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Free Printable Math Worksheets for Geometry

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• Review of equations
• Simplifying square roots
• Adding and subtracting square roots
• Multiplying square roots
• Dividing square roots
• Line segments and their measures inches
• Line segments and their measures cm
• Segment Addition Postulate
• Angles and their measures
• Classifying angles
• Naming angles
• The Angle Addition Postulate
• Angle pair relationships
• Understanding geometric diagrams and notation
• Parallel lines and transversals
• Proving lines parallel
• Points in the coordinate plane
• The Midpoint Formula
• The Distance Formula
• Parallel lines in the coordinate plane
• Classifying triangles
• Triangle angle sum
• The Exterior Angle Theorem
• Triangles and congruence
• SSS and SAS congruence
• ASA and AAS congruence
• SSS, SAS, ASA, and AAS congruences combined
• Right triangle congruence
• Isosceles and equilateral triangles
• Midsegment of a triangle
• Angle bisectors
• The Triangle Inequality Theorem
• Inequalities in one triangle
• Classifying quadrilaterals
• Angles in quadrilaterals
• Properties of parallelograms
• Properties of trapezoids
• Properties of rhombuses
• Properties of kites
• Areas of triangles and quadrilaterals
• Introduction to polygons
• Polygons and angles
• Areas of regular polygons
• Solving proportions
• Similar polygons
• Using similar polygons
• Similar triangles
• Similar right triangles
• Proportional parts in triangles and parallel lines
• The Pythagorean Theorem and its Converse
• Multi-step Pythagorean Theorem problems
• Special right triangles
• Multi-step special right triangle problems
• Trig. ratios
• Inverse trig. ratios
• Solving right triangles
• Multi-step trig. problems
• Rhombuses and kites with right triangles
• Trigonometry and area
• Identifying solid figures
• Volume of prisms and cylinders
• Surface area of prisms and cylinders
• Volume of pyramids and cones
• Surface area of pyramids and cones
• More on nets of solids
• Similar solids
• Arcs and central angles
• Arcs and chords
• Circumference and area
• Inscribed angles
• Tangents to circles
• Secant angles
• Secant-tangent and tangent-tangent angles
• Segment measures
• Equations of circles
• Translations
• Reflections
• All transformations combined
• Sample spaces and The Counting Principle
• Independent and dependent events
• Mutualy exclusive events
• Permutations
• Combinations
• Permutations vs combinations
• Probability using permutations and combinations
• Line segments
• Perpendicular segments
• Medians of triangles
• Altitudes of triangles

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Printable Triangle Worksheets

This extensive collection of worksheets on triangles for grades 3 through high-school is incredibly useful in imparting a clear understanding of a variety of topics like classifying triangles, similar triangles, congruence of triangles, median and centroid of a triangle, inequality theorem, Pythagorean inequalities, area, perimeter and angles in a triangle and much more.

List of Triangle Worksheets

Coloring, Tracing, and Drawing Triangles

• Classifying Triangles
• Perimeter of a Triangle
• Area of a Triangle
• Angles in a Triangle
• Medians and Centroid of a Triangle
• Triangle Inequality Theorem
• Pythagorean Inequality Theorem
• Congruent Triangles
• Similar Triangles
• Ordering the Sides and Angles of a Triangle

Explore the Triangle Worksheets in Detail

Take kiddos into the world of triangles with this compilation of worksheets featuring exercises like identifying triangles in real-life, coloring, tracing, and drawing triangles, etc., and a lot more fascinating activities.

Classifying Triangles Worksheets

This cluster of classifying triangles worksheets consists of skills like identifying triangles based on the sides as isosceles, equilateral and scalene and based on angles as acute, obtuse and equilateral. Find exercises to classify triangles offering a mix of angles and sides.

Perimeter of a Triangle Worksheets

Bolster skills with worksheets on finding the perimeter of a triangle with dimensions presented as integers, decimals and fractions. Find the perimeter of scalene, isosceles and equilateral triangles, determine the measure of the sides using the perimeter and solve word problems.

