The area of a regular polygon is measured by half of the product of its perimeter by the radius of the inscribed circle.--The area of a circle is measured by half of the product of the circumference by the radius.--The areas of circles are to each other as the squares of the radii. The area of a sector of a circle is measured by half of the product of the arc by the radius.--Measure of the area of a segment of a circle. 2. OF PLANES AND BODIES TERMINATED BY PLANE SURFACES. Conditions required to render a right line and a plane respectively perpendicular. Of all the lines which can be drawn from a given point to a given plane, the perpendicular is the shortest, and the oblique lines are longer in proportion to their divergence from the foot of the perpendicular. Parallel right lines and planes.--Angles which have their sides parallel, and their openings turned in the same direction, are equal, although situated in different planes. Dihedral angle.--How to measure the ratio of any dihedral angle to the right dihedral angle. Planes perpendicular to each other.--The intersection of two planes perpendicular to a third plane, is perpendicular to this third plane. Parallel planes.--when two parallel planes are cut by a third plane the intersections are parallel.--Two parallel planes have their perpendiculars common to both. The shortest distance between two right lines, not intersecting and not parallel. Two right lines comprised between two parallel planes are always divided into proportional parts by a third plane parallel to the first two. Trihedral angle.--The sum of any two of the plane angles which compose a trihedral angle is always greater than the third. The sum of the plane angles which form a convex polyhedral angle is always less than four right angles. If two trihedral angles are formed by the same plane angles, the dihedral angles comprised between the equal plane angles are equal.--There may be absolute equality or simple symmetry between the two trihedral angles. _Of polyhedrons._ If two tetrahedrons have each a trihedral angle composed of equal and similarly arranged triangles, these tetrahedrons are equal. They are also equal if two faces of the one are equal to two faces of the other, are arranged in the same manner, and form with each other the same dihedral angle. When the triangles which form two homologous trihedral angles of two tetrahedrons are similar, each to each, and similarly disposed, these tetrahedrons are similar. They are also similar if two faces of the one, making with each other the same angle as two faces of the other, are also similar to these latter, and are united by homologous sides and summits. Similar pyramids.--A plane parallel to the base of a pyramid cuts off from it a pyramid similar to it.--To find the height of a pyramid when we know the dimension of its trunk with parallel bases.