How to Find the Area of a triangle

How to Find Angles and Area of Triangles

 Read,3 minutes

For triangles, two ACT facts appear often: the angle sum is \(180^\circ\), and area is half the product of a base and its perpendicular height.

Core Formulas

  • Area: \(A=\frac{1}{2}bh\).
  • Missing height: \(h=\frac{2A}{b}\).
  • Triangle angle sum: \(x+y+z=180^\circ\).
  • Right-triangle area: the two legs can be used as base and height.

Worked Example

A triangle with base \(14\) cm and height \(9\) cm has area \(\frac12(14)(9)=63\) square centimeters.

Reference Diagram

Area of Triangles

Angle of Triangle

Video Lesson

Original Practice Figures

These saved figures are kept with the lesson for continuity. The exercise text below gives all needed measurements.

Area of a triangle Area of a triangle1 Area of a triangle2 Area of a triangle3 Area of a triangle4 Area of a triangle5 Area of a triangle6 Area of a triangle7 Area of a triangle8 Area of a triangle9 Area of a triangle Area of a triangle1 Area of a triangle2 Area of a triangle3 Area of a triangle4 Area of a triangle5 Area of a triangle6 Area of a triangle7 Area of a triangle8 Area of a triangle9

Angle and Area of Triangles

Think of this lesson as more than a rule to memorize. Angle and Area of Triangles is about shape relationships, formulas, and units. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Geometry formulas work because they measure a specific feature: length around, space inside, or space enclosed by a solid. Match the question to the measurement first.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Sketch or label the shape.
  • Decide whether the question asks for length, area, volume, or surface area.
  • Substitute values into the matching formula.
  • Keep units squared for area and cubed for volume.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) Find the area of a triangle with base 10 cm and height 6 cm.

2) Find the area of a triangle with base 14 ft and height 9 ft.

3) A triangle has area 48 square inches and base 12 inches. Find the height.

4) Two angles of a triangle are \(45^\circ\) and \(65^\circ\). Find the third angle.

5) A right triangle has legs 8 m and 15 m. Find its area.

6) Find the area of a triangle with base 7.5 in and height 4 in.

7) A triangle has angles \(x^\circ\), \(2x^\circ\), and \(3x^\circ\). Find the angles.

8) An isosceles triangle has vertex angle \(40^\circ\). Find each base angle.

9) A triangle has area 84 square cm and height 14 cm. Find the base.

10) A triangle has base 18 yd and height 11 yd. Find its area.

11) The area of a right triangle is 96 square ft and one leg is 12 ft. Find the other leg.

12) Two angles of a triangle are \(3x^\circ\) and \(4x^\circ\), and the third is \(40^\circ\). Find \(x\).

13) Find the area of a triangle with base 16 and height 16.

14) A triangle has side used as base \(25\) cm and perpendicular height \(18\) cm. Find its area.

15) The angles of a triangle are \(x^\circ\), \((x+20)^\circ\), and \((2x)^\circ\). Find all angles.

16) A triangle has area 135 square meters and base 15 meters. Find the height.

17) An equilateral triangle has side length 12. Find its exact area.

18) A triangle has base 30 and height \(2x\). Its area is 180. Find \(x\).

19) A triangle has angles in the ratio \(2:3:4\). Find the largest angle.

20) A triangular sign has base 42 cm and height 35 cm. Paint costs $0.02 per square cm. What is the paint cost?

 
1) Use \(A=\frac12bh\).
\(A=\frac12(10)(6)=30\).
The area is \(30\) square cm.
2) Use \(A=\frac12bh\).
\(A=\frac12(14)(9)=63\).
The area is \(63\) square ft.
3) Start with \(A=\frac12bh\).
\(48=\frac12(12)h=6h\).
So \(h=8\) inches.
4) Triangle angles sum to \(180^\circ\).
Third angle \(=180^\circ-45^\circ-65^\circ=70^\circ\).
5) The legs are perpendicular, so use them as base and height.
\(A=\frac12(8)(15)=60\).
The area is \(60\) square m.
6) Use \(A=\frac12bh\).
\(A=\frac12(7.5)(4)=15\).
The area is \(15\) square in.
7) Add the angle expressions: \(x+2x+3x=180\).
\(6x=180\), so \(x=30\).
The angles are \(30^\circ,60^\circ,90^\circ\).
8) The two base angles are equal.
Remaining measure is \(180^\circ-40^\circ=140^\circ\).
Each base angle is \(140^\circ/2=70^\circ\).
9) Use \(b=\frac{2A}{h}\).
\(b=\frac{2(84)}{14}=12\).
The base is \(12\) cm.
10) Use \(A=\frac12bh\).
\(A=\frac12(18)(11)=99\).
The area is \(99\) square yd.
11) For a right triangle, the legs are base and height.
\(96=\frac12(12)x=6x\).
Thus \(x=16\) ft.
12) Angle sum: \(3x+4x+40=180\).
\(7x=140\), so \(x=20\).
13) Use \(A=\frac12bh\).
\(A=\frac12(16)(16)=128\).
The area is \(128\) square units.
14) Use the perpendicular height, not a slanted side.
\(A=\frac12(25)(18)=225\).
The area is \(225\) square cm.
15) Set up the angle sum: \(x+(x+20)+2x=180\).
\(4x+20=180\), so \(x=40\).
The angles are \(40^\circ,60^\circ,80^\circ\).
16) Use \(h=\frac{2A}{b}\).
\(h=\frac{2(135)}{15}=18\).
The height is \(18\) m.
17) An equilateral triangle area is \(A=\frac{s^2\\sqrt{3}}{4}\).
\(A=\frac{12^2\\sqrt{3}}{4}=36\\sqrt{3}\).
The exact area is \(36\\sqrt{3}\) square units.
18) Use \(A=\frac12bh\).
\(180=\frac12(30)(2x)=30x\).
Thus \(x=6\).
19) The ratio has \(2+3+4=9\) total parts.
Each part is \(180^\circ/9=20^\circ\).
The largest angle is \(4(20^\circ)=80^\circ\).
20) First find area: \(A=\frac12(42)(35)=735\) square cm.
Cost is \(735(0.02)=\$14.70\).

More Practice

Free printable Worksheets

Related Lessons