Area of a Triangle Worksheets

Intensify practice with this compilation of area of a triangle worksheets featuring skills like finding the area of scalene, isosceles and equilateral triangles, find the missing base or height, find the area with measures offered as integers, decimals, fractions and algebraic expressions to mention just a few.

Angles in a Triangle Worksheets

Featuring myriad exercises, this set of angles in a triangle worksheets help learn the application of angle sum property and exterior angle theorem to find the indicated angles with whole numbers and algebraic expressions. Solve for 'x' and try a set of challenging problems as well.

Medians and Centroid of a Triangle Worksheets

Walk through the median and centroid worksheets offering a wide variety of exercises like finding the indicated length, find the midpoint, centroid, equations of the medians and much more.

Triangle Inequality Theorem Worksheets

The triangle inequality theorem worksheets encompass ample skills like check if the side measures form a triangle or not, find the range of possible measures, the lowest and greatest possible whole number measures of the third side. Also, included are multiple response revision worksheets.

Pythagorean Inequality Theorem Worksheets

A triangle can easily be classified if its angle measures are known. But, can you determine a triangle’s type with its side lengths? Plug into our Pythagorean Inequality theorem worksheets and excavate the answer!

Congruent Triangles Worksheets

The worksheets on congruent triangles help master the concept of congruence with ease. This assortment provides adequate PDFs on identifying congruent parts such as angles and sides, writing the congruence statement, identifying the postulates, proving the right triangle congruence and more.

Similar Triangles Worksheets

The similar triangles worksheets provide practice in determining the scale factors, identifying similar triangles, calculating side lengths, writing the similarity statements; finding similarity based on SSS, SAS and AA theorems, solving algebraic expressions and comprehending similarity of right triangles.

Ordering the Sides and Angles of a Triangle Worksheets

Plunge into practice with this set of pdf worksheets and get the hang of ordering the sides of a triangle by analyzing its angle measures, and ordering the angles by analyzing its side lengths.

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Geometry Worksheets

Welcome to the geometry worksheets page at Math-Drills.com where we believe that there is nothing wrong with being square! This page includes Geometry Worksheets on angles, coordinate geometry, triangles, quadrilaterals, transformations and three-dimensional geometry worksheets.

Get out those rulers, protractors and compasses because we've got some great worksheets for geometry! The quadrilaterals are meant to be cut out, measured, folded, compared, and even written upon. They can be quite useful in teaching all sorts of concepts related to quadrilaterals. Just below them, you'll find worksheets meant for angle geometry. Also see the measurement page for more angle worksheets. The bulk of this page is devoted to transformations. Transformational geometry is one of those topics that can be really interesting for students and we've got enough worksheets for that geometry topic to keep your students busy for hours.

Don't miss the challenging, but interesting world of connecting cubes at the bottom of this page. You might encounter a few future artists when you use these worksheets with students.

Most Popular Geometry Worksheets this Week

Lines and Angles

In this section, there are worksheets for two of the basic concepts of geometry: lines and angles.

Lines (or straight lines to be precise) in geometry are continuous and extend in both directions to infinity. They have no width, depth or curvature. In math activities, they are often represented by a drawn straight path with some width. To show that they are lines, arrows are drawn on each end to show they extend to infinity. A line segment is a finite section of a line. Line segments are often represented with points at each end of a drawn straight path. Rays start at a point and extend in a straight line to infinity. This is shown with a point at one end of a drawn straight path and an arrow at the other end.

• Identifying Lines, Line Segments and Rays Identify Lines, Segments and Rays

Angles can be classified into six different types. Acute angles are greater than 0 degrees but less than 90 degrees. Right angles are exactly 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Straight angles are exactly 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees. Complete/Full angles are exactly 360 degrees.

• Identifying Angle Types Worksheets Identifying Acute and Obtuse Angles Identifying Acute and Obtuse Angles (No Angle Marks) Identifying Acute, Obtuse and Right Angles Identifying Acute, Obtuse and Right Angles (No Angle Marks) Identifying Acute, Obtuse, Right and Straight Angles Identifying Acute, Obtuse, Right and Straight Angles (No Angle Marks) Identifying Acute, Obtuse, Right, Straight and Reflex Angles Identifying Acute, Obtuse, Right, Straight, Reflex and Complete/Full Angles

There are several angle relationships of which students should be aware. Complementary angles are two angles that together form a 90 degree angle; supplementary angles are two angles that together form a 180 degree angle; and explementary angles are two angles that together form a 360 degree angle. Vertical angles are found at line intersections; angles opposite each other are equal. Students can practice determining and/or calculating the unknown angle(s) in the following angle relationships worksheets.

