An Equilateral Polygon has all its sides equal. An Equiangular Polygon has all its angles equal. A Regular Polygon has equal sides and equal angles.

A Diagonal is a straight line joining the vertices of two angles of a polygon which are not adjacent.

The Base of any plane figure is the side upon which it is supposed to stand. In a triangle the base lies opposite the angular point chosen as the vertex. The base of a conic or cylindrical surface is the intersection of the surface by a plane.

The Altitude of a triangle is the perpendicular distance from the base to the vertex of the opposite angle.

The Altitude of a parallelogram, or trapezoid, is the perpendicular distance between its parallel bases.

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704. A Circle is a plane figure bounded by a curved line called the circumference, which is everywhere equidistant from a point within called the centre; as, A B C D.

B

A

D

The Radius of a circle is a right line drawn from the centre to the circumference; as, A E. The Diameter is a right line passing through the centre and terminated both ways by the circumference; as, A C.

705. An Ellipse is an oblong curvilinear plane figure, which corresponds to an oblique projection of a circle, or a section of a cone cut by a plane which passes obliquely through its two slant sides; as, A B C D.

A

B

It has two fixed points (ƒ,ƒ,) called the foci, from which to every point of the circumference the sum of the distances is the same, and is equal to the transverse diameter.

The Transverse Diameter, or axis, is a line passing through the foci. A line perpendicular to this, passing through the centre, is the Conjugate Diameter or Axis.

B

SOLIDS.

706. A Prism is a solid whose ends are two equal, parallel, and similar plane figures, and its sides parallelograms; as, Figures A, B.

A prism is called a triangular prism when its ends are triangles; a square prism, when its ends are square, etc.

A Cube is a square prism having six sides which are all

squares.

D

A Parallelopipedon is a solid having six rectangular sides, every opposite two of which are equal and parallel; as, Figure C.

A Pyramid is a solid whose sides are all triangles, meeting at a common point called the vertex, and whose base is a polygon; as, Figure D

E

F

A Cylinder is a solid described by the revolution of a rectangle about one of its sides as an axis. It as also been defined as a round prism; as, Figure E.

A Cone is a solid described by the revolution of a right-angled triangle about one of its legs as an axis. It is also defined as a solid which has a circular base, and tapers regularly to a point called the vertex; as, Figure F.

A Frustum of a cone or pyramid is the part that remains after cutting off the top by a plane parallel to the plane of its base; as, Figure G.

A Sphere is a solid bounded by one continued surface, every point of which is equally distant from a point within called the centre; as, Figure H.

H

The Altitude of a prism, cube, or parallelopipedon, is the perpendicular distance between its bases.

The Altitude of a pyramid or cone is the perpendicular distance from the vertex to the plane of the base.

The Axis of a solid is a line passing from its extremities through the central point. The axis of a sphere is a line passing through the centre, and terminating in two opposite points of its surface.

MENSURATION OF SURFACES.

707. Mensuration of Surfaces is the process of determining their areas.

The Area of a plane figure is its quantity of surface, ex. pressed in square measure.

The Unit of Measure is a square whose side is some known length; as, an inch, a foot, etc.

PROBLEM I.

708. To find the area of any parallelogram, whether it be a Rectangle, Square, Rhomboid, or Rhombus.

RULE.-Multiply the base by the altitude.

EXAMPLES.

1. Required the area of a square whose side is 8 feet.

Ans. 64 sq. feet.

2. How many square yards are there in the floor of a room 20 feet long, and 15 feet wide?

Ans. 331.

3. The base of a rhombus is 7 feet, and its perpendicular height 5 feet required the area.

Ans. 35 sq. feet.

PROBLEM II.

709. To find the area of a triangle when the base and altitude are given.

RULE. Take one half the product of the base and altitude.

EXAMPLES.

1. What is the area of a triangle whose altitude is 14 feet, and base 27 feet? Ans. 189 sq. ft.

2. Find the area of a triangle whose base is 13 inches, and altitude 10 inches.

3. What is the area of a triangle whose base is 42 rods, and altitude 28 rods?

4. What is the area of a field in the form of a right-angled triangle, of which the base is 27.5 chains, and the perpendicular 18.25 chains?

PROBLEM III.

710. To find the area of a triangle when three sides are given.

RULE. From half the sum of the three sides subtract each side separately. Then multiply the half sum and the three remainders continually together, and the square root of the product will be the area.

EXAMPLES.

1. Required the area of a triangle whose three sides are, respectively, 24, 36, and 40 feet. Ans. 426.62 + sq. ft.

2. Find the area of a triangle whose three sides are, respectively, 8, 10, and 13 feet.

3. What is the area of a field in the form of an equilateral triangle, each side being 75 rods?

NOTE. The area of an equilateral triangle is equal to the square of the side multiplied by .433013.

PROBLEM IV.

711. Any two sides of a right-angled triangle being given, to find the third side.

THEOREM. The square described on the hypothenuse of a right-angled triangle is equal to the sum of the squares described on the other two sides.

COROLLARY.-The square of either side about the right angle is equal to the square of the hypothenuse diminished by the square of the other side.

A

From this theorem, which is demonstrated by geometry, and illustrated by figure A, the following rules are derived :-

I. To find the hypothenuse:

RULE.-Extract the square root of the SUM of the square of the base and the square of the perpendicular.

II. To find the base or perpendicular:

RULE.-Extract the square root of the DIFFERENCE between the square of the hypothenuse and square of the given side.

NOTE.--The square of the Difference may be found by the following simple rule; Multiply the sum of the given sides by their difference; or, multiply the smaller of the two numbers by twice their difference, and add the square of their difference.

EXAMPLES.

1. The base of a right-angled triangle is 24 feet, and the perpendicular 18 feet: what is the hypothen use? Ans. 30 feet.

2. The hypothenuse of a right-angled triangle is 41 rods, and its base 9 rods: what is the perpendicular?

Ans. 40 rods.

3. There is a park 80 rods square: what is the distance from the centre to one of the four corners ?

Ans. 56.57 - rods.