Area of a triangle

Q: What is the formula for the area of a triangle?

A = \tfrac{1}{2} \times \text{base} \times \text{height}, where the height is measured at right angles to the base.

A triangle is half of a rectangle (or a parallelogram) with the same base and height, so its area is half of base × height. When the height is not given but all three sides are, Heron's formula does the job.

Formulas

Area of a triangle

A = \frac{1}{2} \times b \times h

Missing height

h = \frac{2 A}{b}

Heron's formula (sides a, b, c and s = half the perimeter)

A = s (s - a) (s - b) (s - c), s = \frac{a + b + c}{2}

Worked examples

Example 1. A triangle has a base of 10 cm and a height of 6 cm. Find its area.

$A = \frac{1}{2} \times 10 \times 6$ .
$A = 30$ .

Answer:

30 cm^{2}

Example 2. A triangle has an area of

48 cm^{2}

and a base of 12 cm. Find its height.

$h = \frac{2 A}{b} = \frac{2 \times 48}{12}$ .
$h = 96 \div 12 = 8$ .

Answer: 8 cm

Example 3. Find the area of a triangle with sides 13 cm, 14 cm and 15 cm.

$s = (13 + 14 + 15) \div 2 = 21$ .
$A = 21 \times 8 \times 7 \times 6 = 7056$ .
$7056 = 84$ .

Answer:

84 cm^{2}

Common mistakes

Forgetting the ½. Without it you have the area of the rectangle around the triangle.
Using a slanted side as the height. The height must be perpendicular to the base.
Mixing units. Base and height must be in the same unit before multiplying.

Practice questions with answers

1. What is the formula for the area of a triangle?

A $\frac{1}{2} x$ base $x$ height
Bbase $x$ height
C2 $x$ base $x$ height
Dbase + height

Show answer

Answer: A)

\frac{1}{2} x

base

x

height

The area of a triangle is half the product of its base and the corresponding height: Area =

\frac{1}{2} x

base

x

height.

2. In Heron's formula, what does the letter s usually stand for?

AThe semi-perimeter of the triangle
BThe shortest side of the triangle
CThe full perimeter of the triangle
DThe area of the triangle

Show answer

Answer: A) The semi-perimeter of the triangle

Heron'

s

formula uses

s = \frac{a + b + c}{2}

, the semi-perimeter (half of the perimeter).

3. What is the formula for the area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3)?

A $\frac{1}{2} ∣ x 1 (y 2 - y 3) + x 2 (y 3 - y 1) + x 3 (y 1 - y 2) ∣$
B $\frac{1}{2} ∣ x 1 (y 2 + y 3) + x 2 (y 3 + y 1) + x 3 (y 1 + y 2) ∣$
C|x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
D $\frac{1}{2} ∣ x 1 + x 2 + x 3 - y 1 - y 2 - y 3∣$

Show answer

Answer: A)

\frac{1}{2} ∣ x 1 (y 2 - y 3) + x 2 (y 3 - y 1) + x 3 (y 1 - y 2) ∣

The standard coordinate area formula is Area =

\frac{1}{2} ∣ x 1 (y 2 - y 3) + x 2 (y 3 - y 1) + x 3 (y 1 - y 2) ∣

4. The base of a triangle is 20 cm and its area is 130 sq cm.
A student claims the height is 13 cm. Verify whether the claim is correct, and if not, state the correct height.

ACorrect, height = 13 cm
BWrong, height = 6.5 cm
CWrong, height = 26 cm
DWrong, height = 10 cm

Show answer

Answer: A) Correct, height = 13 cm

130 =

\frac{1}{2} x 20 x h = 10 h

, so

h = 13

cm. The student's claim is correct.

5. For the triangle with sides 13, 14, 15, the term (s-a)(s-b)(s-c) equals (with a=13, b=14, c=15):

A672
B336
C301
D371

Show answer

Answer: B) 336

s = 21. (21 - 13) (21 - 14) (21 - 15) = 8 \cdot 7 \cdot 6 = 336

6. Triangle ABC has A(1,1), B(7,1), C(3,5). A point P lies on AB so that area of triangle APC = 6 sq units.
Given area of ABC = 12 sq units, what fraction of AB is AP?

A $\frac{1}{2}$
B $\frac{1}{3}$
C $\frac{2}{3}$
D $\frac{1}{4}$

Show answer

Answer: A)

\frac{1}{2}

Triangles APC and ABC share the same height from C to line AB, so their areas are proportional to bases AP and AB. AP/AB = area(APC)/area(ABC) =

\frac{6}{12} = \frac{1}{2}

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Frequently asked questions

What is the formula for the area of a triangle?

$A = \frac{1}{2} \times base \times height$ , where the height is measured at right angles to the base.

Why is there a half in the formula?

Two identical triangles fit together to make a parallelogram with the same base and height, and the parallelogram's area is base × height. One triangle is half of that.

When should I use Heron's formula?

When you know all three sides but not the height. It is also handy for triangles that are not right-angled.

How do you find the area of a right-angled triangle?

Use the two sides that make the right angle as base and height: $A = \frac{1}{2} \times leg_{1} \times leg_{2}$ .

Area of a triangle

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

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