• Angle Relationships Worksheets Complementary Angles Complementary Angles (Diagrams Rotated) Supplementary Angles Supplementary Angles (Diagrams Rotated) Mixed Complementary and Supplementary Angles Questions (Diagrams Rotated) Explementary Angles Explementary Angles (Diagrams Rotated) Mixed Adjacent Angles Questions (Diagrams Rotated) Vertical/Opposite Angles Vertical/Opposite Angles (Diagrams Rotated) Mixed Angle Relationships Questions(Diagrams Rotated)
• Angles of Transversals Intersecting Parallel Lines Interior Alternate Angles Exterior Alternate Angles Alternate Angles Corresponding Angles Co-Interior Angles Transversals

Measuring angles worksheets, can be found on the Measurement Page

Triangles, Quadrilaterals and Other Shapes

The quadrilaterals set can be used for a number of activities that involve classifying and recognizing quadrilaterals or for finding the properties of quadrilaterals (e.g. that the interior angles add up to 360 degrees). The tangram printables are useful in tangram activities. There are several options available for the tangram printables depending on your printer, and each option includes a large version and smaller versions. If you know someone with a suitable saw, you can use the tangram printable as a template on material such as quarter inch plywood; then simply sand and paint the pieces.

• Shape Sets Quadrilaterals Set Tangrams
• Identifying Regular Polygons Identifying Regular Shapes from Triangles to Octagons

Worksheets for classifying triangles by side and angle properties and for working with Pythagorean theorem.

If you are interested in students measuring angles and sides for themselves, it is best to use the versions with no marks. The marked versions will indicate the right and obtuse angles and the equal sides.

• Classifying Triangles Worksheets Classifying Triangles by Side Properties Classifying Triangles by Angle Properties Classifying Triangles by Side and Angle Properties Classifying Triangles by Side Properties (No Marks) Classifying Triangles by Angle Properties (No Marks) Classifying Triangles by Side and Angle Properties (No Marks)

A cathetus (plural catheti) refers to a side of a right-angle triangle other than the hypotenuse.

• Calculating Triangle Dimensions Using Pythagorean Theorem Calculate the Hypotenuse Using Pythagorean Theorem (No Rotation) Calculate the Hypotenuse Using Pythagorean Theorem Calculate a Cathetus Using Pythagorean Theorem (No Rotation) Calculate a Cathetus Using Pythagorean Theorem Calculate any Side Using Pythagorean Theorem (No Rotation) Calculate any Side Using Pythagorean Theorem

Trigonometric ratios are useful in determining the dimensions of right-angled triangles. The three basic ratios are summarized by the acronym SOHCAHTOA. The SOH part refers to the ratio: sin(α) = O/H where α is an angle measurement; O refers the length of the side (O)pposite the angle measurement and H refers to the length of the (H)ypotenuse of the right-angled triangle. The CAH part refers to the ratio: cos(α) = A/H where A refers to the length of the (A)djacent side to the angle. The TOA refers to the ratio: tan(α) = O/A.

• Calculating Angles and Sides Using Trigonometric Ratios Calculating Angles Using the Sine Ratio Calculating Sides Using the Sine Ratio Calculating Angles and Sides Using the Sine Ratio Calculating Angles Using the Cosine Ratio Calculating Sides Using the Cosine Ratio Calculating Angles and Sides Using the Cosine Ratio Calculating Angles Using the Tangent Ratio Calculating Sides Using the Tangent Ratio Calculating Angles and Sides Using the Tangent Ratio Calculating Angles Using Trigonometric Ratios Calculating Sides Using Trigonometric Ratios Calculating Angles and Sides Using Trigonometric Ratios

