## Side perimeter of a triangle

Enter the lengths of the sides of the triangle

### The formula for the perimeter of a triangle on the sides

P = a + b + c

Where a, b and c are the sides of the triangle

## Perimeter of a triangle along its midlines

Enter the lengths of the midlines

### Formula of the perimeter of a triangle along the midlines

P = MN × 2 + NK × 2 + KM × 2

Where MN, NK and KM are the midlines of the triangle

## Perimeter of a triangle along two sides and the angle between them

Enter the sides and the angle between them

### The formula for the perimeter of a triangle on two sides and the angle between them

Where a, b are the sides of the triangle, α is the angle between the sides

## Perimeter of a right triangle along the leg and hypotenuse

### Formula of the perimeter of a right-angled triangle along the leg and hypotenuse

Where a - hypotenuse, b - leg

## Perimeter of a right triangle along the legs

### Formula of the perimeter of a right-angled triangle for two legs

Where a and b are legs

## Perimeter of an isosceles triangle in base and height

### Formula of the perimeter of an isosceles triangle by base and height

Where h is the height, a is the base

## Perimeter of an isosceles triangle along the lateral side and base

### Formula of the perimeter of an isosceles triangle along the lateral side and base

Where b - sides, a - base

## Perimeter of an equilateral triangle in height

### Equilateral triangle perimeter height formula

Where h is the height

## Perimeter of an equilateral triangle by the area of the inscribed circle

### Formula of the perimeter of an equilateral triangle by the area of the inscribed circle

Where S is the area of the inscribed circle

## Hypotenuse and angle perimeter of a right triangle

### Formula of the perimeter of a right triangle by hypotenuse and angle

P = c × sin (α) + c × cos (α) + c

Where c is the hypotenuse, α is the angle

## Perimeter of a right triangle along the leg and the adjacent corner

### Formula of the perimeter of a right-angled triangle along the leg and the adjacent angle

P = b × tan (α) + b + b / cos (α)

Where b - leg, α - included angle

## Perimeter of a right-angled triangle along the leg and the opposite corner

### Formula of the perimeter of a right-angled triangle along the leg and the opposite angle

P = a + a / tg (α) + a / sin (α)

Where a - leg, α - opposite angle

## 1. How to find the perimeter of a triangle, knowing the three sides

Just count the sum of all sides.

- P is the required perimeter;
- a, b, c - sides of the triangle.

## 2. How to find the perimeter of a triangle, knowing its area and the radius of the inscribed circle

Multiply the area of the triangle by 2.

Divide the result by the radius of the inscribed circle.

## 3. How to calculate the perimeter of a triangle, knowing the two sides and the angle between them

First, find the unknown side of the triangle using the cosine theorem:

- Multiply one side by the other, by the cosine of the angle between them, and by 2.
- Calculate the sum of the squares of the known sides and subtract from it the number obtained in the previous step.
- Find the root of the result.

Now add the two previously known sides to the found side.

- P is the required perimeter;
- b, c - known sides of the triangle;
- ɑ is the angle between the known sides;
- a - unknown side of the triangle.

## 4. How to find the perimeter of an equilateral triangle, knowing one side

Multiply the side by 3.

- P is the required perimeter;
- a - any side of the triangle (recall that in an equilateral triangle all sides are equal).

## 5. How to calculate the perimeter of an isosceles triangle, knowing the side and base

Multiply the side by 2.

Add base to the result.

- P is the required perimeter;
- a - the side of the triangle (in an isosceles triangle, the sides are equal);
- b - the base of the triangle (this is the side that differs in length from the rest).

## 6. How to find the perimeter of an isosceles triangle, knowing the side and height

Find the side and height squares.

Subtract the second from the first number.

Find the root of the result and multiply it by 2.

Add the two sides to the resulting number.

- P is the required perimeter;
- a - lateral side of the triangle;
- h is the height (the perpendicular dropped to the base of the triangle from the side of the opposite vertex; in an isosceles triangle, the height divides the base in half).

## 7. How to calculate the perimeter of a right triangle, knowing the legs

Find the squares of the legs and count their sum.

Extract the root of the resulting number.

Add both legs to the result.

- P is the required perimeter;
- a, b - legs of a triangle (sides that form a right angle).

## 8. How to find the perimeter of a right triangle, knowing the leg and hypotenuse

Count the squares of the hypotenuse and leg.

Subtract the second from the first number.

Find the root of the result.

Add leg and hypotenuse.

- P is the required perimeter;
- a - any leg of the rectangle;
- c - hypotenuse (the side that lies opposite the right angle).

