DIMENSIONS OF PLANE FIGURES Machinery's Handbook, 31st Edition
76
Right Triangle (one angle is a 90-degree angle):
A ab 2 = ---
c
From the Pythagorean theorem, a 2 + b 2 = c 2 , thus
a
a 2 b 2 = +
a c 2 b 2 = –
b c 2 a 2 = –
c
b
Example: Side a is 6 in. and side b is 8 in. Find side c and area A : c a 2 b 2 + 6 2 8 2 + = = = =
36 + 64 100 10 in. =
A ab 2 = = = Example: Side c = 10 and side a = 6. Find side b : b c 2 a 2 – 10 2 6 2 – ------ 6 8 × 2 = ------ 48 2 = in. Acute Triangle (all three angles measure less than 90 degrees): --- 24 in 2 = =
100 – 36 64 8 in. = =
a b c b + − 2 ( ) 2 2 2
a b c b + − 2 ( ) 2 2 2
2
2
b 2
bh
2
2
h a
A
a
= −
=
−
so
A
,
,
=
a
c h
2
a b c + +
b
)( )( ), S a S b S c − − −
S
A S =
=
(
where
2
Example: Side b = 7 inches, h = 4 inches, so A = bh /2 = (7 in. × 4 in.)/2 = 28 in 2 /2 = 14 in 2 Example: Side a = 10 cm, b = 9 cm, and c = 8 cm 2 . Find the area. A b 2 -- a 2 a 2 b 2 c 2 + – 2 b -------------- 2 – 9 2 -- 10 2 10 2 9 2 8 2 + – 2 9 × ---------------- 2 – 4.5 100 117 18 ----- – = = = 4.5 100 – 42.25 4.5 57.75 4.5 7.60 × 34.20cm 2 = = = =
2
Obtuse Triangle (one angle measures greater than 90 degrees):
(
)
(
)
2
2
2
2
2
2
2
2
a b c b + − 2
b 2
a b c b + − 2
bh
2
2
h a
A
a
= −
=
−
so
A
,
,
=
2
c
a h
a b c + +
A S =
)( )( ), S a S b S c − − −
S
=
(
where
b
2
Example: If b = 5 cm and h = 3 cm, then A = bh /2 = (5 cm × 3 cm)/2 = 15 cm 2 /2 = 7.5 cm 2 Example: Side a = 5 in., side b = 4 in., and side c = 8 in. Find the area. S a b c + + ( )/2 5+4+8 ( )/2 17/2 8.5 = = = = A S S a – ( ) S b – ( ) S c – ( ) 8.5 8.5– 5 ( ) 8.5– 4 ( ) 8.5– 8 ( ) = = 8.5 3.5 × 4.5 0.5 × × 66.937 8.18 in 2 = = =
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