Quadrilaterals are interesting shapes to classify. Their classification relies on a few attributes and most quadrilaterals can be classified as more than one shape. A square, for example, is also a parallelogram, rhombus, rectangle and kite. A quick summary of all quadrilaterals is as follows: quadrilaterals have four sides. A square has 90 degree corners and equal length sides. A rectangle has 90 degree corners, but the side lengths don't have to be equal. A rhombus has equal length sides, but the angles don't have to be 90 degrees. A parallelogram has both pairs of opposite sides equal and parallel and both pairs of opposite angles are equal. A trapezoid only needs to have one pair of opposite sides parallel. A kite has two pairs of equal length sides where each pair is joined/adjacent rather than opposite to one other. A bowtie is sometimes included which is a complex quadrilateral with two sides that crossover one another, but they are readily recognizable. Any other four-sided polygon can safely be called a quadrilateral if it doesn't meet any of the criteria for a more specific classification.

• Classifying Quadrilaterals Classifying Simple Quadrilaterals Classifying All Quadrilaterals Classifying All Quadrilaterals (+ Rotation)

Coordinate Plane Worksheets

Coordinate point geometry worksheets to help students learn about the Cartesian plane.

• Plotting Random Coordinate Points Plotting Coordinate Points in All Quadrants Plotting Coordinate Points in Positive x Quadrants Plotting Coordinate Points in Positive y Quadrants

There are many other Cartesian Art plots scattered around the Math-Drills website as many of them are associated with a holiday. To find them quickly, use the search box.

• Cartesian Art Cartesian Art Maple Leaf
• Coordinate Plane Distance and Area Calculating Pythagorean Distances of Coordinate Points Calculating Perimeter and Area of Triangles on Coordinate Planes Calculating Perimeter and Area of Quadrilaterals on Coordinate Planes Calculating Perimeter and Area of Triangles and Quadrilaterals on Coordinate Planes

Transformations Worksheets

Transformations worksheets for translations, reflections, rotations and dilations practice.

Here are two quick and easy ways to check students' answers on the transformational geometry worksheets below. First, you can line up the student's page and the answer page and hold it up to the light. Moving/sliding the pages slightly will show you if the student's answers are correct. Keep the student's page on top and mark it or give feedback as necessary. The second way is to photocopy the answer page onto an overhead transparency. Overlay the transparency on the student's page and flip it up as necessary to mark or give feedback.

Also known as sliding, translations are a way to mathematically describe how something moves on a Cartesian plane. In translations, every vertex and line segment moves the same, so the resulting shape is congruent to the original.

• Translations Worksheets Translation of 3 vertices by up to 3 units. Translation of 3 vertices by up to 6 units. Translation of 3 vertices by up to 25 units. Translation of 4 vertices by up to 6 units. Translation of 5 vertices by up to 6 units.
• Translations Worksheets (Multi-Step) Two-Step Translation of 3 vertices by up to 6 units. Two-Step Translation of 4 vertices by up to 6 units. Three-Step Translation of 3 vertices by up to 6 units. Three-Step Translation of 4 vertices by up to 6 units.

Reflect on this: reflecting shapes over horizontal or vertical lines is actually quite straight-forward, especially if there is a grid involved. Start at one of the original points/vertices and measure the distance to the reflecting line. Note that you should measure perpendicularly or 90 degrees toward the line which is why it is easier with vertical or horizontal reflecting lines than with diagonal lines. Measure out 90 degrees on the other side of the reflecting line, the same distance of course, and make a point to represent the reflected vertex. Once you've done this for all of the vertices, you simply draw in the line segments and your reflected shape will be finished.

Reflecting can also be as simple as paper-folding. Fold the paper on the reflecting line and hold the paper up to the light. On a window is best because you will also have a surface on which to write. Only mark the vertices, don't try to draw the entire shape. Unfold the paper and use a pencil and ruler to draw the line segments between the vertices.