## Definition

The perimeter is the length of all sides of the polygon. The perimeter is denoted by a capital Latin letter P. Under "P" it is convenient to write the name of the figure in small letters so as not to get confused in the problems and the course of the solution.

It is important that all parameters are passed in one unit of length, otherwise we will not be able to calculate the result. Therefore, for the correct solution, it is necessary to convert all data to one unit of measurement.

**What is the perimeter measured in: **

- square millimeter (
*mm*);^{2} - square centimeter (
*cm*);^{2} - square decimeter (
*dm*);^{2} - square meter (
*м*);^{2} - square kilometre (
*km*);^{2} - hectare (ha).

## How to find out the perimeter of a triangle

Let's consider what formulas exist, and under what known initial data they can be applied.

**If three sides are known **, then the perimeter of the triangle is equal to their sum. This method is passed in the second grade.

P = a + b + c, where a, b, c is the side length.

**If the area and radius of the inscribed circle are known: **

P = 2 * S: r, where S is the area, r is the radius of the inscribed circle.

**If you know two sides and the angle between them, you can calculate the perimeter of the triangle like this: **

P = √ b ^{2}+ with ^{2}- 2 * b * c * cosα + (b + c), where b, c are known sides, α is the angle between known sides.

**If one side in an equilateral triangle is known: **

P = 3 * a, where a is the side length.

All sides in an equilateral figure are equal.

**If the side and base are known in an isosceles triangle: **

P = 2 * a + b, where a is the side, b is the base.

The sides in an isosceles figure are equal.

**If the lateral side and the height in an isosceles triangle are known: **

P = 2 * (√ a ^{2}+ h ^{2}) + 2 * a, where a is the side, h is the height.

It is customary to call the segment height, which came out of the top and sank to the bottom. In an isosceles figure, the height bisects the base.

**If the legs in a right-angled triangle are known: **

P = √ a ^{2}+ b ^{2}+ (a + b), where a, b - legs.

The leg is one of two sides that form a right angle.

**If the leg and hypotenuse in a right triangle are known: **

P = √ c ^{2}- a ^{2}+ (a + c), where a is any leg, c is the hypotenuse.

The hypotenuse is the side that lies opposite the right angle.

## Download online spreadsheet

Each geometric figure has many formulas - it can be really difficult to remember everything at once. Regular problem solving and frequent viewing of formulas will help in this matter. You can print this table and use it as a bookmark in a notebook or textbook, and refer to it as needed.

To make your child even better at school, enroll him in math lessons. Summer is a great time to do it with pleasure, at a comfortable pace, without tests and grades for a quarter, lying at home on the floor or on the grass outside the city.

Instead of boring paragraphs, the child is waiting for interactive exercises with instant automatic checking. Our teachers will explain everything from fractions to sines clearly and answer questions that can be awkward to ask in front of the whole class.

We learn to find the perimeter of a triangle in different ways, and also train the knowledge gained on examples of tasks.

## Perimeter of a triangle

DefinitionThe perimeter of a triangle is the sum of the lengths of all its sides.

DefinitionA triangle is a geometric figure that consists of three points (vertices) that do not lie on one straight line. These points are connected in pairs by three segments, which are called sides (edges) of the polygon.

Consider several ways to find the perimeter of the figure in question. Each of the proposed formulas is based on those values that we already know.

## Methods of finding

### On three sides

If we already know the length of each edge of the shape, the calculation of the perimeter will be as follows:

\ (P = a + b + c \)

Where *a*, *b* и *с*Are the sides of the triangle.

In case we know the sides of an isosceles triangle (which has two edges equal), the formula for calculating the perimeter is as follows:

\ (P = a + 2b \) or \ (P = a + 2c \)

Where *a*Is the base of the figure, and *b* и *с*- equal ribs.

A triangle can also be equilateral (when all sides are equal). Then P will be found in accordance with the calculations:

\ (P = 3a \)

Where *a*Is either side of the figure.

### By area and radius of the inscribed circle

When we know the area of a given polygon and the radius of the circle inscribed in it, the calculation of P looks like this:

\ (P = \ frac {2S} r \)

where S is the area of the figure, r is the radius of the inscribed circle.

### On two sides and the corner between them

Since we know the angle and the two sides by which it is formed, we can find the third side of the triangle by the cosine theorem. And then calculate the sum of the lengths of all the edges of the figure.

The cosine theorem looks like this:

\ (a ^ 2 = b ^ 2 + c ^ 2-2bc \ times \ cos \ alpha \)

where α is a known angle.

