area of the shaded region is 4π π = 3π square units.
What is the (exact) area of the shaded region? Unit
Solution The area of the shaded region is still the area left when the small circle is removed from the large circle. So the shaded area is 3π square units.
What is the (exact) area of the shaded region? Unit
Solution The shaded region is a quarter of a circle with radius 4 linear units. The area of the entire circle would be π(4 2 ) = 16π. Therefore, the area of the shaded region is 16π 4 = 4π.
What is the (exact) area of the shaded region? Unit
Solution The shaded area is the area that remains when the quarter circle is removed from the square. The area of the square is 16 square units. The area of the quarter circle is 4π square units. Therefore, the area of the shaded region is 16 4π square units.
Yet another way to see that the area of a trapezoid is the average of the bases times the height.
What is the area of the shaded region? 5 units 10 units
Solution 5 units b 1 b 2 b 3 10 units The height of each triangle is the same: 5 units. So the total shaded area is 1 2 b 1(5) + 1 2 b 2(5) + 1 2 b 3(5) = 1 2 (b 1 + b 2 + b 3 )(5) square units. But b 1 + b 2 + b 3 = 10. So the shaded area is 1 2 (10)(5) = 25 square units, half the area of the rectangle.
What portion of the parallelogram is shaded?
Solution Half of it, since the sum of the bases of the triangles is the base of the parallelogram and the height of the triangles is the height of the parallelogram.