• Reflections Worksheets Reflection of 3 Vertices Over x = 0 and y = 0 Reflection of 4 Vertices Over x = 0 and y = 0 Reflection of 5 Vertices Over x = 0 and y = 0 Reflection of 3 Vertices Over Various Lines Reflection of 4 Vertices Over Various Lines Reflection of 5 Vertices Over Various Lines
• Reflections Worksheets (Multi-Step) Two-Step Reflection of 3 Vertices Over Various Lines Two-Step Reflection of 4 Vertices Over Various Lines Three-Step Reflection of 3 Vertices Over Various Lines Three-Step Reflection of 4 Vertices Over Various Lines

Here's an idea on how to complete rotations without measuring. It works best on a grid and with 90 or 180 degree rotations. You will need a blank overhead projector sheet or other suitable clear plastic sheet and a pen that will work on the page. Non-permanent pens are best because the plastic sheet can be washed and reused. Place the sheet over top of the coordinate axes with the figure to be rotated. With the pen, make a small cross to show the x and y axes being as precise as possible. Also mark the vertices of the shape to be rotated. Using the plastic sheet, perform the rotation, lining up the cross again with the axes. Choose one vertex and mark it on the paper by holding the plastic sheet in place, but flipping it up enough to get a mark on the paper. Do this for the other vertices, then remove the plastic sheet and join the vertices with line segments using a ruler.

• Rotations Worksheets Rotation of 3 Vertices around the Origin Starting in Quadrant I Rotation of 4 Vertices around the Origin Starting in Quadrant I Rotation of 5 Vertices around the Origin Starting in Quadrant I Rotation of 3 Vertices around the Origin Rotation of 4 Vertices around the Origin Rotation of 5 Vertices around the Origin Rotation of 3 Vertices around Any Point Rotation of 4 Vertices around Any Point Rotation of 5 Vertices around Any Point
• Rotations Worksheets (Multi-Step) Two-Step Rotations of 3 Vertices around Any Point Two-Step Rotations of 4 Vertices around Any Point Two-Step Rotations of 5 Vertices around Any Point Three-Step Rotations of 3 Vertices around Any Point Three-Step Rotations of 4 Vertices around Any Point Three-Step Rotations of 5 Vertices around Any Point
• Dilations Worksheets Dilations Using Center (0, 0) Dilations Using Various Centers
• Determining Scale Factors Worksheets Determine Scale Factors of Rectangles (Whole Numbers) Determine Scale Factors of Rectangles (0.5 Intervals) Determine Scale Factors of Rectangles (0.1 Intervals) Determine Scale Factors of Triangles (Whole Numbers) Determine Scale Factors of Triangles (0.5 Intervals) Determine Scale Factors of Triangles (0.1 Intervals) Determine Scale Factors of Rectangles and Triangles (Whole Numbers) Determine Scale Factors of Rectangles and Triangles (0.5 Intervals) Determine Scale Factors of Rectangles Triangles (0.1 Intervals)
• Mixed Transformations Worksheets (Multi-Step) Two-Step Transformations Three-Step Transformations

Constructions Worksheets

Constructions worksheets for constructing bisectors, perpendicular lines and triangle centers.

It is amazing what one can accomplish with a compass, a straight-edge and a pencil. In this section, students will do math like Euclid did over 2000 years ago. Not only will this be a lesson in history, but students will gain valuable skills that they can use in later math studies.

• Constructing Midpoints And Bisectors On Line Segments And Angles Midpoints on Horizontal Line Segments Perpendicular Bisectors on Horizontal Line Segments Perpendicular Bisectors on Rotated Line Segments Angle Bisectors (Angles not Rotated) Angle Bisectors (Angles Randomly Rotated)
• Constructing Perpendicular Lines Construct Perpendicular Lines Through Points on a Line Segment Construct Perpendicular Lines Through Points Not on Line Segment Construct Perpendicular Lines Through Points on Line Segment (Segments are randomly rotated) Construct Perpendicular Lines Through Points Not on Line Segment (Segments are randomly rotated)
• Constructing Triangle Centers Centroids for Acute Triangles Centroids for Mixed Acute and Obtuse Triangles Orthocenters for Acute Triangles Orthocenters for Mixed Acute and Obtuse Triangles Incenters for Acute Triangles Incenters for Mixed Acute and Obtuse Triangles Circumcenters for Acute Triangles Circumcenters for Mixed Acute and Obtuse Triangles All Centers for Acute Triangles All Centers for Mixed Acute and Obtuse Triangles

Three-Dimensional Geometry

Three-dimensional geometry worksheets that are based on connecting cubes and worksheets for classifying three-dimensional figures.

