RULE. From half the sum of the sides subtract each side separately. Multiply the half sum and these remainders together, and extract the square root of the product.

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

EXAMPLES.

434. 1. Find the area of a triangle whose sides are, respectively, 12, 16, and 20 feet.

2. How many acres in a triangular field whose sides are, respectively, 50, 60, and 70 rods?

3. Find the area of a triangular field whose sides are each 60 rods.

THE RIGHT TRIANGLE.

435. A Right Triangle is a triangle which has

one right angle.

436. The Hypotenuse is the side opposite the right angle.

437. The Perpendicular is the side which forms a right angle with the base.

438. To find the Hypotenuse.

It is seen that the square described on the hypotenuse is equal to the sum of the squares described on the other two sides. Hence,

RULE. Add the square of the base to the square of the perpendicular, and extract the square root of their sum.

439. To find the Base or Perpendicular.

A

B

RULE.-From the square of the hypotenuse take the square of the given side, and extract the square root of the remainder.

EXAMPLES.

1. The base of a right triangle is 15 inches, and the perpendicular is 8 inches. What is the hypotenuse?

Solution Hypotenuse the hypotenuse is 17 in.

=

√152 + 82 = √225 + 64 = √289 = 17. Hence,

2. The hypotenuse of a right triangle is 30 ft., and the perpenaicular is 24 feet. Find the base.

Solution: 302 — 24a :
=V900-576 =

=

324 18. Hence, the base is 18

feet.

3. The hypotenuse of a right triangle is 80 ft., and the base is 64 feet. What is the perpendicular?

4. A rectangular field is 70 rd. long and 45 rd. wide. What is the distance between the opposite corners?

5. How far apart are the opposite corners of a square farm containing 160 acres?

6. How far is it from a lower corner to the opposite upper corner of a room 18 ft. long, 16 ft. wide, and 10 ft. high?

7. I wish to build a house 36 ft. wide with the ridge of the roof 14 ft. above the body of the house. What must be the 'ength of the rafters allowing 8 in. for extension?

440. To find the area of a Trapezoid.

441. A Trapezoid is a quadrilateral which has only two of its sides parallel.

The Altitude of a trapezoid is the perpendicular distance between its parallel sides.

A

D

B

E

It is seen that any trapezoid may be divided into two triangles by a diagonal; as DA. The area of triangle ABD is the length of the base AB multiplied by half the altitude DE (Art. 431), and the area of ACD is the length of the upper base CD multiplied by half the altitude DE. Hence,

RULE.-Multiply the sum of the parallel sides by half the

altitude.

EXAMPLES.

442. 1. What is the area of a trapezoid whose bases are 18 and 16 in., respectively, and altitude 14 in. ?

2. A field in the form of a trapezoid, whose parallel sides are 30 rd. and 36 rd., respectively, and which has an altitude of 25 rd., contains how many acres?

3. A walk in the form of a trapezoid is 30 ft. long, 6 ft. wide at one end and 4 ft. at the other. How many square feet in the walk?

443. To find the area of a Trapezium.

444. A Trapezium is a quadrilateral which has no two sides parallel.

It will be seen that any trapezium may be divided by a diagonal, as AC, into two triangles whose areas may be found by Arts. 431 and 433. Hence,

RULE.-Divide the trapezium into two triangles by a diagonal. Find the area of each triangle and take the sum.

The diagonal may also be made the base of each triangle and perpendiculars may be drawn; as, DE. If one of the triangles is a right triangle, then either of its sides about the right angle may be regarded as the base and the other as the altitude; as, AB and BC.

EXAMPLES.

445. 1. Find the area of a trapezium whose diagonal is 75 ft., and the altitudes of the triangles, the diagonal being the base, 20 and 30 ft., respectively.

2. What is the area of a trapezium whose diagonal is 100 rd., and the perpendiculars, 36 and 44 rd., respectively?

3. How many acres in a farm in the form of a trapezium the length of whose sides are respectively 40, 50, 60, and 70 chains; and the length of the diagonal 80 chains?

446. To find the area of a Regular Polygon.

447. A Regular Polygon is a polygon which is equilateral and equiangular.

G

F

E

D

It will be seen that a regular polygon may be divided into a number of triangles equal to the number of sides of the polygon, having equal bases and altitudes. The area of each triangle is equal to the base multiplied by half the altitude, and the area of all the triangles, or the polygon, is equal to the sum of all the bases multiplied by half the altitude. Hence, RULE.-Multiply the perimeter by half the perpendicular falling from the center of the polygon upon one of its sides.

EXAMPLES.

A

X

B

448. 1. What is the area of a pentagon each of whose sides is 12 ft. and the perpendicular distance from the center to one of its sides is 9 ft. ?

2. Find the area of a regular octagon whose side is 35 rd., if the perpendicular distance from its center to a side is 42 rd.

THE CIRCLE.

449. A Circle is a plane figure bounded by a curved line, every part of which is equally distant from a point

within called the center.

The Circumference of a circle is the bounding line; as, ABDE.

An Arc is any part of the circumference; as, EF. A Chord is any straight line having its extremities in the circumference; as, AD, or GH.

E

B

H

The Diameter of a circle is any straight line passing through the center and terminating in the circumference; as AD, or EB. The Radius of a circle is any straight line drawn from the center to the circumference, as CB, CD, CF, etc.

The radius is half of the diameter of the circle.

450. To find the Circumference or Diameter of a Circle.

If we take a circle 4 inches in diameter and measure accurately the circumference, we shall find it to be about 12.5664 inches. Dividing the circumference, 12.5664 in. by the diameter, 4 in., we have for a quotient 3.1416. Hence,

RULE.-1. To find the circumference, multiply the diameter by

3.1416.

2. To find the diameter, divide the circumference by 3.1416.

This number, 3.1416, is often represented by the Greek letter π, called pi. If we take a circle whose circumference is 1, its diameter will be 13.1416, or .3183. Hence,

3. To find the diameter, multiply the circumference by .3183. If we let represent the circumference, D the diameter, and R the radius, the rules may be briefly expressed as follows:

451. 1. What is the circumference of a circle whose diameter is 12 ft.?

2. What is the circumference of a wheel which has a diameter of 38 inches?

3. The circumference of circle is 100 ft. What is its diameter?

4. What is the diameter of a circle whose circumference is 314.16 rd.?

5. Find the radius of a circle whose circumference is 94.248 ft.

6. The front wheel of a buggy is 41 inches in diameter. How many revolutions does it make in going a mile?

452. To find the area of a Circle.

A circle may be regarded as composed of a very large number of triangles, the sum of whose bases forms the circumference of the circle, and whose altitude is the radius of the circle. Hence,

RULE.-Multiply the circumference by half the radius, or one-fourth of the diameter.

The area of a circle is also equal to the square of the radius multiplied by 3.1416; or areaπ R2; or, π D2 = .7854 X D2.

EXAMPLES.

453. 1. What is the area of a circle whose diameter is 20 in. and circumference 62.832 in. ?

2. What is the area of a circle whose diameter is 25 ft.?

3. Find the area of a circle whose radius is 18 ft.

4. Find the area of a circle whose radius is a rod. 5. The circumference of a circular field is 120 rd. acres in the field?

How many

6. A has a field that is 40 rd. square, and B has a circular field 40 rd. in diameter. What is the distance around each field, and how many acres does each contain ?

454. To find the Diameter or Circumference of a Circle from the area.

C = × D, or π D; and C × D

.. πDX or

=Area.

D, π D2

4

4