Geometry · Area

How to Find the Area of a Triangle
Formula A=b×h÷2 + 15 Solved Examples

Learn how to find the area of any triangle using A=base×height÷2. With 15 solved examples, all triangle types, and Heron's formula when you only have the sides.

⚡ RESPUESTA RÁPIDA

Area of a triangle: A = base × height ÷ 2. For b=10cm and h=6cm: A=10×6÷2=30cm². The height must be perpendicular to the base — not the slant side.

Why Divide by 2? — Visual Explanation

Triangle = half rectangle other half Rectangle: b×h Triangle: b×h÷2 Any triangle = exactly half a rectangle

Step-by-Step Method

Identify the base and perpendicular heightThe height is PERPENDICULAR (90°) to the base. In a right triangle, the two legs are base and height.
Apply the formula: A = base × height ÷ 2b=10, h=6: A=10×6÷2=60÷2=30cm².
Check your answerThe area should be less than base×height (the rectangle). 30<60 ✓

15 Solved Examples

b=6,h=4
12cm²
b=10,h=5
25cm²
b=8,h=3
12cm²
b=15,h=6
45cm²
b=7,h=7
24.5cm²
b=12,h=4
24cm²
b=3.5,h=8
14cm²
b=20,h=2.5
25cm²
A=30,b=10. h?
6cm
A=24,h=8. b?
6cm
Right: legs 6,8
24cm²
Right: legs 5,12
30cm²
Equil. l=6
15.59cm²
A=36,h=6. b?
12cm
A=50,b=5. h?
20cm

Heron's Formula — When You Only Have the 3 Sides

1
Calculate s (semi-perimeter)s=(a+b+c)÷2. For sides 5,7,8: s=(5+7+8)÷2=10.
2
Apply: A=√(s(s-a)(s-b)(s-c))A=√(10×5×3×2)=√300≈17.32cm².
Classic 3-4-5 right triangle

s=6. A=√(6×3×2×1)=√36=6cm². Verify: 3×4÷2=6 ✓

Preguntas Frecuentes

Does the base have to be horizontal?

No. Any side can be the base, but the height must be perpendicular to whichever side you choose as the base.

Can a triangle have the same area but different shapes?

Yes. A triangle with b=10,h=6 (A=30) and one with b=15,h=4 (A=30) have equal area but look different.

What is the largest area triangle for a given perimeter?

The equilateral triangle maximizes area for a given perimeter, just like the circle does for all shapes.

Real-World Applications of Triangle Area

Roof calculation

A triangular roof section with base=8m and height=5m: A=8×5÷2=20m². Multiply by 2 for both sides=40m² of roofing material needed.

Land area

A triangular plot with base=50m and height=30m: A=50×30÷2=750m².

Sail area

A triangular sail with base=4m and height=7m: A=4×7÷2=14m² of material.

Common Mistakes to Avoid

Using the slant side instead of height

Height is always perpendicular (90°) to the base. The slant side of a triangle is NOT the height (unless it's a right triangle where one angle is 90°).

Forgetting to divide by 2

A=b×h not b×h. The ÷2 is what makes it a triangle and not a rectangle.

Using the wrong base-height pair

If you change which side is the base, the height changes too. But the area stays the same — verify this.

b=6,h=4
12cm²
b=10,h=5
25cm²
b=8,h=3
12cm²
b=15,h=6
45cm²
b=14,h=9
63cm²
b=20,h=7
70cm²
A=30,b=10. h
6cm
A=24,h=8. b
6cm
A=50,b=5. h
20cm
Right: 3,4
6cm²
Right: 6,8
24cm²
Right: 5,12
30cm²

Preguntas Frecuentes

Is the formula different for obtuse triangles?

No. A=b×h÷2 works for ALL triangles. For obtuse triangles, the height may fall outside the triangle, but the formula still applies.

What units does area use?

Always squared: cm², m², km². If base and height are in cm, area is in cm².

Can I use any side as the base?

Yes. The area stays the same regardless of which side you choose as the base, as long as you use the corresponding perpendicular height.

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