Connecting cubes can be a powerful tool for developing spatial sense in students. The first two worksheets below are difficult to do even for adults, but with a little practice, students will be creating structures much more complex than the ones below. Use isometric grid paper and square graph paper or dot paper to help students create three-dimensional sketches of connecting cubes and side views of structures.

• Connecting Cube Structures Side Views of Connecting Cube Structures Build Connecting Cube Structures
• Classifying Three-Dimensional Figures Classify Prisms Classify Pyramids Classify Prisms and Pyramids

This section includes a number of nets that students can use to build the associated 3D solids. All of the Platonic solids and many of the Archimedean solids are included. A pair of scissors, a little tape and some dexterity are all that are needed. For something a little more substantial, copy or print the nets onto cardstock first. You may also want to check your print settings to make sure you print in "actual size" rather than fitting to the page, so there is no distortion.

• Nets of Three-Dimensional Figures Nets of Platonic and Archimedean Solids Nets of All Platonic Solids Nets of Some Archimedean Solids Net of a Tetrahedron Net of a Cube Net of an Octahedron Net of a Dodecahedron (Version 1) Net of a Dodecahedron (Version 2) Net of an Icosahedron Net of a Truncated Tetrahedron Net of a Cuboctahedron Net of a Truncated Cube Net of a Truncated Octahedron Net of a Rhombicuboctahedron Net of a Truncated Cuboctahedron Net of a Snub Cube Net of an Icosidodecahedron

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How to Solve Triangles Problems? (+FREE Worksheet!)

Learn how to solve mathematics problems related to triangle area and angles using common triangle formulas.

Related Topics

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A step-by-step guide to solving Triangles

• In any triangle, the sum of all angles is \(180\) degrees.

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Triangles – example 1:.

Use the area formula: Area \(= \color{blue}{\frac{1}{2 }(base \ × \ height)}\) base \(=10\) and height \(=6\) Area \(= \frac{1}{2} (10×6)=\frac{1}{2} (60)=30\)

Triangles – Example 2:

What is the missing angle of the following triangle?

All angles in a triangle sum up to \(180\) degrees. Then: : \(45+60+x=180 → 105+x=180 → x=180-105=75\)

Triangles – Example 3:

What is the area of the following triangle?

Use the area formula: Area \(= \color{blue}{\frac{1}{2 }(base \ × \ height)}\) base \(=12\) and height \(=8\) Area \(= \frac{1}{2} (12×8)= \frac{1}{2} (96)=48 \)

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Triangles – example 4:.

What is the area of a triangle with base \(6\) and height \(5\)?

Use the area formula: Area \(= \color{blue}{\frac{1}{2 }(base \ × \ height)}\) base \(=6\) and height \(=5\) Area \(= \frac{1}{2 } (5×6)= \frac{1}{2} (30)=15 \)

Exercises for Solving Triangles

Find the measure of the unknown angle in each triangle., download triangles worksheet.

• \(\color{blue}{45^{\circ}}\)
• \( \color{blue}{ 15^{\circ}}\)
• \( \color{blue}{ 40^{\circ}}\)

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by: Effortless Math Team about 5 years ago (category: Articles , Free Math Worksheets )

Effortless Math Team

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How to Find the Area of a Triangle

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With its three sides and three angles, the triangle is one of the most basic shapes in geometry. This means calculating the area of a triangle is a fundamental skill in geometry, with multiple formulas available depending on the type of triangle and the given data.

But knowing how to find the area of a triangle has plenty of applications beyond mathematics. For example, understanding right-angled triangles is essential for finding accurate measurements in construction and navigation. Isosceles triangles are crucial in structural engineering , aerospace design, and optics, where precision is absolutely a requirement. And understanding the area of an equilateral triangle is essential in architecture and even art.

Let's review the basic formula for finding the area of a triangle, plus formulas for other scenarios.

Basic Formula for Area

4 types of triangles and their formulas, using heron's formula, other formulas for specific cases.