Then the formula for calculating the perimeter of the entire figure in this case:

\ (P = \ sqrt {b ^ 2 + c ^ 2-2bc \ times \ cos \ alpha} + b + c \)

### Lateral and height (for isosceles)

Returning to the properties of an isosceles triangle, we recall that the height drawn to the base of the triangle from the opposite vertex is simultaneously the height, bisector and median. This means that both right-angled triangles that it forms are equal to each other.

The formula for finding the perimeter of our isosceles will be based on the Pythagorean theorem. Let 1/2 of the base ( *c) = d. *Then:

\ (d ^ 2 = a ^ 2-h ^ 2 \)

\ (d = \ sqrt {a ^ 2-h ^ 2} \)

Where *a *- side of an isosceles triangle and the hypotenuse of a right-angled one, *h - *the height of the isosceles and rectangular legs.

Do not forget that *d *- this is only half of the base of an isosceles triangle, so to find the perimeter, the result will need to be multiplied by 2.

\ (P = 2 \ sqrt {a ^ 2-h ^ 2} + 2a \)

### On two legs (for rectangular)

Let us once again recall the Pythagorean theorem for finding the hypotenuse (we denote it by the letter *с*).

\ (c ^ 2 = a ^ 2 + b ^ 2 \)

\ (c = \ sqrt {a ^ 2 + b ^ 2} \)

Where *a* и *b*- the legs of the triangle.

Substitute the value *c*into the formula for finding the perimeter and we get:

\ (P = \ sqrt {a ^ 2 + b ^ 2} + a + b \)

## Examples of problem solving

To train the knowledge gained, we will consider several examples of solving problems to find the perimeter of a triangle.

### Problem number 1

What is the P of a triangle if its sides are 6 cm, 7 cm and 3 cm.

**Decision: **

We substitute the known values into the formula P = a + b + c and we get: P = 6 + 7 + 3 = 16 cm.

Answer: 16 cm.

### Problem number 2

It is known that the base of an isosceles triangle is 6 cm, and its lateral side is 4 cm. Find the P figure.

**Decision: **

For this case, the formula P = a + 2b is suitable, we substitute the values: \ (P = 6 + 4 \ times2 = 14 \) cm.

Answer: 14 cm.

### Problem number 3

We know that the area of a triangle is 24 cm ^{2}, and the radius of the inscribed circle is 8 cm.Find P.

**Decision: **

In this case, we will calculate P as follows: \ (P = \ frac {2S} r \) ... With the values already known to us, we get: \ (P = \ frac {2 \ times24} 8 = 6 \) cm.

Answer: 6 cm.

### Problem number 4

An isosceles triangle is given. We know its lateral side (4 cm) and the height lowered to the base (2 cm). You need to calculate the perimeter of the shape.

**Decision: **

We know that in this case P is calculated as \ (P = 2 \ sqrt {a ^ 2-h ^ 2} + 2a \) ... With the existing values, it turns out: \ (P = 2 \ sqrt {4 ^ 2-2 ^ 2} +2 \ times2 = 4 \ sqrt3 + 4 \) cm.

Answer: P = 4 \ sqrt3 + 4 cm.

### Problem number 5

Given a right-angled triangle with legs 5 cm and 7 cm. Determine the perimeter of the figure.

**Decision: **

Into the formula \ (P = \ sqrt {a ^ 2 + b ^ 2} + a + b \) substitute the known values: \ (P = \ sqrt {5 ^ 2 + 7 ^ 2} + 5 + 7 = \ sqrt {74} +12 \) cm.

Answer: \ (P = \ sqrt {74} +12 \) cm.

Before answering the question of how to find the perimeter of a triangle, let's repeat what is called the perimeter of a triangle.

**Definition. **

*The perimeter of a triangle is the sum of the lengths of its sides. *

Triangle perimeter formula for triangle ABC

If you call the triangle with other letters, the formula for the perimeter of the triangle, respectively, will also look different.

For example, the formula for the perimeter of a triangle is MNP:

In general, the formula for the perimeter of a triangle is written as follows:

where a, b and c are the lengths of the sides of the triangle.

Thus, *to find the perimeter of a triangle, add the lengths of all its sides. *

Examples.

1) Find the perimeter of a triangle with sides 3 cm, 4 cm, 5 cm.

Decision:

By the formula for finding the perimeter of a triangle

we have:

2) Find the perimeter of the triangle ABC if AB = 10 cm, BC = 12 cm, AC = 15 cm.

Decision:

According to the formula

we have:

How to find the perimeter of triangles of individual types - isosceles and equilateral - we will see later.