Geometry – accel.
Worksheet: Area of a Triangle
April 2013
I. Calculating the area of a right triangle.
1
c
The area, 𝐾, of a right triangle with legs 𝑎 and 𝑏 is given by the formula 𝐾 = 2 𝑎𝑏.
a
b
II. Finding the area of a triangle with SAS specified.
In the triangle at right, the length of the altitude, ℎ, can be found using the
right triangle on the right:
sin 𝑥° =
ℎ
𝑎
B
ℎ = 𝑎 ∙ sin 𝑥°
Using this relation we can derive the area, 𝐾, of the triangle in terms of the
two sides, 𝑎 and 𝑏, and the included angle 𝑥°:
𝐾=
1
𝑎𝑏
2
h
a
sin 𝑥°
x
III. Finding the area of a triangle with SSS specified (Heron’s Formula).
A
C
b
This famous formula is credited to Heron of Alexandria and a proof can be found in his book, Metrica, written
c. A.D. 60. It was probably known to the famous mathematician and philosopher Archimedes. A modern proof
uses the Law of Cosines and will not be presented here, but can be found on Wikipedia.
Given the three sides of a triangle, 𝑎, 𝑏, and 𝑐, the area, 𝐾, of the triangle can be found using Heron’s Formula
(sometimes called Hero’s Formula):
𝐾 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where 𝑠, the semiperimeter, is given by
𝑠=
𝑎+𝑏+𝑐
2
1. Find the area of each of the triangles below using any appropriate method.
36°
38°
14
22
12
11
15
40
32
78°
14
30
2. Find the area of the triangle with sides 5, 12, and 13 two different ways.
3. Find the area of the triangle with sides of length 7 and 10 and included angle of 50°.
4. A triangle with area 83 has two sides of lengths 24 and 17. Find the angle between these sides to the
nearest degree.
5. A triangle with area 1280 has a 77° angle with one adjacent side of length 60. Find the length of the other
adjacent side to the nearest tenth.
Find the area of each triangle to the nearest hundredth, given the three side lengths.
6. 5, 7 𝑎𝑛𝑑 9
7. 16, 34 𝑎𝑛𝑑 30
8. 15, 25 𝑎𝑛𝑑 30
9. 16, 18 𝑎𝑛𝑑 9
9. 3.5, 2.5 𝑎𝑛𝑑 4
10. 4.2, 5.6 𝑎𝑛𝑑 8.2