Honors Geometry Section 5.2

Areas of Triangles and Quadrilaterals

WHY?

RHL E F

You can see that the area of parallelogram ABCD is equal to the area of rectangle EBCF.

For a parallelogram with base b and height h, the area is given by

the formula: A parallelogram = ______

Note that the height is the length of the segment perpendicular to the base from a point on the opposite side which is called

the altitude of the parallelogram.

hb

+

s2

s

3s 34

2 3603415 uA

60610 A608 x

ux 5.78/60

Any triangle is half of a parallelogram. For a triangle with base b and height h, the area is given by the formula:

A triangle = ________

The height is the length of the ____________ to the base

bh21

altitude

Example: Find the area of to the nearest 1000th.

226.410

25sin

AC

AC

063.910

25cos

BC

BC

2uA 150.19063.9226.45.

Example: A triangle has an area of 56 and a base of 10. Find its height.

h

hbA

102156

21

2.11h

2uA 15)3)(10(5.

Trigonometry and the Area of a Triangle

Using your knowledge of trigonometry, express h in terms of sinC.

Substituting this into the formula , and using a as the base we get

b

hC sin hCb sin

bhA2

1

CabA sin2

1

We have just discovered that the area of a triangle can be expressed using the lengths of two sides and

the sine of the included angle.

Example: Use what you have learned above to find the area of parallelogram ABCD to the nearest 1000th.

)(2// trianglegram AA

50sin2515

2

12// gramA

50sin2515// gramA

2// 267.287 cmA gram

An altitude of a trapezoid is a segment perpendicular to the two bases with an endpoint in each of

the bases.

The length of an altitude will be the height of the trapezoid.

For a trapezoid with bases b1 and b2 and height h, the area of a trapezoid is given by the formula: 21. 2

1 bbhAtrap

hb221

hb121

hbhb 12 21

21 )(2

112 bbh

Recall that the diagonals of both rhombuses and kites are

perpendicular.

E

AEBC21

DEBC21

DEBCAEBCAkite 21

21

DEAEBCAkite 21

21hom 21 ddAA busrkite