Area of Triangles Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Area of Triangles.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The area of a triangle is half the product of its base and height: $A = \frac{1}{2}bh$ .

Every triangle is exactly half of a rectangle with the same base and height — cut the rectangle along the diagonal.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A triangle's area is half its base times its perpendicular height, because every triangle is half a rectangle of the same base and height.

Common stuck point: The procedure for area of triangles is the easy part; the trap is forgetting the $\frac{1}{2}$ . Asking "Do I have a base and a height that meets it at a right angle, and do I remember to take half?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have a base and a height that meets it at a right angle, and do I remember to take half?

Worked Examples

Example 1

medium

A triangle has vertices

(0,0)

(8,0)

(3,6)

. Find its area.

Answer

First step

Base along

x

-axis:

8

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Example 2

medium

Use Heron's formula to find the area of a triangle with sides

5

12

13

Example 3

medium

A triangle has vertices

(2,3)

(8,3)

(5,9)

. Find its area.

Example 4

hard

Use the shoelace formula to find the area of the triangle with vertices

(0,0)

(5,2)

(3,7)

Example 5

hard

Find the area of the triangle with sides

9

10

17

using Heron's formula.

Example 6

hard

A triangle has base

b

on the

x

-axis and a vertex at

(x_0,y_0)

with

y_0>0

. Explain why its area equals

\tfrac{1}{2}b\cdot y_0

Example 7

hard

A triangle has vertices

(0,0)

(6,0)

(0,8)

. Find the inradius given area

24

and perimeter

24

(sides

6

8

10

Example 8

hard

A triangle's vertices are

(1,1)

(6,1)

(4,8)

. Find its area.

Example 9

challenge

A triangle has sides

3

4

5

inscribed with a smaller similar triangle whose sides are half. Compare areas.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the area of a triangle with base

7

and height

4

Example 2

easy

Find the area of a triangle with base

12

and height

5

Example 3

easy

A right triangle has legs

5

and

12

. Find its area.

Example 4

easy

A right triangle has legs

9

and

40

. Find its area.

Example 5

easy

Why must we divide by

2

in the triangle area formula?

Example 6

medium

A triangle's base is doubled and height is halved. How does its area change?

Example 7

medium

Find the area of an equilateral triangle with side

8

(use

\sqrt 3\approx 1.73

Example 8

medium

A triangle has sides

7

and

10

with included angle

30°

. Find its area.

Example 9

medium

A triangular flag has base

24

in and height

18

in. How many square feet of fabric does it need?

Example 10

medium

A triangle has area

48

. A new triangle has base

4\times

as big and height

\tfrac{1}{4}

as big. Find its area.

Example 11

hard

A triangle and a parallelogram share base

12

and the triangle's area is

30

. What height makes the parallelogram have the same area?

Example 12

hard

In a triangle, the three medians all split it into two equal-area triangles. True or false?

Example 13

hard

An isosceles triangle has two sides

13

and base

10

. Find its area.

Example 14

hard

A triangle has sides

a=8

b=15

, included angle

C=90°

. Find its area.

Example 15

challenge

A triangle has area

A

. If a cevian splits the opposite side in ratio

2:3

, find the areas of the two pieces.

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Related Concepts

Area Triangles Multiplication Area of Parallelograms

Background Knowledge

These ideas may be useful before you work through the harder examples.

areatrianglesmultiplication