To find the area of any triangle when the three sides only are given.

RULE:-From half the sum of the three sides subtract each side separately, then multiply the half sum and the three remainders together. The square root of the product is the area of the triangle.

THE EQUILATERAL TRIANGLE. Definition:-An equilateral triangle is one which has all its sides equal, as M N P. To find the area of an equilateral triangle. RULE:-Multiply the fourth part of the square of the side by the square root of 3 for the area.

M

THE TRAPEZIUM.

Definition :-A trapezium is a figure contained by four straight lines, no two of which are parallel to each other, as A B CD; where A c is the diagonal, and D E, B F perpendiculars to it.

D

F

E

B

To find the area of a trapezium.

RULE:-Divide the trapezium into two triangles by joining two of its opposite angles; find the area of each triangle, and the sum of both areas will give the area of the trapezium.

RULE:-Multiply half the sum of the parallel sides by the pernendicular distance between them; the product is the area.

D D

THE CIRCLE

Definition :-A circle is a plane figure bounded by one line

called the circumference, and is such that all straight lines drawn from a certain point within the circle to the circumference are equal to one another. This point is called the centre of the circle.

NOTE: If c be the centre of the circle, C D is called a radius, and B E a diameter of the circle-C B, C E are also radii.

From the diameter of a circle to find the circumference. RULE:-Say, As 7 : 22 :: diameter: circumference: or, as 1 : 3.1416 :: diameter: circumference.

To find the area of a circle.

RULE:-Multiply half the circumference by half the diameter for the area.

Or, multiply the square of the diameter by 7854 for the

area.

Or, multiply the square of the circumference by '07958 for

the area.

THE CUBE.

A cube is a solid having six equal sides at right-angles to one another.

To find the solid content of a cube.

RULE:-Multiply the length of the side twice by itself; the product is the solid content of the cube.

THE PARALLELOPIPEDON.

Definition :-A parallelopipedon is a solid having six rectangular sides, every opposite pair of which are equal and parallel.

To find the solid content of a parallelopipedon.

RULE:-Multiply the length, breadth, and depth together; the product will be the solidity.

THE CYLINDER.

Definition :-A cylinder is a round solid, having circular ends, as A B C D (see below).

To find the convex surface of a cylinder.

RULE:-Multiply the circumference of the cylinder by its perpendicular height for the area of its convex surface.

To find the solid content of a cylinder.

RULE:-Multiply the area of the base by the height of the cylinder, and the product will be its solid content.

Definition :-A right cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right-angle, which side re

mains fixed. The base of a cone is a circle. See fig. M N P above.

E

THE PYRAMID.

Definition :-A pyramid is a solid having a plane figure for its base; and whose sides are triangles meeting in a point, called the vertex. When the base is a triangle, the solid is called a triangular pyramid; when the base is a square, it is called a square pyramid, and so on.

B

D

To find the convex surface of a right cone or pyramid. RULE:-Multiply the circumference of the base by the slant height or length of the side of the cone or pyramid, and half the product will be the surface required. And if this be added to the area of the base, it will give the whole surface.

To find the solidity of a cone or pyramid.

RULE:-Multiply the area of the base by the perpendicular height of the cone or pyramid, and one-third of the product will be the solidity.

THE SPHERE.

Definition:-A sphere is a solid figure described by the revolution of a semicircle about its diameter which remains unmoved.

To find the convex surface of a sphere.

RULE:-Multiply the diameter of the sphere by its circumference, or 31416 by the square of the diameter, and the product will be the convex superficies required.

To find the solid content of a sphere.

RULE:-Multiply the cube of the diameter by 5236 (which is one-sixth of 3·1416), and the product will be the solidity.

BRICKLAYER'S WORK MEASURED BY THE ROD.

Bricklayers always estimate their work at a brick-and-a-half or three half bricks thick, which is called standard measure.

To find the number of rods in a piece of brickwork.

RULE:-Multiply the length and height of the wall in feet together, and this product by the number of half bricks in the thickness of it. Divide the result by 3. This will give the standard measure.

Now a square rod is 30 yards or 2724 square feet. Therefore if the standard measure in feet be divided by 2721, the quotient will be the number of rods required.

NOTE:-Theis usually rejected in favour of the workmen.

ROOFING AND SLATING.

In roofing and slating it is usual to take one-and-a-half times the width of the house within the walls as the measure of the roof when it is a true pitch,' i.e. when each rafter is threefourths of the width of the house. If the roof be not of the true pitch it is measured by a string passed over it or otherwise.

SPECIFIC GRAVITY

Is the weight of any body compared with that of an equal volume of water. The weight of a cubic foot of water is 1,000 ounces avoirdupois. Therefore when we say that the specific gravity of iron is 7.788; we mean that it weighs so many times an equal volume of water, i.e. 7,788 ounces. We may thus estimate the weight of bodies which it would be impossible actually to weigh. A cubic foot of water is 62.5 lbs. av. A gallon of water is generally said to weigh 10 lbs., and contains 277.274 cubic inches.