The most basic triangle formula for finding area involves the shape's base and height. For any triangle with a base b and a height h , you can find the area A by using:

This formula represents half the product of the base and the height, which is the perpendicular drawn from the base to the opposite vertex.

Although all triangles have three sides, not all triangles are exactly the same. Here are four different types and additional ways to find the area for each one.

1. Right Triangle

A right triangle has one angle equal to 90 degrees. The two sides that form this right angle are the base b and the height h . To find the area, you can use:

In a right-angled triangle, the relationship between the sides and the angles is fundamental. The side opposite the right angle is called the hypotenuse, and it is the longest side of the right-angled triangle. The other two sides are known as the legs.

For example, in a right-angled triangle with legs measuring 3 units and 4 units, the hypotenuse can be found using the Pythagorean theorem.

2. Isosceles Triangle

An isosceles triangle has two sides of the same length and two angles of the same measure. If the equal sides have length a , and the base has length b , you can apply the same triangle area formula you might use for a right-angled triangle.

First, you determine the height h , which you can calculate using the Pythagorean theorem , splitting the base into two equal parts:

Now use the triangle area formula to find A :

By plugging the known dimensions into this triangle area formula, you can calculate the area of the isosceles triangle accurately.

3. Equilateral Triangle

An equilateral triangle has all sides equal and all angles equal to 60 degrees. This symmetry means you have to use a special formula to calculate the area. For an equilateral triangle with side length a , you can calculate the area using:

Being perfectly symmetrical, an equilateral triangle is a fundamental shape in both geometry and the real world. Whether you're looking to create a perfect aesthetic, such as in designing a triangular garden plot, or attempting to distribute weight evenly, such as in the construction of a truss for a bridge , the principles governing the area of an equilateral triangle can be valuable.

4. Scalene Triangle

To find the area of a scalene triangle, where all the sides have different lengths, you could use a tool called Heron's formula:

First, you calculate the semi-perimeter, or half the sum of all sides. Mathematicians refer to this figure as s. To find the semi-perimeter, you use:

Now continue with Heron's formula to find the area A .

Heron's formula is a powerful tool in geometry that allows you to calculate the area of a triangle when you know the lengths of all three sides:

Named after the ancient Greek mathematician Hero of Alexandria, this formula is particularly useful because it does not require you to know the height of the triangle or any of its angles opposite the given sides.

Suppose you have a scalene triangle with the following side lengths:

First, you determine the semi-perimeter, or s . Again, the formula for semi-perimeter is:

Calculating the semi-perimeter in this example looks like this:

Now, you find the area A using:

To apply this formula to the above example, you make the following calculations:

Advantages of Heron's Formula

Mathematicians have used Heron's formula for centuries because it offers certain advantages:

• It's versatile. Heron's formula works for all types of triangles, whether they are scalene (no equal sides), isosceles (two equal sides), or equilateral (all sides equal).
• You don't need to know the height or angles. Unlike other triangle area formulas that require the height or specific angles, Heron's formula only needs the side lengths.
• It results in an exact figure. Heron's formula provides an exact area measurement without the need for additional geometric constructions.

Connection to Other Triangle Formulas

Heron's formula complements other methods of finding a triangle's area. For example, while the basic triangle formula — A = (1/2) x base x height — is straightforward for right triangles or when you know the height, Heron's formula provides a way to calculate the area solely from side lengths. You could use both methods to be confident in your resulting figures.

Sometimes a scenario calls for a special formula, such as the following cases.

SAS (Side-Angle-Side)

When you know the two sides and the included angle of a triangle, you can calculate the area using the formula:

where a and b are the sides, and C is the included angle.

ASA (Angle-Side-Angle)

For a given triangle where you know two angles and the included side, you can find the area using:

where a is the known side, and A , B and C are the angles.

We created this article in conjunction with AI technology, then made sure it was fact-checked and edited by a HowStuffWorks editor.

Please copy/paste the following text to properly cite this HowStuffWorks.com article:

Right Similar Triangles Worksheet and Answer Key

Students will practice solving for side lengths of right similar triangles .

Example Questions

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This is a 4 part worksheet:

• Part I Model Problems
• Part II Practice
• Part III Challenge Problems
• Part IV Answer Key

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