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Concave is a word that is applied to polygons and curves. A concave polygon is a polygon that has at least one interior angle greater than 180Â°. In other words, a concave
polygon has at least one interior angle that is a reflex angle. A concave polygon is sometimes described as a reentrant polygon.
Figure a is a plan of Amandaâ€™s garden. The lawn is a concave octagon, and has two angles greater than 180Â°. All triangles are convex, because it is impossible to have a concave triangle, since none of its angles can be greater than 180Â°.
A convex polygon has none of its interior angles greater than 180Â°.
Curved graphs have concavity which is described in the following way. Graphs which are the shapes shown in figure b are said to be concave down. Graphs which are the shapes shown in figure c are said to be concave up.
In figure d the parabola y concave up.
-x2 is concave down and the parabola y = x2 is
y â€” â€”X
Points on a curve where a curve changes concavity are called points of inflection. In figure e the cubic curve y = x3 is concave down for x < 0 and concave up for x > 0. The origin, where this curve changes concavity, is a point of inflection.
y = Xs
A rocket is fired into the air and its second stage is fired just as the first stage reaches its maximum height. In figure f the point A where the second stage is fired is a point of inflection.
References: Angle, Cubic Graphs, Graphs, Octagon, Parabola, Polygon, Reflex Angle, Triangle.
CONCAVE DOWN / UP
Circles are concentric when they have the same center (see figure). When a small stone is dropped into the center of a bowl of still water the waves travel outward from the center of the bowl in the form of concentric circles.
The target that is used in the game of darts is made up of concentric circles made of wire and fastened to a circular board.
References: Annulus, Circle.
Lines are concurrent if they all pass through the same point. An example is given in the entry Orthocenter.
References: Altitude, Collinear, Orthocenter, Simultaneous Equations.
Reference: Cyclic Quadrilateral.
A cone is a three-dimensional shape similar to a pyramid, except that its base is a closed curve. The cone we shall deal with has a circle for its base and it is a right cone, which means its axis of symmetry is at right angles to its base. The parts of the cone are labeled in figure a. A generator is the shortest straight line drawn on the curved surface of the cone from the vertex to a point on the circumference of the base.
The curved surface of the cone is its whole surface, but does not include the circular base. The slant height s is the length of a generator. The perpendicular height h, which is also referred to as the altitude of the cone, is the distance from the vertex to the center of the base. The radius of the circular base is R. The volume of the cone is given by
V = | x area of base x perpendicular height Area of circular base = ttR2
V = \n.R2h
The area of the curved surface, not including the base, is
A = ttRs
The area of the circular base is
A = TtR2
There is a relationship between R, h, and j', using Pythagorasâ€™ theorem:
s2 = R2 + h2
The net of a cone is a sector of a circle. The center O of the circle becomes the vertex of the cone, and the radius R of the sector of the circle becomes the slant height s of the cone. The area of the sector of the circle becomes the curved surface area of the cone. The net of the cone is folded so that the two radii R and R come together as one generator of the cone.
Example. Madge has bought her children an artificial Christmas tree (see figure c). They put it together and discover it is in the shape of a cone. Julie measures the slant height s of the tree to be 45 centimeters and Jane measures the radius of the circular base to be 18 centimeters. Calculate the volume and the area of the curved surface of the conical tree.
Solution. Write, to find the volume,
452 =h2 + 182 h2 = 452 â€” 182
h = VlTOl
I x it x 182 x V'1701
Volume of tree = 13,990 cm3 (to 4 sf )
Using Pythagorasâ€™ theorem
Rewriting with h on the left of the = sign
Taking the square root to obtain h
Formula for volume of a cone
Substituting R = 18, h = V1701
Using value of it in calculator
CONIC SECTIONS 107
For the area
A = ttRs Formula for area of a curved
A = 7i x 18x45 Substituting R = 18 and s = 45
Area of curved surface = 2545 cm2 (to 4 sf) Using value of it in calculator
References: Altitude, Cylinder, Frustum, Net, Pyramid, Pythagorasâ€™ Theorem, Radius, Sector of a Circle.
Two figures are congruent if they are identical in shape and size so that one figure can be laid exactly on top of the other figure. If one figure needs turning over in order to lie exactly on top of the other, then the two figures are oppositely congruent. If one of the figures does not need turning over, the two figures are directly congruent. The two pentominoes in figure a are oppositely congruent, since one of them needs turning over for it to lie exactly on top of the other. The two pentominoes in figure b are directly congruent, because one will lie exactly on top of the other without one of them being turned over, but one of them will need turning around.