Types of curvesTypes of curves

simple curves:simple curves: A curve is simple if it does A curve is simple if it does not cross itself.not cross itself.

Types of CurvesTypes of Curves

closed curvesclosed curves: a closed curve is a : a closed curve is a curve with no endpoints and which completely with no endpoints and which completely encloses an encloses an area

Types of CurvesTypes of Curves

convex curveconvex curve: If a plane closed curve be : If a plane closed curve be such that a straight line can cut it in at such that a straight line can cut it in at most two points, it is called a convex most two points, it is called a convex curve.curve.

Convex Curves

Not Convex Curves

CirclesCircles

Area = Area = ππrr22

Circumference = 2 Circumference = 2 ππr r or or

Circumference = Circumference = ππ d d

CirclesCircles

Otis is drawing a circle with a 4 inch radius. Otis is drawing a circle with a 4 inch radius. He wants to double the radius. How will He wants to double the radius. How will this affect the area of the circle?this affect the area of the circle?

CirclesCircles

Su is selling 12 inch diameter pumpkin pies Su is selling 12 inch diameter pumpkin pies for $6.50. How should she adjust her for $6.50. How should she adjust her price (if she wants to be fair) when she price (if she wants to be fair) when she reduces her pies to a 10 inch diameter?reduces her pies to a 10 inch diameter?

CirclesCircles

Javier’s bicycle tire has a 12 inch radius. Javier’s bicycle tire has a 12 inch radius. How far will he travel. . . How far will he travel. . .

. . . in one rotation of the tire? . . . in one rotation of the tire?

. . . in 10 rotations of the tire?. . . in 10 rotations of the tire?

VolumeVolume Cylinder Cylinder vs. vs. Cone Cone

Cube vs. Square pyramidCube vs. Square pyramid

lwhV

BhV

3131

hrV 2 hrV 2

31

lwhVBhV

Area of CircleArea of Circle draw a square, and inscribe a circle in it, which means to draw the circle inside the square so that draw a square, and inscribe a circle in it, which means to draw the circle inside the square so that

the circle just touches each side of the square. the circle just touches each side of the square.

We can find the area of this square by first finding the area of the four smaller squares—each We can find the area of this square by first finding the area of the four smaller squares—each with sides equal to with sides equal to rr, the radius of the circle—and adding them together., the radius of the circle—and adding them together.

Notice that the sides of the square are twice as long as the radius of the circle. You could also Notice that the sides of the square are twice as long as the radius of the circle. You could also

find the area of the square by multiplying the side times the side, or 2r x 2r, which also equals find the area of the square by multiplying the side times the side, or 2r x 2r, which also equals 4r2.4r2.

You can see that the area of the circle must be less than the area of four of the squares. But how You can see that the area of the circle must be less than the area of four of the squares. But how much less? We could make an educated guess and say that the area of the circle might be a little much less? We could make an educated guess and say that the area of the circle might be a little bit larger than three of the smaller squares.bit larger than three of the smaller squares.

The actual number we're looking for, which is between 3 and 4, is the special number called pi, The actual number we're looking for, which is between 3 and 4, is the special number called pi, represented by the Greek letter . Pi is approximately equal to 3.14. represented by the Greek letter . Pi is approximately equal to 3.14.

The symbol you see here means "approximately equal to." Pi actually has an unending number The symbol you see here means "approximately equal to." Pi actually has an unending number

of decimal points, but 3.14 is usually close enough for our calculation purposes. Pi is the ratio of decimal points, but 3.14 is usually close enough for our calculation purposes. Pi is the ratio between the diameter and circumference of a circle.between the diameter and circumference of a circle.

The final formula for the area of a circle is shown here. The final formula for the area of a circle is shown here.