445. A Plane Surface is a surface that does not change its direction.

446. A Quadrilateral is a plain figure having four straight sides.

447. Parallel Lines are lines having the same direction and equally distant from each other.

448. A Parallelogram is a quadrilateral whose opposite sides are parallel.

449. A Rhomboid is a parallelogram whose angles are not right angles.

What is a parallelogram called whose angles are right angles ?

450. A Rhombus is a rhomboid whose sides are equal.

451. The area of a rhomboid is found by multiplying the base by the altitude.

NOTE. The altitude of a parallelogram is the perpendicular distance between the sides.

1. Find the area of a parallelogram whose base is 24 rods, and altitude 18 rods.

2. Find the area of a rhombus whose base is 15 ft. and altitude 8 ft.

3. Draw a rhomboid whose base is 15 ft. and altitude 10 ft. Find its area.

Draw a rectangle having the same dimensions.

452. A Trapezoid is a quadrilateral having only two

sides parallel.

453. To find the area of a trapezoid, multiply the sum of the parallel sides by the altitude.

4. Find the area of a trapezoid whose altitude is 10 ft., its

A Trapezoid.

longest side 20 ft., and shortest side 15 ft. ?

5. A board 20 inches wide at one end and 12 inches

wide at the other is 16 feet long.

does it contain?

A Trapezium.

How many board feet

454. A Trapezium is a quadrilateral having no two sides parallel.

NOTE. - By drawing a diagonal between any two opposite sides of a trapezium, we have two triangles, the diagonal serving as the base of each. The altitude of each is the

perpendicular distance from its other angle to the diagonal.

455. To find the area of a trapezium, multiply the diagonal by half the sum of the altitudes of the two triangles.

6. The diagonal of a trapezium is 18 ft.; the altitudes of its two triangles are 5 ft. and 3 ft. What is the area?

7. A farm is in the form of a trapezium. The diagonal distance between the northern and southern corners is 108 rods, and the perpendicular distances from the east and west corners to the diagonal are 52 rods and 36 rods respectively. How many acres in the farm?

SOLIDS.

456. A solid whose two bases are equal and parallel, and its other faces parallelograms, is called a Prism.

NOTE. Prisms take their names from the form of their bases, as triangular, quadrangular, pentagonal, hexagonal, etc., according as the bases have three, four, five, or six sides, etc.

A Triangular A Rectangular

Prism.

Prism.

457. To find the contents of a prism, multiply the area of the base by the altitude.

8. Find the contents of a triangular prism whose altitude is 10 in., and the area of its base 7 sq. in.

9. What are the contents of a quadrangular prism whose

base is 5 in. by 8 in., and whose altitude is 12 in. ?

10. What are the contents of a hexagonal prism, the area of whose base is 10 sq. ft., and whose altitude is 15 ft.?

PYRAMIDS AND CONES.

458. A solid whose base is a triangle, square, pentagon, etc., and whose sides are triangles meeting at a vertex, is called a Pyramid.

NOTE.

A pyramid takes its name from the form of its base.

A solid whose base is a circle, and whose convex surface terminates in a point, is called a Cone.

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

The Slant Height is the shortest distance from the vertex to the perimeter of the base.

E

A Pyramid.

6 in.

4 i

A Cone.

460. To find the contents of a pyramid or cone, multiply the area of the base by of the altitude.

To find the convex surface, multiply the perimeter of the base by the slant height.

11. Find the contents of a quadrangular pyramid whose altitude is 10 in., and whose sides of bases are 8 in. and 6 in.

SOLUTION. 8 x 6 x 10 = 160 cu in. Ans.

12. Find the convex surface of a regular hexagonal pyramid whose slant height is 16 in., and whose side of base is 4 in.

13. Find the convex surface of a cone, when the circumference of its base equals 16 ft. and its slant height 18 ft.

14. Find the convex surface and volume of a cone whose radius is 4 in. and altitude 6 in.

461. The Frustum of a cone or pyramid is the part which is left after the top is cut off in a plane parallel to the base.

462. To find the contents of the frustum of a pyramid or cone, multiply

of

the altitude by the sum of the areas

of the two bases plus the square root of their product.

15. Find the con

Frustum of a Pyramid.

Frustum of a Cone.

tents of the frus

tum of a quadrangular pyramid whose altitude is 15 ft., and whose ends are 6 ft. and 4 ft. square.

16. A log 16 ft. long is 30 in. in diameter at one end and 24 in. at the other. Find its cubical contents.

463. A Sphere is a solid bounded by a curved surface, all parts of which are equally distant from the centre.

464. To find the surface of a sphere, multiply the circumference by the diameter.

25. Compare the volume of a 4-ft. cube and a 4-ft. sphere.

26. Compare the surfaces of a 4-ft. cube and a 4-ft. sphere.

REVIEW OF MENSURATION.

466. 1. Find the cubic yards in a cone, the circumference of whose base is 20 ft., and whose altitude is 30 ft. 2. Find area of a semi-circle when its radius equals 14 ft.

3. Find area of a square inscribed in a circle of 4 ft. in diameter.

4. The circumference of a circle and the perimeter of a square are each 300 ft. Which has the greater area? 5. A circle is inscribed in a 6-ft. square. of the circle.

Find the area